Results of Bessel Functions III: Difference between revisions

Latest revision as of 07:04, 25 May 2021

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
10.45.E1	${\displaystyle{\displaystyle x^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}+% x\frac{\mathrm{d}w}{\mathrm{d}x}+(\nu^{2}-x^{2})w=0}}$ `x^{2}\deriv[2]{w}{x}+x\deriv{w}{x}+(\nu^{2}-x^{2})w = 0`		(x)^(2)* diff(w, [x$(2)])+ xdiff(w, x)+((nu)^(2)- (x)^(2))w = 0	(x)^(2)* D[w, {x, 2}]+ xD[w, x]+(\[Nu]^(2)- (x)^(2))w == 0	Failure	Failure	Failed [240 / 300] Result: -1.948557159-.1249999996I Test Values: {nu = 1/23^(1/2)+1/2I, w = 1/23^(1/2)+1/2I, x = 3/2} Result: -.2165063507+.8750000006I Test Values: {nu = 1/23^(1/2)+1/2I, w = 1/23^(1/2)+1/2I, x = 1/2} ... skip entries to safe data	Failed [240 / 300] Result: Complex[-1.948557158514987, -0.12499999999999989] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.9485571585149875, -2.125] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.45.E2	${\displaystyle{\displaystyle\displaystyle\widetilde{I}_{\nu}\left(x\right)=\Re% \left(I_{i\nu}\left(x\right)\right)}}$ `\displaystyle\modBesselIimag{\nu}@{x} = \realpart@{\modBesselI{i\nu}@{x}}`	${\displaystyle{\displaystyle\Re((\mathrm{i}\nu)+k+1)>0}}$	Re(BesselI(I(nu), x)) = Re(BesselI(Inu, x))	Re[BesselI[I\[Nu], x]] == Re[BesselI[I\[Nu], x]]	Successful	Successful	-	Successful [Tested: 30]
10.45.E2	${\displaystyle{\displaystyle\displaystyle\widetilde{K}_{\nu}\left(x\right)=K_{% i\nu}\left(x\right)}}$ `\displaystyle\modBesselKimag{\nu}@{x} = \modBesselK{i\nu}@{x}`		BesselK(I(nu), x) = BesselK(Inu, x)	BesselK[I\[Nu], x] == BesselK[I\[Nu], x]	Successful	Successful	-	Successful [Tested: 30]
10.45.E3	${\displaystyle{\displaystyle\displaystyle\widetilde{I}_{-\nu}\left(x\right)=% \widetilde{I}_{\nu}\left(x\right)}}$ `\displaystyle\modBesselIimag{-\nu}@{x} = \modBesselIimag{\nu}@{x}`		Re(BesselI(I(- nu), x)) = Re(BesselI(I(nu), x))	Re[BesselI[I- \[Nu], x]] == Re[BesselI[I\[Nu], x]]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.45.E3	${\displaystyle{\displaystyle\displaystyle\widetilde{K}_{-\nu}\left(x\right)=% \widetilde{K}_{\nu}\left(x\right)}}$ `\displaystyle\modBesselKimag{-\nu}@{x} = \modBesselKimag{\nu}@{x}`		BesselK(I(- nu), x) = BesselK(I(nu), x)	BesselK[I- \[Nu], x] == BesselK[I\[Nu], x]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.45.E4	${\displaystyle{\displaystyle\mathscr{W}\left\{\widetilde{K}_{\nu}\left(x\right% ),\widetilde{I}_{\nu}\left(x\right)\right\}=1/x}}$ `\Wronskian@{\modBesselKimag{\nu}@{x},\modBesselIimag{\nu}@{x}} = 1/x`		(BesselK(I(nu), x))diff(Re(BesselI(I(nu), x)), x)-diff(BesselK(I(nu), x), x)(Re(BesselI(I(nu), x))) = 1/x	Wronskian[{BesselK[I\[Nu], x], Re[BesselI[I\[Nu], x]]}, x] == 1/x	Failure	Failure	Error	Failed [30 / 30] Result: Plus[-0.6666666666666666, Times[0.5, Plus[Complex[1.0700115379721733, -0.3754447148158467], Times[Complex[0.1636629185333998, 0.09141848176750039], Derivative[1][Re][Complex[2.445786867824693, 0.6492150843755028]]]]]] Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[-0.6666666666666666, Times[0.5, Plus[Complex[0.8415452902387464, 0.2726729041814867], Times[Complex[0.3412924192180222, 0.19179892830603273], Derivative[1][Re][Complex[1.3137906770541619, -0.7251169608509622]]]]]] Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.45.E8	${\displaystyle{\displaystyle\widetilde{K}_{0}\left(x\right)=K_{0}\left(x\right% )}}$ `\modBesselKimag{0}@{x} = \modBesselK{0}@{x}`		BesselK(I*(0), x) = BesselK(0, x)	BesselK[I*0, x] == BesselK[0, x]	Successful	Successful	-	Successful [Tested: 3]
10.47.E1	${\displaystyle{\displaystyle z^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+% 2z\frac{\mathrm{d}w}{\mathrm{d}z}+\left(z^{2}-n(n+1)\right)w=0}}$ `z^{2}\deriv[2]{w}{z}+2z\deriv{w}{z}+\left(z^{2}-n(n+1)\right)w = 0`		(z)^(2)* diff(w, [z$(2)])+ 2zdiff(w, z)+((z)^(2)- n(n + 1))w = 0	(z)^(2)* D[w, {z, 2}]+ 2zD[w, z]+((z)^(2)- n(n + 1))w == 0	Failure	Failure	Failed [210 / 210] Result: -1.732050808+.3733632160e-9I Test Values: {w = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I, n = 1} Result: -5.196152424-2.000000000I Test Values: {w = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I, n = 2} ... skip entries to safe data	Failed [210 / 210] Result: Complex[-1.7320508075688772, 1.1102230246251565*^-16] Test Values: {Rule[n, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-5.196152422706633, -1.9999999999999996] Test Values: {Rule[n, 2], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.47.E2	${\displaystyle{\displaystyle z^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+% 2z\frac{\mathrm{d}w}{\mathrm{d}z}-\left(z^{2}+n(n+1)\right)w=0}}$ `z^{2}\deriv[2]{w}{z}+2z\deriv{w}{z}-\left(z^{2}+n(n+1)\right)w = 0`		(z)^(2)* diff(w, [z$(2)])+ 2zdiff(w, z)-((z)^(2)+ n(n + 1))w = 0	(z)^(2)* D[w, {z, 2}]+ 2zD[w, z]-((z)^(2)+ n(n + 1))w == 0	Failure	Failure	Failed [210 / 210] Result: -1.732050808-2.000000000I Test Values: {w = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I, n = 1} Result: -5.196152424-4.000000000I Test Values: {w = 1/23^(1/2)+1/2I, z = 1/23^(1/2)+1/2I, n = 2} ... skip entries to safe data	Failed [210 / 210] Result: Complex[-1.7320508075688776, -1.9999999999999998] Test Values: {Rule[n, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-5.196152422706632, -3.9999999999999996] Test Values: {Rule[n, 2], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.47.E3	${\displaystyle{\displaystyle\mathsf{j}_{n}\left(z\right)=\sqrt{\tfrac{1}{2}\pi% /z}J_{n+\frac{1}{2}}\left(z\right)}}$ `\sphBesselJ{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\BesselJ{n+\frac{1}{2}}@{z}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0}}$	Error	SphericalBesselJ[n, z] == Sqrt[Divide[1,2]Pi/z]BesselJ[n +Divide[1,2], z]	Missing Macro Error	Failure	Skip - symbolical successful subtest	Successful [Tested: 21]
10.47.E3	${\displaystyle{\displaystyle\sqrt{\tfrac{1}{2}\pi/z}J_{n+\frac{1}{2}}\left(z% \right)=(-1)^{n}\sqrt{\tfrac{1}{2}\pi/z}Y_{-n-\frac{1}{2}}\left(z\right)}}$ `\sqrt{\tfrac{1}{2}\pi/z}\BesselJ{n+\frac{1}{2}}@{z} = (-1)^{n}\sqrt{\tfrac{1}{2}\pi/z}\BesselY{-n-\frac{1}{2}}@{z}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0}}$	sqrt((1)/(2)Pi/z)BesselJ(n +(1)/(2), z) = (- 1)^(n)sqrt((1)/(2)Pi/z)*BesselY(- n -(1)/(2), z)	Sqrt[Divide[1,2]Pi/z]BesselJ[n +Divide[1,2], z] == (- 1)^(n)Sqrt[Divide[1,2]Pi/z]*BesselY[- n -Divide[1,2], z]	Failure	Failure	Successful [Tested: 21]	Successful [Tested: 21]
10.47.E4	${\displaystyle{\displaystyle\mathsf{y}_{n}\left(z\right)=\sqrt{\tfrac{1}{2}\pi% /z}Y_{n+\frac{1}{2}}\left(z\right)}}$ `\sphBesselY{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\BesselY{n+\frac{1}{2}}@{z}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-(n+\frac{1}{2}))+% k+1)>0}}$	Error	SphericalBesselY[n, z] == Sqrt[Divide[1,2]Pi/z]BesselY[n +Divide[1,2], z]	Missing Macro Error	Failure	Skip - symbolical successful subtest	Successful [Tested: 21]
10.47.E4	${\displaystyle{\displaystyle\sqrt{\tfrac{1}{2}\pi/z}Y_{n+\frac{1}{2}}\left(z% \right)=(-1)^{n+1}\sqrt{\tfrac{1}{2}\pi/z}J_{-n-\frac{1}{2}}\left(z\right)}}$ `\sqrt{\tfrac{1}{2}\pi/z}\BesselY{n+\frac{1}{2}}@{z} = (-1)^{n+1}\sqrt{\tfrac{1}{2}\pi/z}\BesselJ{-n-\frac{1}{2}}@{z}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-(n+\frac{1}{2}))+% k+1)>0,\Re((-n-\frac{1}{2})+k+1)>0}}$	sqrt((1)/(2)Pi/z)BesselY(n +(1)/(2), z) = (- 1)^(n + 1)sqrt((1)/(2)Pi/z)*BesselJ(- n -(1)/(2), z)	Sqrt[Divide[1,2]Pi/z]BesselY[n +Divide[1,2], z] == (- 1)^(n + 1)Sqrt[Divide[1,2]Pi/z]*BesselJ[- n -Divide[1,2], z]	Failure	Failure	Successful [Tested: 21]	Successful [Tested: 21]
10.47.E5	${\displaystyle{\displaystyle{\mathsf{h}^{(1)}_{n}}\left(z\right)=\sqrt{\tfrac{% 1}{2}\pi/z}{H^{(1)}_{n+\frac{1}{2}}}\left(z\right)}}$ `\sphHankelh{1}{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\HankelH{1}{n+\frac{1}{2}}@{z}`		Error	SphericalHankelH1[n, z] == Sqrt[Divide[1,2]Pi/z]HankelH1[n +Divide[1,2], z]	Missing Macro Error	Failure	-	Successful [Tested: 21]
10.47.E5	${\displaystyle{\displaystyle\sqrt{\tfrac{1}{2}\pi/z}{H^{(1)}_{n+\frac{1}{2}}}% \left(z\right)=(-1)^{n+1}\mathrm{i}\sqrt{\tfrac{1}{2}\pi/z}{H^{(1)}_{-n-\frac{% 1}{2}}}\left(z\right)}}$ `\sqrt{\tfrac{1}{2}\pi/z}\HankelH{1}{n+\frac{1}{2}}@{z} = (-1)^{n+1}\iunit\sqrt{\tfrac{1}{2}\pi/z}\HankelH{1}{-n-\frac{1}{2}}@{z}`		sqrt((1)/(2)Pi/z)HankelH1(n +(1)/(2), z) = (- 1)^(n + 1)* Isqrt((1)/(2)Pi/z)*HankelH1(- n -(1)/(2), z)	Sqrt[Divide[1,2]Pi/z]HankelH1[n +Divide[1,2], z] == (- 1)^(n + 1)* ISqrt[Divide[1,2]Pi/z]*HankelH1[- n -Divide[1,2], z]	Successful	Failure	-	Successful [Tested: 21]
10.47.E6	${\displaystyle{\displaystyle{\mathsf{h}^{(2)}_{n}}\left(z\right)=\sqrt{\tfrac{% 1}{2}\pi/z}{H^{(2)}_{n+\frac{1}{2}}}\left(z\right)}}$ `\sphHankelh{2}{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\HankelH{2}{n+\frac{1}{2}}@{z}`		Error	SphericalHankelH2[n, z] == Sqrt[Divide[1,2]Pi/z]HankelH2[n +Divide[1,2], z]	Missing Macro Error	Failure	-	Successful [Tested: 21]
10.47.E6	${\displaystyle{\displaystyle\sqrt{\tfrac{1}{2}\pi/z}{H^{(2)}_{n+\frac{1}{2}}}% \left(z\right)=(-1)^{n}\mathrm{i}\sqrt{\tfrac{1}{2}\pi/z}{H^{(2)}_{-n-\frac{1}% {2}}}\left(z\right)}}$ `\sqrt{\tfrac{1}{2}\pi/z}\HankelH{2}{n+\frac{1}{2}}@{z} = (-1)^{n}\iunit\sqrt{\tfrac{1}{2}\pi/z}\HankelH{2}{-n-\frac{1}{2}}@{z}`		sqrt((1)/(2)Pi/z)HankelH2(n +(1)/(2), z) = (- 1)^(n)* Isqrt((1)/(2)Pi/z)*HankelH2(- n -(1)/(2), z)	Sqrt[Divide[1,2]Pi/z]HankelH2[n +Divide[1,2], z] == (- 1)^(n)* ISqrt[Divide[1,2]Pi/z]*HankelH2[- n -Divide[1,2], z]	Successful	Failure	-	Successful [Tested: 21]
10.47.E7	${\displaystyle{\displaystyle{\mathsf{i}^{(1)}_{n}}\left(z\right)=\sqrt{\tfrac{% 1}{2}\pi/z}I_{n+\frac{1}{2}}\left(z\right)}}$ `\modsphBesseli{1}{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\modBesselI{n+\frac{1}{2}}@{z}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0}}$	Error	Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)(n + 1/2), n] == Sqrt[Divide[1,2]Pi/z]*BesselI[n +Divide[1,2], z]	Missing Macro Error	Failure	-	Failed [20 / 21] Result: Complex[0.06771919180965624, -0.29579816936516184] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.4498252419402129, -0.19064547195046921] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.47.E8	${\displaystyle{\displaystyle{\mathsf{i}^{(2)}_{n}}\left(z\right)=\sqrt{\tfrac{% 1}{2}\pi/z}I_{-n-\frac{1}{2}}\left(z\right)}}$ `\modsphBesseli{2}{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\modBesselI{-n-\frac{1}{2}}@{z}`	${\displaystyle{\displaystyle\Re((-n-\frac{1}{2})+k+1)>0}}$	Error	Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)(n + 1/2), n] == Sqrt[Divide[1,2]Pi/z]*BesselI[- n -Divide[1,2], z]	Missing Macro Error	Failure	-	Failed [20 / 21] Result: Complex[-0.41419719140728084, -0.8850762711170854] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.1065867555175597, 2.4569570135519543] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.47.E9	${\displaystyle{\displaystyle\mathsf{k}_{n}\left(z\right)=\sqrt{\tfrac{1}{2}\pi% /z}K_{n+\frac{1}{2}}\left(z\right)}}$ `\modsphBesselK{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\modBesselK{n+\frac{1}{2}}@{z}`		Error	Sqrt[1/2 Pi /z] BesselK[n + 1/2, z] == Sqrt[Divide[1,2]Pi/z]BesselK[n +Divide[1,2], z]	Missing Macro Error	Successful	-	Successful [Tested: 21]
10.47.E9	${\displaystyle{\displaystyle\sqrt{\tfrac{1}{2}\pi/z}K_{n+\frac{1}{2}}\left(z% \right)=\sqrt{\tfrac{1}{2}\pi/z}K_{-n-\frac{1}{2}}\left(z\right)}}$ `\sqrt{\tfrac{1}{2}\pi/z}\modBesselK{n+\frac{1}{2}}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\modBesselK{-n-\frac{1}{2}}@{z}`		sqrt((1)/(2)Pi/z)BesselK(n +(1)/(2), z) = sqrt((1)/(2)Pi/z)BesselK(- n -(1)/(2), z)	Sqrt[Divide[1,2]Pi/z]BesselK[n +Divide[1,2], z] == Sqrt[Divide[1,2]Pi/z]BesselK[- n -Divide[1,2], z]	Successful	Successful	-	Successful [Tested: 21]
10.47#Ex1	${\displaystyle{\displaystyle{\mathsf{h}^{(1)}_{n}}\left(z\right)=\mathsf{j}_{n% }\left(z\right)+i\mathsf{y}_{n}\left(z\right)}}$ `\sphHankelh{1}{n}@{z} = \sphBesselJ{n}@{z}+i\sphBesselY{n}@{z}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0,\Re((-(n+\frac{1}{2}))+k+1)>0}}$	Error	SphericalHankelH1[n, z] == SphericalBesselJ[n, z]+ I*SphericalBesselY[n, z]	Missing Macro Error	Successful	-	Successful [Tested: 21]
10.47#Ex2	${\displaystyle{\displaystyle{\mathsf{h}^{(2)}_{n}}\left(z\right)=\mathsf{j}_{n% }\left(z\right)-i\mathsf{y}_{n}\left(z\right)}}$ `\sphHankelh{2}{n}@{z} = \sphBesselJ{n}@{z}-i\sphBesselY{n}@{z}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0,\Re((-(n+\frac{1}{2}))+k+1)>0}}$	Error	SphericalHankelH2[n, z] == SphericalBesselJ[n, z]- I*SphericalBesselY[n, z]	Missing Macro Error	Successful	-	Successful [Tested: 21]
10.47.E11	${\displaystyle{\displaystyle\mathsf{k}_{n}\left(z\right)=(-1)^{n+1}\tfrac{1}{2% }\pi\left({\mathsf{i}^{(1)}_{n}}\left(z\right)-{\mathsf{i}^{(2)}_{n}}\left(z% \right)\right)}}$ `\modsphBesselK{n}@{z} = (-1)^{n+1}\tfrac{1}{2}\pi\left(\modsphBesseli{1}{n}@{z}-\modsphBesseli{2}{n}@{z}\right)`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0}}$	Error	Sqrt[1/2 Pi /z] BesselK[n + 1/2, z] == (- 1)^(n + 1)Divide[1,2]Pi(Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)(n + 1/2), n]- Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n])	Missing Macro Error	Failure	-	Failed [20 / 21] Result: Complex[-0.7569924845794465, -0.925635877692591] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.0316385731075524, -4.1588442590402455] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.47#Ex3	${\displaystyle{\displaystyle{\mathsf{i}^{(1)}_{n}}\left(z\right)=i^{-n}\mathsf% {j}_{n}\left(iz\right)}}$ `\modsphBesseli{1}{n}@{z} = i^{-n}\sphBesselJ{n}@{iz}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0}}$	Error	Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)(n + 1/2), n] == (I)^(- n) SphericalBesselJ[n, I*z]	Missing Macro Error	Failure	-	Failed [20 / 21] Result: Complex[0.06771919180965624, -0.2957981693651618] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.44982524194021284, -0.19064547195046921] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.47#Ex4	${\displaystyle{\displaystyle{\mathsf{i}^{(2)}_{n}}\left(z\right)=i^{-n-1}% \mathsf{y}_{n}\left(iz\right)}}$ `\modsphBesseli{2}{n}@{z} = i^{-n-1}\sphBesselY{n}@{iz}`	${\displaystyle{\displaystyle\Re((-n-\frac{1}{2})+k+1)>0,\Re((n+\frac{1}{2})+k+% 1)>0,\Re((-(n+\frac{1}{2}))+k+1)>0}}$	Error	Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)(n + 1/2), n] == (I)^(- n - 1) SphericalBesselY[n, I*z]	Missing Macro Error	Failure	-	Failed [20 / 21] Result: Complex[-0.41419719140728045, -0.8850762711170859] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.1065867555175588, 2.456957013551956] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.47.E13	${\displaystyle{\displaystyle\mathsf{k}_{n}\left(z\right)=-\tfrac{1}{2}\pi i^{n% }{\mathsf{h}^{(1)}_{n}}\left(iz\right)}}$ `\modsphBesselK{n}@{z} = -\tfrac{1}{2}\pi i^{n}\sphHankelh{1}{n}@{iz}`		Error	Sqrt[1/2 Pi /z] BesselK[n + 1/2, z] == -Divide[1,2]Pi(I)^(n)* SphericalHankelH1[n, I*z]	Missing Macro Error	Failure	-	Successful [Tested: 21]
10.47.E13	${\displaystyle{\displaystyle-\tfrac{1}{2}\pi i^{n}{\mathsf{h}^{(1)}_{n}}\left(% iz\right)=-\tfrac{1}{2}\pi i^{-n}{\mathsf{h}^{(2)}_{n}}\left(-iz\right)}}$ `-\tfrac{1}{2}\pi i^{n}\sphHankelh{1}{n}@{iz} = -\tfrac{1}{2}\pi i^{-n}\sphHankelh{2}{n}@{-iz}`		Error	-Divide[1,2]Pi(I)^(n)* SphericalHankelH1[n, Iz] == -Divide[1,2]Pi(I)^(- n) SphericalHankelH2[n, - I*z]	Missing Macro Error	Failure	-	Successful [Tested: 21]
10.47.E14	${\displaystyle{\displaystyle\displaystyle\mathsf{j}_{n}\left(-z\right)=(-1)^{n% }\mathsf{j}_{n}\left(z\right)}}$ `\displaystyle\sphBesselJ{n}@{-z} = (-1)^{n}\sphBesselJ{n}@{z}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0}}$	Error	SphericalBesselJ[n, - z] == (- 1)^(n)* SphericalBesselJ[n, z]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.47.E14	${\displaystyle{\displaystyle\displaystyle\mathsf{y}_{n}\left(-z\right)=(-1)^{n% +1}\mathsf{y}_{n}\left(z\right)}}$ `\displaystyle\sphBesselY{n}@{-z} = (-1)^{n+1}\sphBesselY{n}@{z}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-(n+\frac{1}{2}))+% k+1)>0,\Re((-n-\frac{1}{2})+k+1)>0}}$	Error	SphericalBesselY[n, - z] == (- 1)^(n + 1)* SphericalBesselY[n, z]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.47.E15	${\displaystyle{\displaystyle\displaystyle{\mathsf{h}^{(1)}_{n}}\left(-z\right)% =(-1)^{n}{\mathsf{h}^{(2)}_{n}}\left(z\right)}}$ `\displaystyle\sphHankelh{1}{n}@{-z} = (-1)^{n}\sphHankelh{2}{n}@{z}`		Error	SphericalHankelH1[n, - z] == (- 1)^(n)* SphericalHankelH2[n, z]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.47.E15	${\displaystyle{\displaystyle\displaystyle{\mathsf{h}^{(2)}_{n}}\left(-z\right)% =(-1)^{n}{\mathsf{h}^{(1)}_{n}}\left(z\right)}}$ `\displaystyle\sphHankelh{2}{n}@{-z} = (-1)^{n}\sphHankelh{1}{n}@{z}`		Error	SphericalHankelH2[n, - z] == (- 1)^(n)* SphericalHankelH1[n, z]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.47.E16	${\displaystyle{\displaystyle\displaystyle{\mathsf{i}^{(1)}_{n}}\left(-z\right)% =(-1)^{n}{\mathsf{i}^{(1)}_{n}}\left(z\right)}}$ `\displaystyle\modsphBesseli{1}{n}@{-z} = (-1)^{n}\modsphBesseli{1}{n}@{z}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0}}$	Error	Sqrt[Divide[Pi, - z]/2] BesselI[(-1)^(1-1)(n + 1/2), n] == (- 1)^(n) Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.47.E16	${\displaystyle{\displaystyle\displaystyle{\mathsf{i}^{(2)}_{n}}\left(-z\right)% =(-1)^{n+1}{\mathsf{i}^{(2)}_{n}}\left(z\right)}}$ `\displaystyle\modsphBesseli{2}{n}@{-z} = (-1)^{n+1}\modsphBesseli{2}{n}@{z}`	${\displaystyle{\displaystyle\Re((-n-\frac{1}{2})+k+1)>0}}$	Error	Sqrt[Divide[Pi, - z]/2] BesselI[(-1)^(2-1)(n + 1/2), n] == (- 1)^(n + 1) Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.47.E17	${\displaystyle{\displaystyle\mathsf{k}_{n}\left(-z\right)=-\tfrac{1}{2}\pi% \left({\mathsf{i}^{(1)}_{n}}\left(z\right)+{\mathsf{i}^{(2)}_{n}}\left(z\right% )\right)}}$ `\modsphBesselK{n}@{-z} = -\tfrac{1}{2}\pi\left(\modsphBesseli{1}{n}@{z}+\modsphBesseli{2}{n}@{z}\right)`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0}}$	Error	Sqrt[1/2 Pi /- z] BesselK[n + 1/2, - z] == -Divide[1,2]Pi(Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)(n + 1/2), n]+ Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)(n + 1/2), n])	Missing Macro Error	Failure	-	Failed [21 / 21] Result: Complex[-0.5442463690831921, -1.8549132335154932] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[2.444806248586177, 3.5599138449204935] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.49.E2	${\displaystyle{\displaystyle\mathsf{j}_{n}\left(z\right)=\sin\left(z-\tfrac{1}% {2}n\pi\right)\sum_{k=0}^{\left\lfloor n/2\right\rfloor}(-1)^{k}\frac{a_{2k}(n% +\tfrac{1}{2})}{z^{2k+1}}+\cos\left(z-\tfrac{1}{2}n\pi\right)\sum_{k=0}^{\left% \lfloor(n-1)/2\right\rfloor}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{2k+2}}}}$ `\sphBesselJ{n}@{z} = \sin@{z-\tfrac{1}{2}n\pi}\sum_{k=0}^{\floor{n/2}}(-1)^{k}\frac{a_{2k}(n+\tfrac{1}{2})}{z^{2k+1}}+\cos@{z-\tfrac{1}{2}n\pi}\sum_{k=0}^{\floor{(n-1)/2}}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{2k+2}}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0,k\geq 1}}$	Error	SphericalBesselJ[n, z] == Sin[z -Divide[1,2]nPi]Sum[(- 1)^(k)Divide[Subscript[a, 2k](n +Divide[1,2]),(z)^(2k + 1)], {k, 0, Floor[n/2]}, GenerateConditions->None]+ Cos[z -Divide[1,2]nPi]Sum[(- 1)^(k)Divide[Subscript[a, 2k + 1](n +Divide[1,2]),(z)^(2k + 2)], {k, 0, Floor[(n - 1)/2]}, GenerateConditions->None]	Missing Macro Error	Failure	-	Skipped - Because timed out
10.49#Ex1	${\displaystyle{\displaystyle\mathsf{j}_{0}\left(z\right)=\frac{\sin z}{z}}}$ `\sphBesselJ{0}@{z} = \frac{\sin@@{z}}{z}`	${\displaystyle{\displaystyle\Re((0+\frac{1}{2})+k+1)>0,\Re((-0-\frac{1}{2})+k+% 1)>0,\Re((-(-0-\frac{1}{2}))+k+1)>0}}$	Error	SphericalBesselJ[0, z] == Divide[Sin[z],z]	Missing Macro Error	Successful	-	Successful [Tested: 7]
10.49#Ex2	${\displaystyle{\displaystyle\mathsf{j}_{1}\left(z\right)=\frac{\sin z}{z^{2}}-% \frac{\cos z}{z}}}$ `\sphBesselJ{1}@{z} = \frac{\sin@@{z}}{z^{2}}-\frac{\cos@@{z}}{z}`	${\displaystyle{\displaystyle\Re((1+\frac{1}{2})+k+1)>0,\Re((-1-\frac{1}{2})+k+% 1)>0,\Re((-(-1-\frac{1}{2}))+k+1)>0}}$	Error	SphericalBesselJ[1, z] == Divide[Sin[z],(z)^(2)]-Divide[Cos[z],z]	Missing Macro Error	Successful	-	Successful [Tested: 7]
10.49#Ex3	${\displaystyle{\displaystyle\mathsf{j}_{2}\left(z\right)=\left(-\frac{1}{z}+% \frac{3}{z^{3}}\right)\sin z-\frac{3}{z^{2}}\cos z}}$ `\sphBesselJ{2}@{z} = \left(-\frac{1}{z}+\frac{3}{z^{3}}\right)\sin@@{z}-\frac{3}{z^{2}}\cos@@{z}`	${\displaystyle{\displaystyle\Re((2+\frac{1}{2})+k+1)>0,\Re((-2-\frac{1}{2})+k+% 1)>0,\Re((-(-2-\frac{1}{2}))+k+1)>0}}$	Error	SphericalBesselJ[2, z] == (-Divide[1,z]+Divide[3,(z)^(3)])Sin[z]-Divide[3,(z)^(2)]Cos[z]	Missing Macro Error	Successful	-	Successful [Tested: 7]
10.49.E4	${\displaystyle{\displaystyle\mathsf{y}_{n}\left(z\right)=-\cos\left(z-\tfrac{1% }{2}n\pi\right)\sum_{k=0}^{\left\lfloor n/2\right\rfloor}(-1)^{k}\frac{a_{2k}(% n+\tfrac{1}{2})}{z^{2k+1}}+\sin\left(z-\tfrac{1}{2}n\pi\right)\sum_{k=0}^{% \left\lfloor(n-1)/2\right\rfloor}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{2k% +2}}}}$ `\sphBesselY{n}@{z} = -\cos@{z-\tfrac{1}{2}n\pi}\sum_{k=0}^{\floor{n/2}}(-1)^{k}\frac{a_{2k}(n+\tfrac{1}{2})}{z^{2k+1}}+\sin@{z-\tfrac{1}{2}n\pi}\sum_{k=0}^{\floor{(n-1)/2}}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{2k+2}}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-(n+\frac{1}{2}))+% k+1)>0,\Re((-n-\frac{1}{2})+k+1)>0,k\geq 1}}$	Error	SphericalBesselY[n, z] == - Cos[z -Divide[1,2]nPi]Sum[(- 1)^(k)Divide[Subscript[a, 2k](n +Divide[1,2]),(z)^(2k + 1)], {k, 0, Floor[n/2]}, GenerateConditions->None]+ Sin[z -Divide[1,2]nPi]Sum[(- 1)^(k)Divide[Subscript[a, 2k + 1](n +Divide[1,2]),(z)^(2k + 2)], {k, 0, Floor[(n - 1)/2]}, GenerateConditions->None]	Missing Macro Error	Failure	-	Skipped - Because timed out
10.49#Ex4	${\displaystyle{\displaystyle\mathsf{y}_{0}\left(z\right)=-\frac{\cos z}{z}}}$ `\sphBesselY{0}@{z} = -\frac{\cos@@{z}}{z}`	${\displaystyle{\displaystyle\Re((0+\frac{1}{2})+k+1)>0,\Re((-(0+\frac{1}{2}))+% k+1)>0,\Re((-0-\frac{1}{2})+k+1)>0}}$	Error	SphericalBesselY[0, z] == -Divide[Cos[z],z]	Missing Macro Error	Successful	-	Successful [Tested: 7]
10.49#Ex5	${\displaystyle{\displaystyle\mathsf{y}_{1}\left(z\right)=-\frac{\cos z}{z^{2}}% -\frac{\sin z}{z}}}$ `\sphBesselY{1}@{z} = -\frac{\cos@@{z}}{z^{2}}-\frac{\sin@@{z}}{z}`	${\displaystyle{\displaystyle\Re((1+\frac{1}{2})+k+1)>0,\Re((-(1+\frac{1}{2}))+% k+1)>0,\Re((-1-\frac{1}{2})+k+1)>0}}$	Error	SphericalBesselY[1, z] == -Divide[Cos[z],(z)^(2)]-Divide[Sin[z],z]	Missing Macro Error	Successful	-	Successful [Tested: 7]
10.49#Ex6	${\displaystyle{\displaystyle\mathsf{y}_{2}\left(z\right)=\left(\frac{1}{z}-% \frac{3}{z^{3}}\right)\cos z-\frac{3}{z^{2}}\sin z}}$ `\sphBesselY{2}@{z} = \left(\frac{1}{z}-\frac{3}{z^{3}}\right)\cos@@{z}-\frac{3}{z^{2}}\sin@@{z}`	${\displaystyle{\displaystyle\Re((2+\frac{1}{2})+k+1)>0,\Re((-(2+\frac{1}{2}))+% k+1)>0,\Re((-2-\frac{1}{2})+k+1)>0}}$	Error	SphericalBesselY[2, z] == (Divide[1,z]-Divide[3,(z)^(3)])Cos[z]-Divide[3,(z)^(2)]Sin[z]	Missing Macro Error	Successful	-	Successful [Tested: 7]
10.49.E6	${\displaystyle{\displaystyle{\mathsf{h}^{(1)}_{n}}\left(z\right)=e^{iz}\sum_{k% =0}^{n}i^{k-n-1}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}}}$ `\sphHankelh{1}{n}@{z} = e^{iz}\sum_{k=0}^{n}i^{k-n-1}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}`	${\displaystyle{\displaystyle k\geq 1}}$	Error	SphericalHankelH1[n, z] == Exp[Iz]Sum[(I)^(k - n - 1)Divide[Subscript[a, k](n +Divide[1,2]),(z)^(k + 1)], {k, 0, n}, GenerateConditions->None]	Missing Macro Error	Failure	-	Failed [210 / 210] Result: Complex[-0.3966692432410339, 0.7497610210111748] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.3157223500929769, 0.5313692545383957] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.49.E7	${\displaystyle{\displaystyle{\mathsf{h}^{(2)}_{n}}\left(z\right)=e^{-iz}\sum_{% k=0}^{n}(-i)^{k-n-1}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}}}$ `\sphHankelh{2}{n}@{z} = e^{-iz}\sum_{k=0}^{n}(-i)^{k-n-1}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}`	${\displaystyle{\displaystyle k\geq 1}}$	Error	SphericalHankelH2[n, z] == Exp[- Iz]Sum[(- I)^(k - n - 1)Divide[Subscript[a, k](n +Divide[1,2]),(z)^(k + 1)], {k, 0, n}, GenerateConditions->None]	Missing Macro Error	Failure	-	Skipped - Because timed out
10.49.E8	${\displaystyle{\displaystyle{\mathsf{i}^{(1)}_{n}}\left(z\right)=\tfrac{1}{2}e% ^{z}\sum_{k=0}^{n}(-1)^{k}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}+(-1)^{n+1}\% \tfrac{1}{2}e^{-z}\sum_{k=0}^{n}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}}}$ `\modsphBesseli{1}{n}@{z} = \tfrac{1}{2}e^{z}\sum_{k=0}^{n}(-1)^{k}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}+(-1)^{n+1}\\tfrac{1}{2}e^{-z}\sum_{k=0}^{n}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,k\geq 1}}$	Error	Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)(n + 1/2), n] == Divide[1,2]Exp[z]Sum[(- 1)^(k)Divide[Subscript[a, k](n +Divide[1,2]),(z)^(k + 1)], {k, 0, n}, GenerateConditions->None]+(- 1)^(n + 1)Divide[1,2](E)^(- z) Sum[Divide[Subscript[a, k]*(n +Divide[1,2]),(z)^(k + 1)], {k, 0, n}, GenerateConditions->None]	Missing Macro Error	Failure	-	Skipped - Because timed out
10.49#Ex7	${\displaystyle{\displaystyle{\mathsf{i}^{(1)}_{0}}\left(z\right)=\frac{\sinh z% }{z}}}$ `\modsphBesseli{1}{0}@{z} = \frac{\sinh@@{z}}{z}`	${\displaystyle{\displaystyle\Re((0+\frac{1}{2})+k+1)>0}}$	Error	Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(0 + 1/2), 0] == Divide[Sinh[z],z]	Missing Macro Error	Failure	-	Failed [7 / 7] Result: Complex[-1.0789668887893185, -0.15155203743332835] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.9126970224666039, 0.13712305377128448] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.49#Ex8	${\displaystyle{\displaystyle{\mathsf{i}^{(1)}_{1}}\left(z\right)=-\frac{\sinh z% }{z^{2}}+\frac{\cosh z}{z}}}$ `\modsphBesseli{1}{1}@{z} = -\frac{\sinh@@{z}}{z^{2}}+\frac{\cosh@@{z}}{z}`	${\displaystyle{\displaystyle\Re((1+\frac{1}{2})+k+1)>0}}$	Error	Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(1 + 1/2), 1] == -Divide[Sinh[z],(z)^(2)]+Divide[Cosh[z],z]	Missing Macro Error	Failure	-	Failed [7 / 7] Result: Complex[0.06771919180965646, -0.2957981693651617] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.3178790653897484, -0.6062561841669247] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.49#Ex9	${\displaystyle{\displaystyle{\mathsf{i}^{(1)}_{2}}\left(z\right)=\left(\frac{1% }{z}+\frac{3}{z^{3}}\right)\sinh z-\frac{3}{z^{2}}\cosh z}}$ `\modsphBesseli{1}{2}@{z} = \left(\frac{1}{z}+\frac{3}{z^{3}}\right)\sinh@@{z}-\frac{3}{z^{2}}\cosh@@{z}`	${\displaystyle{\displaystyle\Re((2+\frac{1}{2})+k+1)>0}}$	Error	Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)(2 + 1/2), 2] == (Divide[1,z]+Divide[3,(z)^(3)])Sinh[z]-Divide[3,(z)^(2)]*Cosh[z]	Missing Macro Error	Failure	-	Failed [6 / 7] Result: Complex[0.44982524194021334, -0.19064547195046933] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.2843828483915114, -0.37732112452647515] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.49.E10	${\displaystyle{\displaystyle{\mathsf{i}^{(2)}_{n}}\left(z\right)=\tfrac{1}{2}e% ^{z}\sum_{k=0}^{n}(-1)^{k}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}+(-1)^{n}\tfrac{% 1}{2}e^{-z}\sum_{k=0}^{n}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}}}$ `\modsphBesseli{2}{n}@{z} = \tfrac{1}{2}e^{z}\sum_{k=0}^{n}(-1)^{k}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}+(-1)^{n}\tfrac{1}{2}e^{-z}\sum_{k=0}^{n}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}`	${\displaystyle{\displaystyle\Re((-n-\frac{1}{2})+k+1)>0,k\geq 1}}$	Error	Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)(n + 1/2), n] == Divide[1,2]Exp[z]Sum[(- 1)^(k)Divide[Subscript[a, k](n +Divide[1,2]),(z)^(k + 1)], {k, 0, n}, GenerateConditions->None]+(- 1)^(n)Divide[1,2](E)^(- z) Sum[Divide[Subscript[a, k]*(n +Divide[1,2]),(z)^(k + 1)], {k, 0, n}, GenerateConditions->None]	Missing Macro Error	Failure	-	Skipped - Because timed out
10.49#Ex10	${\displaystyle{\displaystyle{\mathsf{i}^{(2)}_{0}}\left(z\right)=\frac{\cosh z% }{z}}}$ `\modsphBesseli{2}{0}@{z} = \frac{\cosh@@{z}}{z}`	${\displaystyle{\displaystyle\Re((-0-\frac{1}{2})+k+1)>0}}$	Error	Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(0 + 1/2), 0] == Divide[Cosh[z],z]	Missing Macro Error	Failure	-	Failed [7 / 7] Result: DirectedInfinity[] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: DirectedInfinity[] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.49#Ex11	${\displaystyle{\displaystyle{\mathsf{i}^{(2)}_{1}}\left(z\right)=-\frac{\cosh z% }{z^{2}}+\frac{\sinh z}{z}}}$ `\modsphBesseli{2}{1}@{z} = -\frac{\cosh@@{z}}{z^{2}}+\frac{\sinh@@{z}}{z}`	${\displaystyle{\displaystyle\Re((-1-\frac{1}{2})+k+1)>0}}$	Error	Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(1 + 1/2), 1] == -Divide[Cosh[z],(z)^(2)]+Divide[Sinh[z],z]	Missing Macro Error	Failure	-	Failed [7 / 7] Result: Complex[-0.41419719140728073, -0.8850762711170859] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.1181398580617885, 1.2868595835312289] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.49#Ex12	${\displaystyle{\displaystyle{\mathsf{i}^{(2)}_{2}}\left(z\right)=\left(\frac{1% }{z}+\frac{3}{z^{3}}\right)\cosh z-\frac{3}{z^{2}}\sinh z}}$ `\modsphBesseli{2}{2}@{z} = \left(\frac{1}{z}+\frac{3}{z^{3}}\right)\cosh@@{z}-\frac{3}{z^{2}}\sinh@@{z}`	${\displaystyle{\displaystyle\Re((-2-\frac{1}{2})+k+1)>0}}$	Error	Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)(2 + 1/2), 2] == (Divide[1,z]+Divide[3,(z)^(3)])Cosh[z]-Divide[3,(z)^(2)]*Sinh[z]	Missing Macro Error	Failure	-	Failed [6 / 7] Result: Complex[1.106586755517561, 2.456957013551956] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-2.803584197807803, -1.2408087832280956] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.49.E12	${\displaystyle{\displaystyle\mathsf{k}_{n}\left(z\right)=\tfrac{1}{2}\pi e^{-z% }\sum_{k=0}^{n}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}}}$ `\modsphBesselK{n}@{z} = \tfrac{1}{2}\pi e^{-z}\sum_{k=0}^{n}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}`	${\displaystyle{\displaystyle k\geq 1}}$	Error	Sqrt[1/2 Pi /z] BesselK[n + 1/2, z] == Divide[1,2]PiExp[- z]Sum[Divide[Subscript[a, k](n +Divide[1,2]),(z)^(k + 1)], {k, 0, n}, GenerateConditions->None]	Missing Macro Error	Failure	-	Failed [210 / 210] Result: Complex[-1.0260307573251746, 0.0967341401667452] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-2.907697530268464, -0.43148595883398677] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.49#Ex13	${\displaystyle{\displaystyle\mathsf{k}_{0}\left(z\right)=\tfrac{1}{2}\pi\frac{% e^{-z}}{z}}}$ `\modsphBesselK{0}@{z} = \tfrac{1}{2}\pi\frac{e^{-z}}{z}`		Error	Sqrt[1/2 Pi /z] BesselK[0 + 1/2, z] == Divide[1,2]PiDivide[Exp[- z],z]	Missing Macro Error	Failure	-	Successful [Tested: 7]
10.49#Ex14	${\displaystyle{\displaystyle\mathsf{k}_{1}\left(z\right)=\tfrac{1}{2}\pi e^{-z% }\left(\frac{1}{z}+\frac{1}{z^{2}}\right)}}$ `\modsphBesselK{1}@{z} = \tfrac{1}{2}\pi e^{-z}\left(\frac{1}{z}+\frac{1}{z^{2}}\right)`		Error	Sqrt[1/2 Pi /z] BesselK[1 + 1/2, z] == Divide[1,2]PiExp[- z]*(Divide[1,z]+Divide[1,(z)^(2)])	Missing Macro Error	Failure	-	Successful [Tested: 7]
10.49#Ex15	${\displaystyle{\displaystyle\mathsf{k}_{2}\left(z\right)=\tfrac{1}{2}\pi e^{-z% }\left(\frac{1}{z}+\frac{3}{z^{2}}+\frac{3}{z^{3}}\right)}}$ `\modsphBesselK{2}@{z} = \tfrac{1}{2}\pi e^{-z}\left(\frac{1}{z}+\frac{3}{z^{2}}+\frac{3}{z^{3}}\right)`		Error	Sqrt[1/2 Pi /z] BesselK[2 + 1/2, z] == Divide[1,2]PiExp[- z]*(Divide[1,z]+Divide[3,(z)^(2)]+Divide[3,(z)^(3)])	Missing Macro Error	Failure	-	Successful [Tested: 7]
10.49#Ex16	${\displaystyle{\displaystyle\mathsf{j}_{n}\left(z\right)=z^{n}\left(-\frac{1}{% z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{n}\frac{\sin z}{z}}}$ `\sphBesselJ{n}@{z} = z^{n}\left(-\frac{1}{z}\deriv{}{z}\right)^{n}\frac{\sin@@{z}}{z}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0}}$	Error	(-Divide[1,z]D[(z)^(n)-Divide[1,z], z])^(n)*Divide[Sin[z],z]	Missing Macro Error	Failure	-	Failed [21 / 21] Result: Complex[0.28766324258243325, 0.13393934480402792] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.302013441049254, 0.9125931496973667] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.49#Ex17	${\displaystyle{\displaystyle\mathsf{y}_{n}\left(z\right)=-z^{n}\left(-\frac{1}% {z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{n}\frac{\cos z}{z}}}$ `\sphBesselY{n}@{z} = -z^{n}\left(-\frac{1}{z}\deriv{}{z}\right)^{n}\frac{\cos@@{z}}{z}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-(n+\frac{1}{2}))+% k+1)>0,\Re((-n-\frac{1}{2})+k+1)>0}}$	Error	SphericalBesselY[n, z] (-Divide[1,z]D[(z)^(n)-Divide[1,z], z])^(n)*Divide[Cos[z],z]	Missing Macro Error	Failure	-	Failed [21 / 21] Result: Complex[-0.9342001374760677, 0.968266641946737] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.14357960272401077, 3.9384338499123404] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.49#Ex18	${\displaystyle{\displaystyle{\mathsf{i}^{(1)}_{n}}\left(z\right)=z^{n}\left(% \frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{n}\frac{\sinh z}{z}}}$ `\modsphBesseli{1}{n}@{z} = z^{n}\left(\frac{1}{z}\deriv{}{z}\right)^{n}\frac{\sinh@@{z}}{z}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0}}$	Error	Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)(n + 1/2), n] (Divide[1,z]D[(z)^(n)Divide[1,z], z])^(n)Divide[Sinh[z],z]	Missing Macro Error	Failure	-	Failed [21 / 21] Result: Complex[0.35534425318828616, -0.09521420567684166] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.19008700336701606, 0.7298484499303669] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.49#Ex19	${\displaystyle{\displaystyle{\mathsf{i}^{(2)}_{n}}\left(z\right)=z^{n}\left(% \frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{n}\frac{\cosh z}{z}}}$ `\modsphBesseli{2}{n}@{z} = z^{n}\left(\frac{1}{z}\deriv{}{z}\right)^{n}\frac{\cosh@@{z}}{z}`	${\displaystyle{\displaystyle\Re((-n-\frac{1}{2})+k+1)>0}}$	Error	Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)(n + 1/2), n] (Divide[1,z]D[(z)^(n)Divide[1,z], z])^(n)Divide[Cosh[z],z]	Missing Macro Error	Failure	-	Failed [21 / 21] Result: Complex[-0.3553442531882861, 0.09521420567684165] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.31198506093225176, 1.0184810034762684] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.49.E16	${\displaystyle{\displaystyle\mathsf{k}_{n}\left(z\right)=(-1)^{n}\tfrac{1}{2}% \pi z^{n}\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{n}\frac{e^{-z% }}{z}}}$ `\modsphBesselK{n}@{z} = (-1)^{n}\tfrac{1}{2}\pi z^{n}\left(\frac{1}{z}\deriv{}{z}\right)^{n}\frac{e^{-z}}{z}`		Error	Sqrt[1/2 Pi /z] BesselK[n + 1/2, z] == (- 1)^(n)Divide[1,2](Divide[1,z]D[(z)^(n)Divide[1,z], z])^(n)*Divide[Exp[- z],z]	Missing Macro Error	Failure	-	Failed [21 / 21] Result: Complex[0.3593544107322247, -1.2247601267643444] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.45891810409859557, -4.100723067341411] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.49.E18	${\displaystyle{\displaystyle{\mathsf{j}_{n}^{2}}\left(z\right)+{\mathsf{y}_{n}% ^{2}}\left(z\right)=\sum_{k=0}^{n}\frac{s_{k}(n+\frac{1}{2})}{z^{2k+2}}}}$ `\sphBesselJ{n}^{2}@{z}+\sphBesselY{n}^{2}@{z} = \sum_{k=0}^{n}\frac{s_{k}(n+\frac{1}{2})}{z^{2k+2}}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0,\Re((-(n+\frac{1}{2}))+k+1)>0}}$	Error	(SphericalBesselJ[n, z])^(2)+ (SphericalBesselY[n, z])^(2) == Sum[Divide[Subscript[s, k](n +Divide[1,2]),(z)^(2k + 2)], {k, 0, n}, GenerateConditions->None]	Missing Macro Error	Failure	-	Failed [210 / 210] Result: Complex[-1.2990381056766571, 0.5179491924311224] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[s, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-9.999999999999996, 1.5358983848622398] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[s, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.49#Ex20	${\displaystyle{\displaystyle{\mathsf{j}_{0}^{2}}\left(z\right)+{\mathsf{y}_{0}% ^{2}}\left(z\right)=z^{-2}}}$ `\sphBesselJ{0}^{2}@{z}+\sphBesselY{0}^{2}@{z} = z^{-2}`	${\displaystyle{\displaystyle\Re((0+\frac{1}{2})+k+1)>0,\Re((-0-\frac{1}{2})+k+% 1)>0,\Re((-(-0-\frac{1}{2}))+k+1)>0,\Re((-(0+\frac{1}{2}))+k+1)>0}}$	Error	(SphericalBesselJ[0, z])^(2)+ (SphericalBesselY[0, z])^(2) == (z)^(- 2)	Missing Macro Error	Successful	-	Successful [Tested: 7]
10.49#Ex21	${\displaystyle{\displaystyle{\mathsf{j}_{1}^{2}}\left(z\right)+{\mathsf{y}_{1}% ^{2}}\left(z\right)=z^{-2}+z^{-4}}}$ `\sphBesselJ{1}^{2}@{z}+\sphBesselY{1}^{2}@{z} = z^{-2}+z^{-4}`	${\displaystyle{\displaystyle\Re((1+\frac{1}{2})+k+1)>0,\Re((-1-\frac{1}{2})+k+% 1)>0,\Re((-(-1-\frac{1}{2}))+k+1)>0,\Re((-(1+\frac{1}{2}))+k+1)>0}}$	Error	(SphericalBesselJ[1, z])^(2)+ (SphericalBesselY[1, z])^(2) == (z)^(- 2)+ (z)^(- 4)	Missing Macro Error	Successful	-	Successful [Tested: 7]
10.49#Ex22	${\displaystyle{\displaystyle{\mathsf{j}_{2}^{2}}\left(z\right)+{\mathsf{y}_{2}% ^{2}}\left(z\right)=z^{-2}+3z^{-4}+9z^{-6}}}$ `\sphBesselJ{2}^{2}@{z}+\sphBesselY{2}^{2}@{z} = z^{-2}+3z^{-4}+9z^{-6}`	${\displaystyle{\displaystyle\Re((2+\frac{1}{2})+k+1)>0,\Re((-2-\frac{1}{2})+k+% 1)>0,\Re((-(-2-\frac{1}{2}))+k+1)>0,\Re((-(2+\frac{1}{2}))+k+1)>0}}$	Error	(SphericalBesselJ[2, z])^(2)+ (SphericalBesselY[2, z])^(2) == (z)^(- 2)+ 3(z)^(- 4)+ 9(z)^(- 6)	Missing Macro Error	Successful	-	Successful [Tested: 7]
10.49.E20	${\displaystyle{\displaystyle\left({\mathsf{i}^{(1)}_{n}}\left(z\right)\right)^% {2}-\left({\mathsf{i}^{(2)}_{n}}\left(z\right)\right)^{2}=(-1)^{n+1}\sum_{k=0}% ^{n}(-1)^{k}\frac{s_{k}(n+\frac{1}{2})}{z^{2k+2}}}}$ `\left(\modsphBesseli{1}{n}@{z}\right)^{2}-\left(\modsphBesseli{2}{n}@{z}\right)^{2} = (-1)^{n+1}\sum_{k=0}^{n}(-1)^{k}\frac{s_{k}(n+\frac{1}{2})}{z^{2k+2}}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0}}$	Error	(Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)(n + 1/2), n])^(2)-(Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)(n + 1/2), n])^(2) == (- 1)^(n + 1)* Sum[(- 1)^(k)Divide[Subscript[s, k](n +Divide[1,2]),(z)^(2*k + 2)], {k, 0, n}, GenerateConditions->None]	Missing Macro Error	Failure	-	Failed [210 / 210] Result: Complex[-1.299038105676658, -0.7500000000000001] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[s, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.35182282028742856, 0.20312500000000058] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[s, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.50#Ex1	${\displaystyle{\displaystyle\mathscr{W}\left\{\mathsf{j}_{n}\left(z\right),% \mathsf{y}_{n}\left(z\right)\right\}=z^{-2}}}$ `\Wronskian@{\sphBesselJ{n}@{z},\sphBesselY{n}@{z}} = z^{-2}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0,\Re((-(n+\frac{1}{2}))+k+1)>0}}$	Error	Wronskian[{SphericalBesselJ[n, z], SphericalBesselY[n, z]}, z] == (z)^(- 2)	Missing Macro Error	Successful	-	Successful [Tested: 21]
10.50#Ex2	${\displaystyle{\displaystyle\mathscr{W}\left\{{\mathsf{h}^{(1)}_{n}}\left(z% \right),{\mathsf{h}^{(2)}_{n}}\left(z\right)\right\}=-2iz^{-2}}}$ `\Wronskian@{\sphHankelh{1}{n}@{z},\sphHankelh{2}{n}@{z}} = -2iz^{-2}`		Error	Wronskian[{SphericalHankelH1[n, z], SphericalHankelH2[n, z]}, z] == - 2I(z)^(- 2)	Missing Macro Error	Successful	-	Successful [Tested: 21]
10.50#Ex3	${\displaystyle{\displaystyle\mathscr{W}\left\{{\mathsf{i}^{(1)}_{n}}\left(z% \right),{\mathsf{i}^{(2)}_{n}}\left(z\right)\right\}=(-1)^{n+1}z^{-2}}}$ `\Wronskian@{\modsphBesseli{1}{n}@{z},\modsphBesseli{2}{n}@{z}} = (-1)^{n+1}z^{-2}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0}}$	Error	Wronskian[{Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)(n + 1/2), n], Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)(n + 1/2), n]}, z] == (- 1)^(n + 1)* (z)^(- 2)	Missing Macro Error	Failure	-	Failed [21 / 21] Result: Complex[-0.5000000000000001, 0.8660254037844386] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.5000000000000001, -0.8660254037844386] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.50#Ex4	${\displaystyle{\displaystyle\mathscr{W}\left\{{\mathsf{i}^{(1)}_{n}}\left(z% \right),\mathsf{k}_{n}\left(z\right)\right\}=\mathscr{W}\left\{{\mathsf{i}^{(2% )}_{n}}\left(z\right),\mathsf{k}_{n}\left(z\right)\right\}\\ }}$ `\Wronskian@{\modsphBesseli{1}{n}@{z},\modsphBesselK{n}@{z}} = \Wronskian@{\modsphBesseli{2}{n}@{z},\modsphBesselK{n}@{z}}\\`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0}}$	Error	Wronskian[{Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)(n + 1/2), n], Sqrt[1/2 Pi /z] BesselK[n + 1/2, z]}, z] == Wronskian[{Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)(n + 1/2), n], Sqrt[1/2 Pi /z] BesselK[n + 1/2, z]}, z]	Missing Macro Error	Failure	-	Failed [21 / 21] Result: Complex[0.5384915109869794, 1.7026856201657974] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-2.6544302063904848, -2.4451654315616667] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.50#Ex4	${\displaystyle{\displaystyle\mathscr{W}\left\{{\mathsf{i}^{(2)}_{n}}\left(z% \right),\mathsf{k}_{n}\left(z\right)\right\}\\ =-\tfrac{1}{2}\pi z^{-2}}}$ `\Wronskian@{\modsphBesseli{2}{n}@{z},\modsphBesselK{n}@{z}}\\ = -\tfrac{1}{2}\pi z^{-2}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0}}$	Error	Wronskian[{Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)(n + 1/2), n], Sqrt[1/2 Pi /z] BesselK[n + 1/2, z]}, z] == -Divide[1,2]Pi*(z)^(- 2)	Missing Macro Error	Failure	-	Failed [21 / 21] Result: Complex[0.5161524079039588, -2.211692333258562] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[7.686727830477982, 4.996906619076774] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.50#Ex5	${\displaystyle{\displaystyle\mathsf{j}_{n+1}\left(z\right)\mathsf{y}_{n}\left(% z\right)-\mathsf{j}_{n}\left(z\right)\mathsf{y}_{n+1}\left(z\right)=z^{-2}}}$ `\sphBesselJ{n+1}@{z}\sphBesselY{n}@{z}-\sphBesselJ{n}@{z}\sphBesselY{n+1}@{z} = z^{-2}`	${\displaystyle{\displaystyle\Re(((n+1)+\frac{1}{2})+k+1)>0,\Re((n+\frac{1}{2})% +k+1)>0,\Re((-(n+1)-\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+1)>0,\Re((-(-(n% +1)-\frac{1}{2}))+k+1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0,\Re((-(n+\frac{1}{2}))% +k+1)>0,\Re((-((n+1)+\frac{1}{2}))+k+1)>0}}$	Error	SphericalBesselJ[n + 1, z]SphericalBesselY[n, z]- SphericalBesselJ[n, z]SphericalBesselY[n + 1, z] == (z)^(- 2)	Missing Macro Error	Successful	-	Successful [Tested: 21]
10.50#Ex6	${\displaystyle{\displaystyle\mathsf{j}_{n+2}\left(z\right)\mathsf{y}_{n}\left(% z\right)-\mathsf{j}_{n}\left(z\right)\mathsf{y}_{n+2}\left(z\right)=(2n+3)z^{-% 3}}}$ `\sphBesselJ{n+2}@{z}\sphBesselY{n}@{z}-\sphBesselJ{n}@{z}\sphBesselY{n+2}@{z} = (2n+3)z^{-3}`	${\displaystyle{\displaystyle\Re(((n+2)+\frac{1}{2})+k+1)>0,\Re((n+\frac{1}{2})% +k+1)>0,\Re((-(n+2)-\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+1)>0,\Re((-(-(n% +2)-\frac{1}{2}))+k+1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0,\Re((-(n+\frac{1}{2}))% +k+1)>0,\Re((-((n+2)+\frac{1}{2}))+k+1)>0}}$	Error	SphericalBesselJ[n + 2, z]SphericalBesselY[n, z]- SphericalBesselJ[n, z]SphericalBesselY[n + 2, z] == (2n + 3)(z)^(- 3)	Missing Macro Error	Failure	-	Successful [Tested: 21]
10.50.E4	${\displaystyle{\displaystyle\mathsf{j}_{0}\left(z\right)\mathsf{j}_{n}\left(z% \right)+\mathsf{y}_{0}\left(z\right)\mathsf{y}_{n}\left(z\right)=\cos\left(% \tfrac{1}{2}n\pi\right)\sum_{k=0}^{\left\lfloor n/2\right\rfloor}(-1)^{k}\frac% {a_{2k}(n+\tfrac{1}{2})}{z^{2k+2}}+\sin\left(\tfrac{1}{2}n\pi\right)\sum_{k=0}% ^{\left\lfloor(n-1)/2\right\rfloor}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{% 2k+3}}}}$ `\sphBesselJ{0}@{z}\sphBesselJ{n}@{z}+\sphBesselY{0}@{z}\sphBesselY{n}@{z} = \cos@{\tfrac{1}{2}n\pi}\sum_{k=0}^{\floor{n/2}}(-1)^{k}\frac{a_{2k}(n+\tfrac{1}{2})}{z^{2k+2}}+\sin@{\tfrac{1}{2}n\pi}\sum_{k=0}^{\floor{(n-1)/2}}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{2k+3}}`	${\displaystyle{\displaystyle\Re((0+\frac{1}{2})+k+1)>0,\Re((n+\frac{1}{2})+k+1% )>0,\Re((-0-\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+1)>0,\Re((-(-0-\frac{1}% {2}))+k+1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0,\Re((-(0+\frac{1}{2}))+k+1)>0,\Re(% (-(n+\frac{1}{2}))+k+1)>0,k\geq 1}}$	Error	SphericalBesselJ[0, z]SphericalBesselJ[n, z]+ SphericalBesselY[0, z]SphericalBesselY[n, z] == Cos[Divide[1,2]nPi]Sum[(- 1)^(k)Divide[Subscript[a, 2k](n +Divide[1,2]),(z)^(2k + 2)], {k, 0, Floor[n/2]}, GenerateConditions->None]+ Sin[Divide[1,2]nPi]Sum[(- 1)^(k)Divide[Subscript[a, 2k + 1](n +Divide[1,2]),(z)^(2k + 3)], {k, 0, Floor[(n - 1)/2]}, GenerateConditions->None]	Missing Macro Error	Failure	-	Skipped - Because timed out
10.51#Ex1	${\displaystyle{\displaystyle f_{n-1}(z)+f_{n+1}(z)=((2n+1)/z)f_{n}(z)}}$ `f_{n-1}(z)+f_{n+1}(z) = ((2n+1)/z)f_{n}(z)`		f[n - 1](z)+ f[n + 1](z) = ((2n + 1)/z)f[n](z)	Subscript[f, n - 1][z]+ Subscript[f, n + 1][z] == ((2n + 1)/z)Subscript[f, n][z]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.51#Ex5	${\displaystyle{\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}% \right)^{m}(z^{n+1}f_{n}(z))=z^{n-m+1}f_{n-m}(z)}}$ `\left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{n+1}f_{n}(z)) = z^{n-m+1}f_{n-m}(z)`	${\displaystyle{\displaystyle m=0}}$	(diff((1)/(z), z))^(m)((z)^(n + 1) f[n](z)) = (z)^(n - m + 1)* f[n - m](z)	(D[Divide[1,z], z])^(m)((z)^(n + 1) Subscript[f, n][z]) == (z)^(n - m + 1)* Subscript[f, n - m][z]	Failure	Failure	Error	Failed [288 / 300] Result: Complex[-0.49999999999999994, -1.8660254037844388] Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, Plus[Times[-1, m], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.49999999999999994, -1.8660254037844388] Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, Plus[Times[-1, m], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.51#Ex6	${\displaystyle{\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}% \right)^{m}(z^{-n}f_{n}(z))=(-1)^{m}z^{-n-m}f_{n+m}(z)}}$ `\left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{-n}f_{n}(z)) = (-1)^{m}z^{-n-m}f_{n+m}(z)`		(diff((1)/(z), z))^(m)((z)^(- n) f[n](z)) = (- 1)^(m)* (z)^(- n - m)* f[n + m](z)	(D[Divide[1,z], z])^(m)((z)^(- n) Subscript[f, n][z]) == (- 1)^(m)* (z)^(- n - m)* Subscript[f, n + m][z]	Failure	Failure	Failed [288 / 300] Result: 1.366025403-.3660254033I Test Values: {z = 1/23^(1/2)+1/2I, f[n] = 1/23^(1/2)+1/2I, f[n+m] = 1/23^(1/2)+1/2I, n = 1, m = 3} Result: .9999999993-.9999999984I Test Values: {z = 1/23^(1/2)+1/2I, f[n] = 1/23^(1/2)+1/2I, f[n+m] = 1/23^(1/2)+1/2I, n = 2, m = 3} ... skip entries to safe data	Failed [288 / 300] Result: Complex[0.1339745962155613, 0.49999999999999994] Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, Plus[m, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.3660254037844386, 0.36602540378443865] Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, Plus[m, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.51#Ex7	${\displaystyle{\displaystyle g_{n-1}(z)-g_{n+1}(z)=((2n+1)/z)g_{n}(z)}}$ `g_{n-1}(z)-g_{n+1}(z) = ((2n+1)/z)g_{n}(z)`		g[n - 1](z)- g[n + 1](z) = ((2n + 1)/z)g[n](z)	Subscript[g, n - 1][z]- Subscript[g, n + 1][z] == ((2n + 1)/z)Subscript[g, n][z]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.51#Ex11	${\displaystyle{\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}% \right)^{m}(z^{n+1}g_{n}(z))=z^{n-m+1}g_{n-m}(z)}}$ `\left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{n+1}g_{n}(z)) = z^{n-m+1}g_{n-m}(z)`	${\displaystyle{\displaystyle m=0}}$	(diff((1)/(z), z))^(m)((z)^(n + 1) g[n](z)) = (z)^(n - m + 1)* g[n - m](z)	(D[Divide[1,z], z])^(m)((z)^(n + 1) Subscript[g, n][z]) == (z)^(n - m + 1)* Subscript[g, n - m][z]	Failure	Failure	Error	Failed [288 / 300] Result: Complex[-0.49999999999999994, -1.8660254037844388] Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, Plus[Times[-1, m], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.49999999999999994, -1.8660254037844388] Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, Plus[Times[-1, m], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.51#Ex12	${\displaystyle{\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}% \right)^{m}(z^{-n}g_{n}(z))=z^{-n-m}g_{n+m}(z)}}$ `\left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{-n}g_{n}(z)) = z^{-n-m}g_{n+m}(z)`		(diff((1)/(z), z))^(m)((z)^(- n) g[n](z)) = (z)^(- n - m)* g[n + m](z)	(D[Divide[1,z], z])^(m)((z)^(- n) Subscript[g, n][z]) == (z)^(- n - m)* Subscript[g, n + m][z]	Failure	Failure	Failed [288 / 300] Result: .3660254028+1.366025403I Test Values: {z = 1/23^(1/2)+1/2I, g[n] = 1/23^(1/2)+1/2I, g[n+m] = 1/23^(1/2)+1/2I, n = 1, m = 3} Result: .9999999987+.9999999996I Test Values: {z = 1/23^(1/2)+1/2I, g[n] = 1/23^(1/2)+1/2I, g[n+m] = 1/23^(1/2)+1/2I, n = 2, m = 3} ... skip entries to safe data	Failed [288 / 300] Result: Complex[-1.8660254037844388, 0.49999999999999994] Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, Plus[m, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.3660254037844388, 1.3660254037844386] Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, Plus[m, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.53.E1	${\displaystyle{\displaystyle\mathsf{j}_{n}\left(z\right)=z^{n}\sum_{k=0}^{% \infty}\frac{(-\frac{1}{2}z^{2})^{k}}{k!(2n+2k+1)!!}}}$ `\sphBesselJ{n}@{z} = z^{n}\sum_{k=0}^{\infty}\frac{(-\frac{1}{2}z^{2})^{k}}{k!(2n+2k+1)!!}`	${\displaystyle{\displaystyle\|z\|<\infty,\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-% \frac{1}{2})+k+1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0}}$	Error	SphericalBesselJ[n, z] == (z)^(n)* Sum[Divide[(-Divide[1,2](z)^(2))^(k),(k)!(2n + 2k + 1)!!], {k, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Successful [Tested: 21]
10.53.E2	${\displaystyle{\displaystyle\mathsf{y}_{n}\left(z\right)=-\frac{1}{z^{n+1}}% \sum_{k=0}^{n}\frac{(2n-2k-1)!!(\frac{1}{2}z^{2})^{k}}{k!}+\frac{(-1)^{n+1}}{z% ^{n+1}}\sum_{k=n+1}^{\infty}\frac{(-\frac{1}{2}z^{2})^{k}}{k!(2k-2n-1)!!}}}$ `\sphBesselY{n}@{z} = -\frac{1}{z^{n+1}}\sum_{k=0}^{n}\frac{(2n-2k-1)!!(\frac{1}{2}z^{2})^{k}}{k!}+\frac{(-1)^{n+1}}{z^{n+1}}\sum_{k=n+1}^{\infty}\frac{(-\frac{1}{2}z^{2})^{k}}{k!(2k-2n-1)!!}`	${\displaystyle{\displaystyle 0<\|z\|,\|z\|<\infty.,\Re((n+\frac{1}{2})+k+1)>0,\Re(% (-(n+\frac{1}{2}))+k+1)>0,\Re((-n-\frac{1}{2})+k+1)>0}}$	Error	SphericalBesselY[n, z] == -Divide[1,(z)^(n + 1)]Sum[Divide[(2n - 2k - 1)!!(Divide[1,2](z)^(2))^(k),(k)!], {k, 0, n}, GenerateConditions->None]+Divide[(- 1)^(n + 1),(z)^(n + 1)]Sum[Divide[(-Divide[1,2](z)^(2))^(k),(k)!(2k - 2n - 1)!!], {k, n + 1, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Successful [Tested: 21]
10.53.E3	${\displaystyle{\displaystyle{\mathsf{i}^{(1)}_{n}}\left(z\right)=z^{n}\sum_{k=% 0}^{\infty}\frac{(\frac{1}{2}z^{2})^{k}}{k!(2n+2k+1)!!}}}$ `\modsphBesseli{1}{n}@{z} = z^{n}\sum_{k=0}^{\infty}\frac{(\frac{1}{2}z^{2})^{k}}{k!(2n+2k+1)!!}`	${\displaystyle{\displaystyle\|z\|<\infty,\Re((n+\frac{1}{2})+k+1)>0}}$	Error	Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)(n + 1/2), n] == (z)^(n) Sum[Divide[(Divide[1,2](z)^(2))^(k),(k)!(2n + 2k + 1)!!], {k, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Failed [20 / 21] Result: Complex[0.06771919180965624, -0.29579816936516184] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.4498252419402129, -0.19064547195046921] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.53.E4	${\displaystyle{\displaystyle{\mathsf{i}^{(2)}_{n}}\left(z\right)=\frac{(-1)^{n% }}{z^{n+1}}\sum_{k=0}^{n}\frac{(2n-2k-1)!!(-\frac{1}{2}z^{2})^{k}}{k!}+\frac{1% }{z^{n+1}}\sum_{k=n+1}^{\infty}\frac{(\frac{1}{2}z^{2})^{k}}{k!(2k-2n-1)!!}}}$ `\modsphBesseli{2}{n}@{z} = \frac{(-1)^{n}}{z^{n+1}}\sum_{k=0}^{n}\frac{(2n-2k-1)!!(-\frac{1}{2}z^{2})^{k}}{k!}+\frac{1}{z^{n+1}}\sum_{k=n+1}^{\infty}\frac{(\frac{1}{2}z^{2})^{k}}{k!(2k-2n-1)!!}`	${\displaystyle{\displaystyle 0<\|z\|,\|z\|<\infty.,\Re((-n-\frac{1}{2})+k+1)>0}}$	Error	Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)(n + 1/2), n] == Divide[(- 1)^(n),(z)^(n + 1)]Sum[Divide[(2n - 2k - 1)!!(-Divide[1,2](z)^(2))^(k),(k)!], {k, 0, n}, GenerateConditions->None]+Divide[1,(z)^(n + 1)]Sum[Divide[(Divide[1,2](z)^(2))^(k),(k)!(2k - 2*n - 1)!!], {k, n + 1, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Failed [20 / 21] Result: Complex[-0.4141971914072808, -0.8850762711170854] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.1065867555175597, 2.456957013551954] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.54.E1	${\displaystyle{\displaystyle\mathsf{j}_{n}\left(z\right)=\frac{z^{n}}{2^{n+1}n% !}\int_{0}^{\pi}\cos\left(z\cos\theta\right)(\sin\theta)^{2n+1}\mathrm{d}% \theta}}$ `\sphBesselJ{n}@{z} = \frac{z^{n}}{2^{n+1}n!}\int_{0}^{\pi}\cos@{z\cos@@{\theta}}(\sin@@{\theta})^{2n+1}\diff{\theta}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0}}$	Error	SphericalBesselJ[n, z] == Divide[(z)^(n),(2)^(n + 1)* (n)!]Integrate[Cos[zCos[\[Theta]]](Sin[\[Theta]])^(2n + 1), {\[Theta], 0, Pi}, GenerateConditions->None]	Missing Macro Error	Successful	-	Successful [Tested: 21]
10.54.E2	${\displaystyle{\displaystyle\mathsf{j}_{n}\left(z\right)=\frac{(-i)^{n}}{2}% \int_{0}^{\pi}e^{iz\cos\theta}P_{n}\left(\cos\theta\right)\sin\theta\mathrm{d}% \theta}}$ `\sphBesselJ{n}@{z} = \frac{(-i)^{n}}{2}\int_{0}^{\pi}e^{iz\cos@@{\theta}}\assLegendreP[]{n}@{\cos@@{\theta}}\sin@@{\theta}\diff{\theta}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0}}$	Error	SphericalBesselJ[n, z] == Divide[(- I)^(n),2]Integrate[Exp[IzCos[\[Theta]]]LegendreP[n, 0, 3, Cos[\[Theta]]]*Sin[\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None]	Missing Macro Error	Aborted	-	Successful [Tested: 21]
10.54.E3	${\displaystyle{\displaystyle\mathsf{k}_{n}\left(z\right)=\frac{\pi}{2}\int_{1}% ^{\infty}e^{-zt}P_{n}\left(t\right)\mathrm{d}t}}$ `\modsphBesselK{n}@{z} = \frac{\pi}{2}\int_{1}^{\infty}e^{-zt}\assLegendreP[]{n}@{t}\diff{t}`	${\displaystyle{\displaystyle\|\operatorname{ph}z\|<\tfrac{1}{2}\pi.}}$	Error	Sqrt[1/2 Pi /z] BesselK[n + 1/2, z] == Divide[Pi,2]Integrate[Exp[- zt]*LegendreP[n, 0, 3, t], {t, 1, Infinity}, GenerateConditions->None]	Missing Macro Error	Aborted	-	Skipped - Because timed out
10.54.E4	${\displaystyle{\displaystyle\mathsf{j}_{n}\left(z\right)=\frac{(-i)^{n+1}}{2% \pi}\int_{i\infty}^{(-1+,1+)}e^{izt}Q_{n}\left(t\right)\mathrm{d}t}}$ `\sphBesselJ{n}@{z} = \frac{(-i)^{n+1}}{2\pi}\int_{i\infty}^{(-1+,1+)}e^{izt}\assLegendreQ[]{n}@{t}\diff{t}`	${\displaystyle{\displaystyle\|\operatorname{ph}z\|<\tfrac{1}{2}\pi.,\Re((n+\frac% {1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0}}$	Error	SphericalBesselJ[n, z] == Divide[(- I)^(n + 1),2Pi]Integrate[Exp[Izt]LegendreQ[n, 0, 3, t], {t, IInfinity, (- 1 + , 1 +)}, GenerateConditions->None]	Missing Macro Error	Failure	-	Error
10.54#Ex1	${\displaystyle{\displaystyle{\mathsf{h}^{(1)}_{n}}\left(z\right)=\frac{(-i)^{n% +1}}{\pi}\int_{i\infty}^{(1+)}e^{izt}Q_{n}\left(t\right)\mathrm{d}t}}$ `\sphHankelh{1}{n}@{z} = \frac{(-i)^{n+1}}{\pi}\int_{i\infty}^{(1+)}e^{izt}\assLegendreQ[]{n}@{t}\diff{t}`		Error	SphericalHankelH1[n, z] == Divide[(- I)^(n + 1),Pi]Integrate[Exp[Izt]LegendreQ[n, 0, 3, t], {t, I*Infinity, (1 +)}, GenerateConditions->None]	Missing Macro Error	Failure	-	Error
10.54#Ex2	${\displaystyle{\displaystyle{\mathsf{h}^{(2)}_{n}}\left(z\right)=\frac{(-i)^{n% +1}}{\pi}\int_{i\infty}^{(-1+)}e^{izt}Q_{n}\left(t\right)\mathrm{d}t}}$ `\sphHankelh{2}{n}@{z} = \frac{(-i)^{n+1}}{\pi}\int_{i\infty}^{(-1+)}e^{izt}\assLegendreQ[]{n}@{t}\diff{t}`	${\displaystyle{\displaystyle\|\operatorname{ph}z\|<\tfrac{1}{2}\pi.}}$	Error	SphericalHankelH2[n, z] == Divide[(- I)^(n + 1),Pi]Integrate[Exp[Izt]LegendreQ[n, 0, 3, t], {t, I*Infinity, (- 1 +)}, GenerateConditions->None]	Missing Macro Error	Failure	-	Error
10.56.E1	${\displaystyle{\displaystyle\frac{\cos\sqrt{z^{2}-2zt}}{z}=\frac{\cos z}{z}+% \sum_{n=1}^{\infty}\frac{t^{n}}{n!}\mathsf{j}_{n-1}\left(z\right)}}$ `\frac{\cos@@{\sqrt{z^{2}-2zt}}}{z} = \frac{\cos@@{z}}{z}+\sum_{n=1}^{\infty}\frac{t^{n}}{n!}\sphBesselJ{n-1}@{z}`	${\displaystyle{\displaystyle\Re(((n-1)+\frac{1}{2})+k+1)>0,\Re((-(n-1)-\frac{1% }{2})+k+1)>0,\Re((-(-(n-1)-\frac{1}{2}))+k+1)>0}}$	Error	Divide[Cos[Sqrt[(z)^(2)- 2zt]],z] == Divide[Cos[z],z]+ Sum[Divide[(t)^(n),(n)!]*SphericalBesselJ[n - 1, z], {n, 1, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Failed [42 / 42] Result: Plus[Complex[-1.0653161526495918, 0.32810386977400907], Times[-1.0, NSum[Times[Power[-1.5, n], Power[Factorial[n], -1], SphericalBesselJ[Plus[-1, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]] Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[-1.8246723112251149, 0.13108435615091096], Times[-1.0, NSum[Times[Power[-1.5, n], Power[Factorial[n], -1], SphericalBesselJ[Plus[-1, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]] Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.56.E2	${\displaystyle{\displaystyle\frac{\sin\sqrt{z^{2}-2zt}}{z}=\frac{\sin z}{z}+% \sum_{n=1}^{\infty}\frac{t^{n}}{n!}\mathsf{y}_{n-1}\left(z\right)}}$ `\frac{\sin@@{\sqrt{z^{2}-2zt}}}{z} = \frac{\sin@@{z}}{z}+\sum_{n=1}^{\infty}\frac{t^{n}}{n!}\sphBesselY{n-1}@{z}`	${\displaystyle{\displaystyle\Re(((n-1)+\frac{1}{2})+k+1)>0,\Re((-((n-1)+\frac{% 1}{2}))+k+1)>0,\Re((-(n-1)-\frac{1}{2})+k+1)>0}}$	Error	Divide[Sin[Sqrt[(z)^(2)- 2zt]],z] == Divide[Sin[z],z]+ Sum[Divide[(t)^(n),(n)!]*SphericalBesselY[n - 1, z], {n, 1, Infinity}, GenerateConditions->None]	Missing Macro Error	Aborted	-	Skipped - Because timed out
10.56.E3	${\displaystyle{\displaystyle\frac{\cosh\sqrt{z^{2}+2izt}}{z}=\frac{\cosh z}{z}% +\sum_{n=1}^{\infty}\frac{(it)^{n}}{n!}{\mathsf{i}^{(1)}_{n-1}}\left(z\right)}}$ `\frac{\cosh@@{\sqrt{z^{2}+2izt}}}{z} = \frac{\cosh@@{z}}{z}+\sum_{n=1}^{\infty}\frac{(it)^{n}}{n!}\modsphBesseli{1}{n-1}@{z}`	${\displaystyle{\displaystyle\Re(((n-1)+\frac{1}{2})+k+1)>0}}$	Error	Divide[Cosh[Sqrt[(z)^(2)+ 2Izt]],z] == Divide[Cosh[z],z]+ Sum[Divide[(It)^(n),(n)!]Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)(n - 1 + 1/2), n - 1], {n, 1, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Failed [42 / 42] Result: Plus[Complex[-0.13108435615091052, -1.8246723112251153], Times[-1.0, NSum[Times[Power[Complex[0.0, -1.5], n], Power[Power[E, Times[Complex[0, Rational[-1, 6]], Pi]], Rational[1, 2]], Power[Times[Rational[1, 2], Pi], Rational[1, 2]], BesselI[Plus[Rational[-1, 2], n], Plus[-1, n]], Power[Factorial[n], -1]] Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[-0.022834987510423566, -1.7127448295681993], Times[-1.0, NSum[Times[Power[Complex[0.0, -1.5], n], Power[Power[E, Times[Complex[0, Rational[-2, 3]], Pi]], Rational[1, 2]], Power[Times[Rational[1, 2], Pi], Rational[1, 2]], BesselI[Plus[Rational[-1, 2], n], Plus[-1, n]], Power[Factorial[n], -1]] Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.56.E4	${\displaystyle{\displaystyle\frac{\sinh\sqrt{z^{2}+2izt}}{z}=\frac{\sinh z}{z}% +\sum_{n=1}^{\infty}\frac{(it)^{n}}{n!}{\mathsf{i}^{(2)}_{n-1}}\left(z\right)}}$ `\frac{\sinh@@{\sqrt{z^{2}+2izt}}}{z} = \frac{\sinh@@{z}}{z}+\sum_{n=1}^{\infty}\frac{(it)^{n}}{n!}\modsphBesseli{2}{n-1}@{z}`	${\displaystyle{\displaystyle\Re((-(n-1)-\frac{1}{2})+k+1)>0}}$	Error	Divide[Sinh[Sqrt[(z)^(2)+ 2Izt]],z] == Divide[Sinh[z],z]+ Sum[Divide[(It)^(n),(n)!]Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)(n - 1 + 1/2), n - 1], {n, 1, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Failed [42 / 42] Result: Plus[Complex[-0.12983798012989667, -2.1935922908985273], Times[-1.0, NSum[Times[Power[Complex[0.0, -1.5], n], Power[Power[E, Times[Complex[0, Rational[-1, 6]], Pi]], Rational[1, 2]], Power[Times[Rational[1, 2], Pi], Rational[1, 2]], BesselI[Plus[Rational[1, 2], Times[-1, n]], Plus[-1, n]], Power[Factorial[n], -1]] Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[-1.4886830119296848, -1.839102010336905], Times[-1.0, NSum[Times[Power[Complex[0.0, -1.5], n], Power[Power[E, Times[Complex[0, Rational[-2, 3]], Pi]], Rational[1, 2]], Power[Times[Rational[1, 2], Pi], Rational[1, 2]], BesselI[Plus[Rational[1, 2], Times[-1, n]], Plus[-1, n]], Power[Factorial[n], -1]] Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.56.E5	${\displaystyle{\displaystyle\frac{\exp\left(-\sqrt{z^{2}+2izt}\right)}{z}=% \frac{e^{-z}}{z}+\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{(-it)^{n}}{n!}\mathsf{k% }_{n-1}\left(z\right)}}$ `\frac{\exp@{-\sqrt{z^{2}+2izt}}}{z} = \frac{e^{-z}}{z}+\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{(-it)^{n}}{n!}\modsphBesselK{n-1}@{z}`		Error	Divide[Exp[-Sqrt[(z)^(2)+ 2Izt]],z] == Divide[Exp[- z],z]+Divide[2,Pi]Sum[Divide[(- It)^(n),(n)!]Sqrt[1/2 Pi /z] BesselK[n - 1 + 1/2, z], {n, 1, Infinity}, GenerateConditions->None]	Missing Macro Error	Aborted	-	Skipped - Because timed out
10.57.E1	${\displaystyle{\displaystyle\mathsf{j}_{n}'\left((n+\tfrac{1}{2})z\right)=% \frac{\pi^{\frac{1}{2}}}{((2n+1)z)^{\frac{1}{2}}}J_{n+\frac{1}{2}}'\left((n+% \tfrac{1}{2})z\right)-\frac{\pi^{\frac{1}{2}}}{((2n+1)z)^{\frac{3}{2}}}J_{n+% \frac{1}{2}}\left((n+\tfrac{1}{2})z\right)}}$ `\sphBesselJ{n}'@{(n+\tfrac{1}{2})z} = \frac{\pi^{\frac{1}{2}}}{((2n+1)z)^{\frac{1}{2}}}\BesselJ{n+\frac{1}{2}}'@{(n+\tfrac{1}{2})z}-\frac{\pi^{\frac{1}{2}}}{((2n+1)z)^{\frac{3}{2}}}\BesselJ{n+\frac{1}{2}}@{(n+\tfrac{1}{2})z}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0}}$	Error	D[SphericalBesselJ[n, (n +Divide[1,2])z], {(n +Divide[1,2])z, 1}] == Divide[(Pi)^(Divide[1,2]),((2n + 1)z)^(Divide[1,2])]D[BesselJ[n +Divide[1,2], (n +Divide[1,2])z], {(n +Divide[1,2])z, 1}]-Divide[(Pi)^(Divide[1,2]),((2n + 1)z)^(Divide[3,2])]BesselJ[n +Divide[1,2], (n +Divide[1,2])*z]	Missing Macro Error	Failure	-	Failed [21 / 21] Result: Plus[Complex[0.14653389603833195, -0.029869009956249915], Times[Complex[-0.988457695936884, 0.2648564413786163], D[Complex[0.36567703182522004, 0.24184221354059504] Test Values: {Complex[1.299038105676658, 0.7499999999999999], 1.0}]], D[Complex[0.425509744388485, 0.14219887983348967], {Complex[1.299038105676658, 0.7499999999999999], 1.0}]], {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[0.06710374092328811, 0.007963502819859997], Times[Complex[-0.7656560389588212, 0.20515691731902835], D[Complex[0.2637838125883578, 0.3348231997381719] Test Values: {Complex[2.165063509461097, 1.2499999999999998], 1.0}]], D[Complex[0.27065896459303473, 0.20224233103375913], {Complex[2.165063509461097, 1.2499999999999998], 1.0}]], {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.60.E1	${\displaystyle{\displaystyle\frac{\cos w}{w}=-\sum_{n=0}^{\infty}(2n+1)\mathsf% {j}_{n}\left(v\right)\mathsf{y}_{n}\left(u\right)P_{n}\left(\cos\alpha\right)}}$ `\frac{\cos@@{w}}{w} = -\sum_{n=0}^{\infty}(2n+1)\sphBesselJ{n}@{v}\sphBesselY{n}@{u}\assLegendreP[]{n}@{\cos@@{\alpha}}`	${\displaystyle{\displaystyle\|ve^{+i\alpha}\|<\|u\|,\|ve^{-i\alpha}\|<\|u\|,\Re((n+% \frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0% ,\Re((-(n+\frac{1}{2}))+k+1)>0}}$	Error	Divide[Cos[w],w] == - Sum[(2n + 1)SphericalBesselJ[n, v]SphericalBesselY[n, u]LegendreP[n, 0, 3, Cos[\[Alpha]]], {n, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Failed [300 / 300] Result: Plus[Complex[0.43419403794642014, -0.7090399040477617], NSum[Times[Plus[1, Times[2, n]], LegendreP[n, 0.0707372016677029], SphericalBesselJ[n, -0.5], SphericalBesselY[n, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, -0.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5]} Result: Plus[Complex[0.43419403794642014, -0.7090399040477617], NSum[Times[Plus[1, Times[2, n]], LegendreP[n, 0.8775825618903728], SphericalBesselJ[n, -0.5], SphericalBesselY[n, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, -0.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 0.5]} ... skip entries to safe data
10.60.E2	${\displaystyle{\displaystyle\frac{\sin w}{w}=\sum_{n=0}^{\infty}(2n+1)\mathsf{% j}_{n}\left(v\right)\mathsf{j}_{n}\left(u\right)P_{n}\left(\cos\alpha\right)}}$ `\frac{\sin@@{w}}{w} = \sum_{n=0}^{\infty}(2n+1)\sphBesselJ{n}@{v}\sphBesselJ{n}@{u}\assLegendreP[]{n}@{\cos@@{\alpha}}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0}}$	Error	Divide[Sin[w],w] == Sum[(2n + 1)SphericalBesselJ[n, v]SphericalBesselJ[n, u]LegendreP[n, 0, 3, Cos[\[Alpha]]], {n, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Failed [300 / 300] Result: Plus[Complex[0.912697022466604, -0.13712305377128448], Times[-1.0, NSum[Times[Plus[1, Times[2, n]], LegendreP[n, 0.0707372016677029], Power[SphericalBesselJ[n, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5]} Result: Plus[Complex[0.912697022466604, -0.13712305377128448], Times[-1.0, NSum[Times[Plus[1, Times[2, n]], LegendreP[n, 0.8775825618903728], Power[SphericalBesselJ[n, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 0.5]} ... skip entries to safe data
10.60.E3	${\displaystyle{\displaystyle\frac{e^{-w}}{w}=\frac{2}{\pi}\sum_{n=0}^{\infty}(% 2n+1){\mathsf{i}^{(1)}_{n}}\left(v\right)\mathsf{k}_{n}\left(u\right)P_{n}% \left(\cos\alpha\right)}}$ `\frac{e^{-w}}{w} = \frac{2}{\pi}\sum_{n=0}^{\infty}(2n+1)\modsphBesseli{1}{n}@{v}\modsphBesselK{n}@{u}\assLegendreP[]{n}@{\cos@@{\alpha}}`	${\displaystyle{\displaystyle\|ve^{+i\alpha}\|<\|u\|,\|ve^{-i\alpha}\|<\|u\|,\Re((n+% \frac{1}{2})+k+1)>0}}$	Error	Divide[Exp[- w],w] == Divide[2,Pi]Sum[(2n + 1)Sqrt[Divide[Pi, v]/2] BesselI[(-1)^(1-1)(n + 1/2), n]Sqrt[1/2 Pi /u] BesselK[n + 1/2, u]LegendreP[n, 0, 3, Cos[\[Alpha]]], {n, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Skipped - Because timed out
10.60.E4	${\displaystyle{\displaystyle\mathsf{j}_{n}\left(2z\right)=-n!z^{n+1}\sum_{k=0}% ^{n}\frac{2n-2k+1}{k!(2n-k+1)!}\mathsf{j}_{n-k}\left(z\right)\mathsf{y}_{n-k}% \left(z\right)}}$ `\sphBesselJ{n}@{2z} = -n!z^{n+1}\sum_{k=0}^{n}\frac{2n-2k+1}{k!(2n-k+1)!}\sphBesselJ{n-k}@{z}\sphBesselY{n-k}@{z}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re(((n-k)+\frac{1}{2})% +k+1)>0,\Re((-n-\frac{1}{2})+k+1)>0,\Re((-(n-k)-\frac{1}{2})+k+1)>0,\Re((-(-n-% \frac{1}{2}))+k+1)>0,\Re((-(-(n-k)-\frac{1}{2}))+k+1)>0,\Re((-((n-k)+\frac{1}{% 2}))+k+1)>0}}$	Error	SphericalBesselJ[n, 2z] == - (n)!(z)^(n + 1)* Sum[Divide[2n - 2k + 1,(k)!(2n - k + 1)!]SphericalBesselJ[n - k, z]SphericalBesselY[n - k, z], {k, 0, n}, GenerateConditions->None]	Missing Macro Error	Aborted	-	Failed [6 / 21] Result: Plus[0.3456774997623559, Times[2.25, Plus[Times[-2.0, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[-3, Times[-2, ], Times[2, 1]], Plus[-1, Times[-1, ], Times[2, 1]], Plus[Times[-1, ], Times[2, 1]], Plus[1, Times[-1, ], Times[2, 1]], Power[1.5, 2], []], Times[-1, Plus[-1, Times[-1, ], Times[2, 1]], Plus[Times[-1, ], Times[2, 1]], Plus[Times[-3, ], Times[-14, Power[, 2]], Times[-20, Power[, 3]], Times[-8, Power[, 4]], Times[14, , 1], Times[40, Power[, 2], 1], Times[24, Power[, 3], 1], Times[-20, , Power[1, 2]], Times[-24, Power[, 2], Power[1, 2]], Times[8, , Power[1, 3]], Times[-3, Power[1.5, 2]], Times[2, , Power[1.5, 2]], Times[4, Power[, 2], Power[1.5, 2]], Times[-4, 1, Power[1.5, 2]], Times[-8, , 1, Power[1.5, 2]], Times[4, Power[1, 2], Power[1.5, 2]]], [Plus[1, ]]], Times[, Plus[1, , Times[-2, 1]], Plus[3, Times[4, ], Times[4, , 1], Times[-4, Power[1, 2]]], Plus[3, Times[8, ], Times[4, Power[<syntaxhighlight lang=mathematica>Result: Plus[0.2986374970757335, Times[6.75, Plus[Times[-2.0, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[-3, Times[-2, ], Times[2, 2]], Plus[-1, Times[-1, ], Times[2, 2]], Plus[Times[-1, ], Times[2, 2]], Plus[1, Times[-1, ], Times[2, 2]], Power[1.5, 2], []], Times[-1, Plus[-1, Times[-1, ], Times[2, 2]], Plus[Times[-1, ], Times[2, 2]], Plus[Times[-3, ], Times[-14, Power[, 2]], Times[-20, Power[, 3]], Times[-8, Power[, 4]], Times[14, , 2], Times[40, Power[, 2], 2], Times[24, Power[, 3], 2], Times[-20, , Power[2, 2]], Times[-24, Power[, 2], Power[2, 2]], Times[8, , Power[2, 3]], Times[-3, Power[1.5, 2]], Times[2, , Power[1.5, 2]], Times[4, Power[, 2], Power[1.5, 2]], Times[-4, 2, Power[1.5, 2]], Times[-8, , 2, Power[1.5, 2]], Times[4, Power[2, 2], Power[1.5, 2]]], [Plus[1, ]]], Times[, Plus[1, , Times[-2, 2]], Plus[3, Times[4, ], Times[4, , 2], Times[-4, Power[2, 2]]], Plus[3, Times[8, ], Times[4, Power[, 2]], Times[-8, 2], Times[-8, , 2], Times[4, Power[2, 2]], Times[-1, Power[1.5, 2]]], [Plus[2, ]]], Times[-1, , Plus[1, ], Plus[9, Times[39, ], Times[58, Power[, 2]], Times[36, Power[, 3]], Times[8, Power[, 4]], Times[-48, 2], Times[-146, , 2], Times[-136, Power[, 2], 2], Times[-40, Power[, 3], 2], Times[88, Power[2, 2]], Times[164, , Power[2, 2]], Times[72, Power[, 2], Power[2, 2]], Times[-64, Power[2, 3]], Times[-56, , Power[2, 3]], Times[16, Power[2, 4]], Times[-5, Power[1.5, 2]], Times[-10, , Power[1.5, 2]], Times[-4, Power[, 2], Power[1.5, 2]], Times[12, 2, Power[1.5, 2]], Times[8, , 2, Power[1.5, 2]], Times[-4, Power[2, 2], Power[1.5, 2]]], [Plus[3, ]]], Times[-1, , Plus[1, ], Plus[2, ], Plus[1, Times[2, ], Times[-2, 2]], Power[1.5, 2], [Plus[4, ]]]], 0], Equal[[1], 0], Equal[[2], Times[-1, Plus[-1, Times[-2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[Plus[-1, 2], 1.5], SphericalBesselY[Plus[-1, 2], 1.5]]], Equal[[3], Plus[Times[-1, Plus[-1, Times[-2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[Plus[-1, 2], 1.5], SphericalBesselY[Plus[-1, 2], 1.5]], Times[-2, Plus[-1, Times[-2, 2]], 2, Power[1.5, -2], Power[Factorial[Plus[1, Times[2, 2]]], -1], Plus[Times[-1, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselJ[2, 1.5]]], Plus[Times[-1, SphericalBesselY[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselY[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselY[2, 1.5]]]]]], Equal[[4], Plus[Times[-1, Plus[-1, Times[-2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[Plus[-1, 2], 1.5], SphericalBesselY[Plus[-1, 2], 1.5]], Times[-2, Plus[-1, Times[-2, 2]], 2, Power[1.5, -2], Power[Factorial[Plus[1, Times[2, 2]]], -1], Plus[Times[-1, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselJ[2, 1.5]]], Plus[Times[-1, SphericalBesselY[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselY[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselY[2, 1.5]]]], Times[Rational[1, 12], Power[1.5, -2], Plus[Times[12, Plus[-1, Times[-2, 2]], 2, Plus[-1, Times[2, 2]], 1.5, Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[Plus[-1, 2], 1.5]], Times[-12, Plus[-1, Times[-2, 2]], 2, Plus[-3, Times[2, 2]], Plus[-1, Times[2, 2]], Power[1.5, -1], Power[Factorial[Plus[1, Times[2, 2]]], -1], Plus[Times[-1, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselJ[2, 1.5]]]]], Plus[Times[-1, 1.5, SphericalBesselY[Plus[-1, 2], 1.5]], Times[-3, Power[1.5, -1], Plus[Times[-1, SphericalBesselY[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselY[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselY[2, 1.5]]]], Times[2, 2, Power[1.5, -1], Plus[Times[-1, SphericalBesselY[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselY[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselY[2, 1.5]]]]]]]]}]][3.0]], Times[5.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[-3, Times[-2, ], Times[2, 2]], Plus[-1, Times[-1, ], Times[2, 2]], Plus[Times[-1, ], Times[2, 2]], Plus[1, Times[-1, ], Times[2, 2]], Power[1.5, 2], []], Times[-1, Plus[-1, Times[-1, ], Times[2, 2]], Plus[Times[-1, ], Times[2, 2]], Plus[-3, Times[-17, ], Times[-34, Power[, 2]], Times[-28, Power[, 3]], Times[-8, Power[, 4]], Times[14, 2], Times[54, , 2], Times[64, Power[, 2], 2], Times[24, Power[, 3], 2], Times[-20, Power[2, 2]], Times[-44, , Power[2, 2]], Times[-24, Power[, 2], Power[2, 2]], Times[8, Power[2, 3]], Times[8, , Power[2, 3]], Times[-2, Power[1.5, 2]], Times[4, , Power[1.5, 2]], Times[4, Power[, 2], Power[1.5, 2]], Times[-6, 2, Power[1.5, 2]], Times[-8, , 2, Power[1.5, 2]], Times[4, Power[2, 2], Power[1.5, 2]]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[-1, Times[-1, ], 2], Plus[3, Times[2, 2]], Plus[-1, Times[-1, ], Times[2, 2]], Plus[3, Times[8, ], Times[4, Power[, 2]], Times[-8, 2], Times[-8, , 2], Times[4, Power[2, 2]], Times[-1, Power[1.5, 2]]], [Plus[2, ]]], Times[-1, Plus[1, ], Plus[2, ], Plus[9, Times[39, ], Times[58, Power[, 2]], Times[36, Power[, 3]], Times[8, Power[, 4]], Times[-48, 2], Times[-146, , 2], Times[-136, Power[, 2], 2], Times[-40, Power[, 3], 2], Times[88, Power[2, 2]], Times[164, , Power[2, 2]], Times[72, Power[, 2], Power[2, 2]], Times[-64, Power[2, 3]], Times[-56, , Power[2, 3]], Times[16, Power[2, 4]], Times[-6, Power[1.5, 2]], Times[-12, , Power[1.5, 2]], Times[-4, Power[, 2], Power[1.5, 2]], Times[14, 2, Power[1.5, 2]], Times[8, , 2, Power[1.5, 2]], Times[-4, Power[2, 2], Power[1.5, 2]]], [Plus[3, ]]], Times[Plus[1, ], Plus[2, ], Plus[3, ], Plus[-1, Times[-2, ], Times[2, 2]], Power[1.5, 2], [Plus[4, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[2, 1.5], SphericalBesselY[2, 1.5]]], Equal[[2], Plus[Times[-1, Plus[-1, Times[-2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[Plus[-1, 2], 1.5], SphericalBesselY[Plus[-1, 2], 1.5]], Times[Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[2, 1.5], SphericalBesselY[2, 1.5]]]], Equal[[3], Plus[Times[-1, Plus[-1, Times[-2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[Plus[-1, 2], 1.5], SphericalBesselY[Plus[-1, 2], 1.5]], Times[Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[2, 1.5], SphericalBesselY[2, 1.5]], Times[Rational[1, 2], Power[1.5, -2], Plus[Times[2, Plus[-1, Times[-2, 2]], 2, Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[Plus[-1, 2], 1.5]], Times[-4, Plus[-1, Times[-2, 2]], Power[2, 2], Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[Plus[-1, 2], 1.5]], Times[-2, 2, 1.5, Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[2, 1.5]], Times[-4, Power[2, 2], 1.5, Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[2, 1.5]]], Plus[Times[-1, SphericalBesselY[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselY[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselY[2, 1.5]]]]]]}]][3.0]]]]], {Rule[n, 2], Rule[z, 1.5]} ... skip entries to safe data
10.60.E5	${\displaystyle{\displaystyle\mathsf{y}_{n}\left(2z\right)=n!z^{n+1}\sum_{k=0}^% {n}\frac{n-k+\frac{1}{2}}{k!(2n-k+1)!}{\left({\mathsf{j}_{n-k}^{2}}\left(z% \right)-{\mathsf{y}_{n-k}^{2}}\left(z\right)\right)}}}$ `\sphBesselY{n}@{2z} = n!z^{n+1}\sum_{k=0}^{n}\frac{n-k+\frac{1}{2}}{k!(2n-k+1)!}{\left(\sphBesselJ{n-k}^{2}@{z}-\sphBesselY{n-k}^{2}@{z}\right)}`	${\displaystyle{\displaystyle\Re(((n-k)+\frac{1}{2})+k+1)>0,\Re((-(n-k)-\frac{1% }{2})+k+1)>0,\Re((-(-(n-k)-\frac{1}{2}))+k+1)>0,\Re((n+\frac{1}{2})+k+1)>0,\Re% ((-(n+\frac{1}{2}))+k+1)>0,\Re((-((n-k)+\frac{1}{2}))+k+1)>0,\Re((-n-\frac{1}{% 2})+k+1)>0}}$	Error	SphericalBesselY[n, 2z] == (n)!(z)^(n + 1)* Sum[Divide[n - k +Divide[1,2],(k)!(2n - k + 1)!]*((SphericalBesselJ[n - k, z])^(2)- (SphericalBesselY[n - k, z])^(2)), {k, 0, n}, GenerateConditions->None]	Missing Macro Error	Aborted	-	Failed [6 / 21] Result: Plus[0.06295916360231597, Times[-1.125, Plus[Times[-2.0, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[-3, Times[-2, ], Times[2, 1]], Plus[-1, Times[-1, ], Times[2, 1]], Plus[Times[-1, ], Times[2, 1]], Plus[1, Times[-1, ], Times[2, 1]], Power[1.5, 2], []], Times[-1, Plus[-1, Times[-1, ], Times[2, 1]], Plus[Times[-1, ], Times[2, 1]], Plus[Times[-3, ], Times[-14, Power[, 2]], Times[-20, Power[, 3]], Times[-8, Power[, 4]], Times[14, , 1], Times[40, Power[, 2], 1], Times[24, Power[, 3], 1], Times[-20, , Power[1, 2]], Times[-24, Power[, 2], Power[1, 2]], Times[8, , Power[1, 3]], Times[-3, Power[1.5, 2]], Times[2, , Power[1.5, 2]], Times[4, Power[, 2], Power[1.5, 2]], Times[-4, 1, Power[1.5, 2]], Times[-8, , 1, Power[1.5, 2]], Times[4, Power[1, 2], Power[1.5, 2]]], [Plus[1, ]]], Times[, Plus[1, , Times[-2, 1]], Plus[3, Times[4, ], Times[4, , 1], Times[-4, Power[1, 2]]], Plus[3, Times[8, ], Times[4, Pow<syntaxhighlight lang=mathematica>Result: Plus[-0.26703833526449916, Times[-3.375, Plus[Times[-2.0, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[-3, Times[-2, ], Times[2, 2]], Plus[-1, Times[-1, ], Times[2, 2]], Plus[Times[-1, ], Times[2, 2]], Plus[1, Times[-1, ], Times[2, 2]], Power[1.5, 2], []], Times[-1, Plus[-1, Times[-1, ], Times[2, 2]], Plus[Times[-1, ], Times[2, 2]], Plus[Times[-3, ], Times[-14, Power[, 2]], Times[-20, Power[, 3]], Times[-8, Power[, 4]], Times[14, , 2], Times[40, Power[, 2], 2], Times[24, Power[, 3], 2], Times[-20, , Power[2, 2]], Times[-24, Power[, 2], Power[2, 2]], Times[8, , Power[2, 3]], Times[-3, Power[1.5, 2]], Times[2, , Power[1.5, 2]], Times[4, Power[, 2], Power[1.5, 2]], Times[-4, 2, Power[1.5, 2]], Times[-8, , 2, Power[1.5, 2]], Times[4, Power[2, 2], Power[1.5, 2]]], [Plus[1, ]]], Times[, Plus[1, , Times[-2, 2]], Plus[3, Times[4, ], Times[4, , 2], Times[-4, Power[2, 2]]], Plus[3, Times[8, ], Times[4, Power[, 2]], Times[-8, 2], Times[-8, , 2], Times[4, Power[2, 2]], Times[-1, Power[1.5, 2]]], [Plus[2, ]]], Times[-1, , Plus[1, ], Plus[9, Times[39, ], Times[58, Power[, 2]], Times[36, Power[, 3]], Times[8, Power[, 4]], Times[-48, 2], Times[-146, , 2], Times[-136, Power[, 2], 2], Times[-40, Power[, 3], 2], Times[88, Power[2, 2]], Times[164, , Power[2, 2]], Times[72, Power[, 2], Power[2, 2]], Times[-64, Power[2, 3]], Times[-56, , Power[2, 3]], Times[16, Power[2, 4]], Times[-5, Power[1.5, 2]], Times[-10, , Power[1.5, 2]], Times[-4, Power[, 2], Power[1.5, 2]], Times[12, 2, Power[1.5, 2]], Times[8, , 2, Power[1.5, 2]], Times[-4, Power[2, 2], Power[1.5, 2]]], [Plus[3, ]]], Times[-1, , Plus[1, ], Plus[2, ], Plus[1, Times[2, ], Times[-2, 2]], Power[1.5, 2], [Plus[4, ]]]], 0], Equal[[1], 0], Equal[[2], Times[-1, Plus[-1, Times[-2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselJ[Plus[-1, 2], 1.5], 2]]], Equal[[3], Plus[Times[-1, Plus[-1, Times[-2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselJ[Plus[-1, 2], 1.5], 2]], Times[-2, Plus[-1, Times[-2, 2]], 2, Power[1.5, -2], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[Plus[Times[-1, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselJ[2, 1.5]]], 2]]]], Equal[[4], Plus[Times[-1, Plus[-1, Times[-2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselJ[Plus[-1, 2], 1.5], 2]], Times[-2, Plus[-1, Times[-2, 2]], 2, Power[1.5, -2], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[Plus[Times[-1, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselJ[2, 1.5]]], 2]], Times[-1, Plus[-1, Times[-2, 2]], 2, Power[1.5, -2], Power[Factorial[Plus[1, Times[2, 2]]], -1], Plus[Times[-1, Plus[-1, Times[2, 2]], Plus[3, Times[-8, 2], Times[4, Power[2, 2]], Times[-1, Power[1.5, 2]]], Power[SphericalBesselJ[Plus[-1, 2], 1.5], 2]], Times[Plus[-3, Times[2, 2]], Power[1.5, 2], Power[SphericalBesselJ[2, 1.5], 2]], Times[Plus[-3, Times[2, 2]], Power[1.5, -2], Plus[3, Times[-8, 2], Times[4, Power[2, 2]], Times[-1, Power[1.5, 2]]], Power[Plus[Times[-1, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselJ[2, 1.5]]], 2]]]]]]}]][3.0]], Times[2.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[-3, Times[-2, ], Times[2, 2]], Plus[-1, Times[-1, ], Times[2, 2]], Plus[Times[-1, ], Times[2, 2]], Plus[1, Times[-1, ], Times[2, 2]], Power[1.5, 2], []], Times[-1, Plus[-1, Times[-1, ], Times[2, 2]], Plus[Times[-1, ], Times[2, 2]], Plus[Times[-3, ], Times[-14, Power[, 2]], Times[-20, Power[, 3]], Times[-8, Power[, 4]], Times[14, , 2], Times[40, Power[, 2], 2], Times[24, Power[, 3], 2], Times[-20, , Power[2, 2]], Times[-24, Power[, 2], Power[2, 2]], Times[8, , Power[2, 3]], Times[-3, Power[1.5, 2]], Times[2, , Power[1.5, 2]], Times[4, Power[, 2], Power[1.5, 2]], Times[-4, 2, Power[1.5, 2]], Times[-8, , 2, Power[1.5, 2]], Times[4, Power[2, 2], Power[1.5, 2]]], [Plus[1, ]]], Times[, Plus[1, , Times[-2, 2]], Plus[3, Times[4, ], Times[4, , 2], Times[-4, Power[2, 2]]], Plus[3, Times[8, ], Times[4, Power[, 2]], Times[-8, 2], Times[-8, , 2], Times[4, Power[2, 2]], Times[-1, Power[1.5, 2]]], [Plus[2, ]]], Times[-1, , Plus[1, ], Plus[9, Times[39, ], Times[58, Power[, 2]], Times[36, Power[, 3]], Times[8, Power[, 4]], Times[-48, 2], Times[-146, , 2], Times[-136, Power[, 2], 2], Times[-40, Power[, 3], 2], Times[88, Power[2, 2]], Times[164, , Power[2, 2]], Times[72, Power[, 2], Power[2, 2]], Times[-64, Power[2, 3]], Times[-56, , Power[2, 3]], Times[16, Power[2, 4]], Times[-5, Power[1.5, 2]], Times[-10, , Power[1.5, 2]], Times[-4, Power[, 2], Power[1.5, 2]], Times[12, 2, Power[1.5, 2]], Times[8, , 2, Power[1.5, 2]], Times[-4, Power[2, 2], Power[1.5, 2]]], [Plus[3, ]]], Times[-1, , Plus[1, ], Plus[2, ], Plus[1, Times[2, ], Times[-2, 2]], Power[1.5, 2], [Plus[4, ]]]], 0], Equal[[1], 0], Equal[[2], Times[-1, Plus[-1, Times[-2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselY[Plus[-1, 2], 1.5], 2]]], Equal[[3], Plus[Times[-1, Plus[-1, Times[-2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselY[Plus[-1, 2], 1.5], 2]], Times[-2, Plus[-1, Times[-2, 2]], 2, Power[1.5, -2], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[Plus[Times[-1, SphericalBesselY[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselY[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselY[2, 1.5]]], 2]]]], Equal[[4], Plus[Times[-1, Plus[-1, Times[-2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselY[Plus[-1, 2], 1.5], 2]], Times[-2, Plus[-1, Times[-2, 2]], 2, Power[1.5, -2], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[Plus[Times[-1, SphericalBesselY[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselY[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselY[2, 1.5]]], 2]], Times[-1, Plus[-1, Times[-2, 2]], 2, Power[1.5, -2], Power[Factorial[Plus[1, Times[2, 2]]], -1], Plus[Times[-1, Plus[-1, Times[2, 2]], Plus[3, Times[-8, 2], Times[4, Power[2, 2]], Times[-1, Power[1.5, 2]]], Power[SphericalBesselY[Plus[-1, 2], 1.5], 2]], Times[Plus[-3, Times[2, 2]], Power[1.5, 2], Power[SphericalBesselY[2, 1.5], 2]], Times[Plus[-3, Times[2, 2]], Power[1.5, -2], Plus[3, Times[-8, 2], Times[4, Power[2, 2]], Times[-1, Power[1.5, 2]]], Power[Plus[Times[-1, SphericalBesselY[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselY[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselY[2, 1.5]]], 2]]]]]]}]][3.0]], Times[5.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[-3, Times[-2, ], Times[2, 2]], Plus[-1, Times[-1, ], Times[2, 2]], Plus[Times[-1, ], Times[2, 2]], Plus[1, Times[-1, ], Times[2, 2]], Power[1.5, 2], []], Times[-1, Plus[-1, Times[-1, ], Times[2, 2]], Plus[Times[-1, ], Times[2, 2]], Plus[-3, Times[-17, ], Times[-34, Power[, 2]], Times[-28, Power[, 3]], Times[-8, Power[, 4]], Times[14, 2], Times[54, , 2], Times[64, Power[, 2], 2], Times[24, Power[, 3], 2], Times[-20, Power[2, 2]], Times[-44, , Power[2, 2]], Times[-24, Power[, 2], Power[2, 2]], Times[8, Power[2, 3]], Times[8, , Power[2, 3]], Times[-2, Power[1.5, 2]], Times[4, , Power[1.5, 2]], Times[4, Power[, 2], Power[1.5, 2]], Times[-6, 2, Power[1.5, 2]], Times[-8, , 2, Power[1.5, 2]], Times[4, Power[2, 2], Power[1.5, 2]]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[-1, Times[-1, ], 2], Plus[3, Times[2, 2]], Plus[-1, Times[-1, ], Times[2, 2]], Plus[3, Times[8, ], Times[4, Power[, 2]], Times[-8, 2], Times[-8, , 2], Times[4, Power[2, 2]], Times[-1, Power[1.5, 2]]], [Plus[2, ]]], Times[-1, Plus[1, ], Plus[2, ], Plus[9, Times[39, ], Times[58, Power[, 2]], Times[36, Power[, 3]], Times[8, Power[, 4]], Times[-48, 2], Times[-146, , 2], Times[-136, Power[, 2], 2], Times[-40, Power[, 3], 2], Times[88, Power[2, 2]], Times[164, , Power[2, 2]], Times[72, Power[, 2], Power[2, 2]], Times[-64, Power[2, 3]], Times[-56, , Power[2, 3]], Times[16, Power[2, 4]], Times[-6, Power[1.5, 2]], Times[-12, , Power[1.5, 2]], Times[-4, Power[, 2], Power[1.5, 2]], Times[14, 2, Power[1.5, 2]], Times[8, , 2, Power[1.5, 2]], Times[-4, Power[2, 2], Power[1.5, 2]]], [Plus[3, ]]], Times[Plus[1, ], Plus[2, ], Plus[3, ], Plus[-1, Times[-2, ], Times[2, 2]], Power[1.5, 2], [Plus[4, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselJ[2, 1.5], 2]]], Equal[[2], Plus[Times[Plus[1, Times[2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselJ[Plus[-1, 2], 1.5], 2]], Times[Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselJ[2, 1.5], 2]]]], Equal[[3], Plus[Times[Plus[1, Times[2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselJ[Plus[-1, 2], 1.5], 2]], Times[Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselJ[2, 1.5], 2]], Times[2, Plus[1, Times[2, 2]], Power[1.5, -2], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[Plus[Times[-1, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselJ[2, 1.5]]], 2]]]]}]][3.0]], Times[-5.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[-3, Times[-2, ], Times[2, 2]], Plus[-1, Times[-1, ], Times[2, 2]], Plus[Times[-1, ], Times[2, 2]], Plus[1, Times[-1, ], Times[2, 2]], Power[1.5, 2], []], Times[-1, Plus[-1, Times[-1, ], Times[2, 2]], Plus[Times[-1, ], Times[2, 2]], Plus[-3, Times[-17, ], Times[-34, Power[, 2]], Times[-28, Power[, 3]], Times[-8, Power[, 4]], Times[14, 2], Times[54, , 2], Times[64, Power[, 2], 2], Times[24, Power[, 3], 2], Times[-20, Power[2, 2]], Times[-44, , Power[2, 2]], Times[-24, Power[, 2], Power[2, 2]], Times[8, Power[2, 3]], Times[8, , Power[2, 3]], Times[-2, Power[1.5, 2]], Times[4, , Power[1.5, 2]], Times[4, Power[, 2], Power[1.5, 2]], Times[-6, 2, Power[1.5, 2]], Times[-8, , 2, Power[1.5, 2]], Times[4, Power[2, 2], Power[1.5, 2]]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[-1, Times[-1, ], 2], Plus[3, Times[2, 2]], Plus[-1, Times[-1, ], Times[2, 2]], Plus[3, Times[8, ], Times[4, Power[, 2]], Times[-8, 2], Times[-8, , 2], Times[4, Power[2, 2]], Times[-1, Power[1.5, 2]]], [Plus[2, ]]], Times[-1, Plus[1, ], Plus[2, ], Plus[9, Times[39, ], Times[58, Power[, 2]], Times[36, Power[, 3]], Times[8, Power[, 4]], Times[-48, 2], Times[-146, , 2], Times[-136, Power[, 2], 2], Times[-40, Power[, 3], 2], Times[88, Power[2, 2]], Times[164, , Power[2, 2]], Times[72, Power[, 2], Power[2, 2]], Times[-64, Power[2, 3]], Times[-56, , Power[2, 3]], Times[16, Power[2, 4]], Times[-6, Power[1.5, 2]], Times[-12, , Power[1.5, 2]], Times[-4, Power[, 2], Power[1.5, 2]], Times[14, 2, Power[1.5, 2]], Times[8, , 2, Power[1.5, 2]], Times[-4, Power[2, 2], Power[1.5, 2]]], [Plus[3, ]]], Times[Plus[1, ], Plus[2, ], Plus[3, ], Plus[-1, Times[-2, ], Times[2, 2]], Power[1.5, 2], [Plus[4, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselY[2, 1.5], 2]]], Equal[[2], Plus[Times[Plus[1, Times[2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselY[Plus[-1, 2], 1.5], 2]], Times[Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselY[2, 1.5], 2]]]], Equal[[3], Plus[Times[Plus[1, Times[2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselY[Plus[-1, 2], 1.5], 2]], Times[Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselY[2, 1.5], 2]], Times[2, Plus[1, Times[2, 2]], Power[1.5, -2], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[Plus[Times[-1, SphericalBesselY[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselY[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselY[2, 1.5]]], 2]]]]}]][3.0]]]]], {Rule[n, 2], Rule[z, 1.5]} ... skip entries to safe data
10.60.E6	${\displaystyle{\displaystyle\mathsf{k}_{n}\left(2z\right)=\frac{1}{\pi}n!z^{n+% 1}\sum_{k=0}^{n}(-1)^{k}\frac{2n-2k+1}{k!(2n-k+1)!}{\mathsf{k}_{n-k}^{2}}\left% (z\right)}}$ `\modsphBesselK{n}@{2z} = \frac{1}{\pi}n!z^{n+1}\sum_{k=0}^{n}(-1)^{k}\frac{2n-2k+1}{k!(2n-k+1)!}\modsphBesselK{n-k}^{2}@{z}`		Error	Sqrt[1/2 Pi /2z] BesselK[n + 1/2, 2z] == Divide[1,Pi](n)!(z)^(n + 1)* Sum[(- 1)^(k)Divide[2n - 2k + 1,(k)!(2n - k + 1)!](Sqrt[1/2 Pi /z] BesselK[n - k + 1/2, z])^(2), {k, 0, n}, GenerateConditions->None]	Missing Macro Error	Aborted	-	Failed [21 / 21] Result: Complex[0.10365998143807895, 0.01421463603104145] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.21384035370849797, -0.0374061947505589] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.60.E7	${\displaystyle{\displaystyle e^{iz\cos\alpha}=\sum_{n=0}^{\infty}(2n+1)i^{n}% \mathsf{j}_{n}\left(z\right)P_{n}\left(\cos\alpha\right)}}$ `e^{iz\cos@@{\alpha}} = \sum_{n=0}^{\infty}(2n+1)i^{n}\sphBesselJ{n}@{z}\assLegendreP[]{n}@{\cos@@{\alpha}}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0}}$	Error	Exp[IzCos[\[Alpha]]] == Sum[(2n + 1)(I)^(n)* SphericalBesselJ[n, z]*LegendreP[n, 0, 3, Cos[\[Alpha]]], {n, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Failed [21 / 21] Result: Plus[Complex[0.9634389243184156, 0.05909441627762202], Times[-1.0, NSum[Times[Power[Complex[0, 1], n], Plus[1, Times[2, n]], LegendreP[n, 0.0707372016677029], SphericalBesselJ[n, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5]} Result: Plus[Complex[0.46738148067268087, 0.44423123280344756], Times[-1.0, NSum[Times[Power[Complex[0, 1], n], Plus[1, Times[2, n]], LegendreP[n, 0.8775825618903728], SphericalBesselJ[n, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 0.5]} ... skip entries to safe data
10.60.E8	${\displaystyle{\displaystyle e^{z\cos\alpha}=\sum_{n=0}^{\infty}(2n+1){\mathsf% {i}^{(1)}_{n}}\left(z\right)P_{n}\left(\cos\alpha\right)}}$ `e^{z\cos@@{\alpha}} = \sum_{n=0}^{\infty}(2n+1)\modsphBesseli{1}{n}@{z}\assLegendreP[]{n}@{\cos@@{\alpha}}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0}}$	Error	Exp[zCos[\[Alpha]]] == Sum[(2n + 1)Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)(n + 1/2), n]*LegendreP[n, 0, 3, Cos[\[Alpha]]], {n, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Failed [21 / 21] Result: Plus[Complex[1.0625106169893304, 0.037595191618525974], Times[-1.0, NSum[Times[Power[Power[E, Times[Complex[0, Rational[-1, 6]], Pi]], Rational[1, 2]], Plus[1, Times[2, n]], Power[Times[Rational[1, 2], Pi], Rational[1, 2]], BesselI[Plus[Rational[1, 2], n], n], LegendreP[n, 0.0707372016677029]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5]} Result: Plus[Complex[1.935725445820811, 0.9084451535292719], Times[-1.0, NSum[Times[Power[Power[E, Times[Complex[0, Rational[-1, 6]], Pi]], Rational[1, 2]], Plus[1, Times[2, n]], Power[Times[Rational[1, 2], Pi], Rational[1, 2]], BesselI[Plus[Rational[1, 2], n], n], LegendreP[n, 0.8775825618903728]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 0.5]} ... skip entries to safe data
10.60.E9	${\displaystyle{\displaystyle e^{-z\cos\alpha}=\sum_{n=0}^{\infty}(-1)^{n}(2n+1% ){\mathsf{i}^{(1)}_{n}}\left(z\right)P_{n}\left(\cos\alpha\right)}}$ `e^{-z\cos@@{\alpha}} = \sum_{n=0}^{\infty}(-1)^{n}(2n+1)\modsphBesseli{1}{n}@{z}\assLegendreP[]{n}@{\cos@@{\alpha}}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0}}$	Error	Exp[- zCos[\[Alpha]]] == Sum[(- 1)^(n)(2n + 1)Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)(n + 1/2), n]LegendreP[n, 0, 3, Cos[\[Alpha]]], {n, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Failed [21 / 21] Result: Plus[Complex[0.939990215282077, -0.03326000860415312], Times[-1.0, NSum[Times[Power[-1, n], Power[Power[E, Times[Complex[0, Rational[-1, 6]], Pi]], Rational[1, 2]], Plus[1, Times[2, n]], Power[Times[Rational[1, 2], Pi], Rational[1, 2]], BesselI[Plus[Rational[1, 2], n], n], LegendreP[n, 0.0707372016677029]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5]} Result: Plus[Complex[0.4233587200353881, -0.19868425982147583], Times[-1.0, NSum[Times[Power[-1, n], Power[Power[E, Times[Complex[0, Rational[-1, 6]], Pi]], Rational[1, 2]], Plus[1, Times[2, n]], Power[Times[Rational[1, 2], Pi], Rational[1, 2]], BesselI[Plus[Rational[1, 2], n], n], LegendreP[n, 0.8775825618903728]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 0.5]} ... skip entries to safe data
10.60.E10	${\displaystyle{\displaystyle J_{0}\left(z\sin\alpha\right)=\sum_{n=0}^{\infty}% (4n+1)\frac{(2n)!}{2^{2n}(n!)^{2}}\mathsf{j}_{2n}\left(z\right)P_{2n}\left(% \cos\alpha\right)}}$ `\BesselJ{0}@{z\sin@@{\alpha}} = \sum_{n=0}^{\infty}(4n+1)\frac{(2n)!}{2^{2n}(n!)^{2}}\sphBesselJ{2n}@{z}\assLegendreP[]{2n}@{\cos@@{\alpha}}`	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(((2n)+\frac{1}{2})+k+1)>0,\Re((-(% 2n)-\frac{1}{2})+k+1)>0,\Re((-(-(2n)-\frac{1}{2}))+k+1)>0}}$	Error	BesselJ[0, zSin[\[Alpha]]] == Sum[(4n + 1)Divide[(2n)!,(2)^(2n)((n)!)^(2)]SphericalBesselJ[2n, z]LegendreP[2n, 0, 3, Cos[\[Alpha]]], {n, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Failed [21 / 21] Result: Plus[Complex[0.8683151459050518, -0.20203213835937428], Times[-1.0, NSum[Times[Power[2, Times[-2, n]], Plus[1, Times[4, n]], Power[Factorial[n], -2], Factorial[Times[2, n]], LegendreP[Times[2, n], 0.0707372016677029], SphericalBesselJ[Times[2, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5]} Result: Plus[Complex[0.9708614168197589, -0.04904886793011446], Times[-1.0, NSum[Times[Power[2, Times[-2, n]], Plus[1, Times[4, n]], Power[Factorial[n], -2], Factorial[Times[2, n]], LegendreP[Times[2, n], 0.8775825618903728], SphericalBesselJ[Times[2, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 0.5]} ... skip entries to safe data
10.60.E11	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}{\mathsf{j}_{n}^{2}}\left(z% \right)=\frac{\mathrm{Si}\left(2z\right)}{2z}}}$ `\sum_{n=0}^{\infty}\sphBesselJ{n}^{2}@{z} = \frac{\sinint@{2z}}{2z}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0}}$	Error	Sum[(SphericalBesselJ[n, z])^(2), {n, 0, Infinity}, GenerateConditions->None] == Divide[SinIntegral[2z],2z]	Missing Macro Error	Successful	-	Successful [Tested: 7]
10.60.E12	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}(2n+1){\mathsf{j}_{n}^{2}}\left% (z\right)=1}}$ `\sum_{n=0}^{\infty}(2n+1)\sphBesselJ{n}^{2}@{z} = 1`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0}}$	Error	Sum[(2n + 1)(SphericalBesselJ[n, z])^(2), {n, 0, Infinity}, GenerateConditions->None] == 1	Missing Macro Error	Failure	-	Failed [7 / 7] Result: Plus[-1.0, NSum[Times[Plus[1, Times[2, n]], Power[SphericalBesselJ[n, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[-1.0, NSum[Times[Plus[1, Times[2, n]], Power[SphericalBesselJ[n, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], 2]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.60.E13	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}(-1)^{n}(2n+1){\mathsf{j}_{n}^{% 2}}\left(z\right)=\frac{\sin\left(2z\right)}{2z}}}$ `\sum_{n=0}^{\infty}(-1)^{n}(2n+1)\sphBesselJ{n}^{2}@{z} = \frac{\sin@{2z}}{2z}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0}}$	Error	Sum[(- 1)^(n)(2n + 1)(SphericalBesselJ[n, z])^(2), {n, 0, Infinity}, GenerateConditions->None] == Divide[Sin[2z],2*z]	Missing Macro Error	Failure	-	Failed [7 / 7] Result: Plus[Complex[-0.6123335037567501, 0.46246896224791606], NSum[Times[Power[-1, n], Plus[1, Times[2, n]], Power[SphericalBesselJ[n, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[-1.2536290109103816, -0.6921871649112455], NSum[Times[Power[-1, n], Plus[1, Times[2, n]], Power[SphericalBesselJ[n, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], 2]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.60.E14	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}(2n+1)(\mathsf{j}_{n}'\left(z% \right))^{2}=\tfrac{1}{3}}}$ `\sum_{n=0}^{\infty}(2n+1)(\sphBesselJ{n}'@{z})^{2} = \tfrac{1}{3}`	${\displaystyle{\displaystyle\Re((n+\frac{1}{2})+k+1)>0,\Re((-n-\frac{1}{2})+k+% 1)>0,\Re((-(-n-\frac{1}{2}))+k+1)>0}}$	Error	Sum[(2n + 1)(D[SphericalBesselJ[n, z], {z, 1}])^(2), {n, 0, Infinity}, GenerateConditions->None] == Divide[1,3]	Missing Macro Error	Aborted	-	Skipped - Because timed out
10.61.E1	${\displaystyle{\displaystyle\operatorname{ber}_{\nu}x+i\operatorname{bei}_{\nu% }x=J_{\nu}\left(xe^{3\pi i/4}\right)}}$ `\Kelvinber{\nu}@@{x}+i\Kelvinbei{\nu}@@{x} = \BesselJ{\nu}@{xe^{3\pi i/4}}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	KelvinBer(nu, x)+ IKelvinBei(nu, x) = BesselJ(nu, xexp(3PiI/4))	KelvinBer[\[Nu], x]+ IKelvinBei[\[Nu], x] == BesselJ[\[Nu], xExp[3PiI/4]]	Successful	Failure	Skip - symbolical successful subtest	Successful [Tested: 30]
10.61.E1	${\displaystyle{\displaystyle J_{\nu}\left(xe^{3\pi i/4}\right)=e^{\nu\pi i}J_{% \nu}\left(xe^{-\pi i/4}\right)}}$ `\BesselJ{\nu}@{xe^{3\pi i/4}} = e^{\nu\pi i}\BesselJ{\nu}@{xe^{-\pi i/4}}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	BesselJ(nu, xexp(3PiI/4)) = exp(nuPiI)BesselJ(nu, xexp(- PiI/4))	BesselJ[\[Nu], xExp[3PiI/4]] == Exp[\[Nu]PiI]BesselJ[\[Nu], xExp[- PiI/4]]	Failure	Failure	Successful [Tested: 30]	Successful [Tested: 30]
10.61.E1	${\displaystyle{\displaystyle e^{\nu\pi i}J_{\nu}\left(xe^{-\pi i/4}\right)=e^{% \nu\pi i/2}I_{\nu}\left(xe^{\pi i/4}\right)}}$ `e^{\nu\pi i}\BesselJ{\nu}@{xe^{-\pi i/4}} = e^{\nu\pi i/2}\modBesselI{\nu}@{xe^{\pi i/4}}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	exp(nuPiI)BesselJ(nu, xexp(- PiI/4)) = exp(nuPiI/2)BesselI(nu, xexp(PiI/4))	Exp[\[Nu]PiI]BesselJ[\[Nu], xExp[- PiI/4]] == Exp[\[Nu]PiI/2]BesselI[\[Nu], xExp[PiI/4]]	Failure	Failure	Successful [Tested: 30]	Successful [Tested: 30]
10.61.E1	${\displaystyle{\displaystyle e^{\nu\pi i/2}I_{\nu}\left(xe^{\pi i/4}\right)=e^% {3\nu\pi i/2}I_{\nu}\left(xe^{-3\pi i/4}\right)}}$ `e^{\nu\pi i/2}\modBesselI{\nu}@{xe^{\pi i/4}} = e^{3\nu\pi i/2}\modBesselI{\nu}@{xe^{-3\pi i/4}}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	exp(nuPiI/2)BesselI(nu, xexp(PiI/4)) = exp(3nuPiI/2)BesselI(nu, xexp(- 3PiI/4))	Exp[\[Nu]PiI/2]BesselI[\[Nu], xExp[PiI/4]] == Exp[3\[Nu]PiI/2]BesselI[\[Nu], xExp[- 3PiI/4]]	Failure	Failure	Successful [Tested: 30]	Successful [Tested: 30]
10.61.E2	${\displaystyle{\displaystyle\operatorname{ker}_{\nu}x+i\operatorname{kei}_{\nu% }x=e^{-\nu\pi i/2}K_{\nu}\left(xe^{\pi i/4}\right)}}$ `\Kelvinker{\nu}@@{x}+i\Kelvinkei{\nu}@@{x} = e^{-\nu\pi i/2}\modBesselK{\nu}@{xe^{\pi i/4}}`		KelvinKer(nu, x)+ IKelvinKei(nu, x) = exp(- nuPiI/2)BesselK(nu, xexp(PiI/4))	KelvinKer[\[Nu], x]+ IKelvinKei[\[Nu], x] == Exp[- \[Nu]PiI/2]BesselK[\[Nu], xExp[PiI/4]]	Failure	Failure	Successful [Tested: 30]	Successful [Tested: 30]
10.61.E2	${\displaystyle{\displaystyle e^{-\nu\pi i/2}K_{\nu}\left(xe^{\pi i/4}\right)=% \tfrac{1}{2}\pi i{H^{(1)}_{\nu}}\left(xe^{3\pi i/4}\right)}}$ `e^{-\nu\pi i/2}\modBesselK{\nu}@{xe^{\pi i/4}} = \tfrac{1}{2}\pi i\HankelH{1}{\nu}@{xe^{3\pi i/4}}`		exp(- nuPiI/2)BesselK(nu, xexp(PiI/4)) = (1)/(2)PiIHankelH1(nu, xexp(3Pi*I/4))	Exp[- \[Nu]PiI/2]BesselK[\[Nu], xExp[PiI/4]] == Divide[1,2]PiIHankelH1[\[Nu], xExp[3Pi*I/4]]	Failure	Failure	Successful [Tested: 30]	Successful [Tested: 30]
10.61.E2	${\displaystyle{\displaystyle\tfrac{1}{2}\pi i{H^{(1)}_{\nu}}\left(xe^{3\pi i/4% }\right)=-\tfrac{1}{2}\pi ie^{-\nu\pi i}{H^{(2)}_{\nu}}\left(xe^{-\pi i/4}% \right)}}$ `\tfrac{1}{2}\pi i\HankelH{1}{\nu}@{xe^{3\pi i/4}} = -\tfrac{1}{2}\pi ie^{-\nu\pi i}\HankelH{2}{\nu}@{xe^{-\pi i/4}}`		(1)/(2)PiIHankelH1(nu, xexp(3PiI/4)) = -(1)/(2)PiIexp(- nuPiI)HankelH2(nu, xexp(- PiI/4))	Divide[1,2]PiIHankelH1[\[Nu], xExp[3PiI/4]] == -Divide[1,2]PiIExp[- \[Nu]PiI]HankelH2[\[Nu], xExp[- PiI/4]]	Failure	Failure	Successful [Tested: 30]	Successful [Tested: 30]
10.61.E3	${\displaystyle{\displaystyle x^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}+% x\frac{\mathrm{d}w}{\mathrm{d}x}-(ix^{2}+\nu^{2})w=0}}$ `x^{2}\deriv[2]{w}{x}+x\deriv{w}{x}-(ix^{2}+\nu^{2})w = 0`		(x)^(2)* diff(w, [x$(2)])+ xdiff(w, x)-(I(x)^(2)+ (nu)^(2))*w = 0	(x)^(2)* D[w, {x, 2}]+ xD[w, x]-(I(x)^(2)+ \[Nu]^(2))*w == 0	Failure	Failure	Failed [300 / 300] Result: 1.125000000-2.948557160I Test Values: {nu = 1/23^(1/2)+1/2I, w = 1/23^(1/2)+1/2I, x = 3/2} Result: .1249999997-1.216506352I Test Values: {nu = 1/23^(1/2)+1/2I, w = 1/23^(1/2)+1/2I, x = 1/2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[1.1249999999999996, -2.948557158514987] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.1249999999999996, -0.9485571585149869] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.61.E4	${\displaystyle{\displaystyle x^{4}\frac{{\mathrm{d}}^{4}w}{{\mathrm{d}x}^{4}}+% 2x^{3}\frac{{\mathrm{d}}^{3}w}{{\mathrm{d}x}^{3}}-(1+2\nu^{2})\left(x^{2}\frac% {{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}-x\frac{\mathrm{d}w}{\mathrm{d}x}\right)% +(\nu^{4}-4\nu^{2}+x^{4})w=0}}$ `x^{4}\deriv[4]{w}{x}+2x^{3}\deriv[3]{w}{x}-(1+2\nu^{2})\left(x^{2}\deriv[2]{w}{x}-x\deriv{w}{x}\right)+(\nu^{4}-4\nu^{2}+x^{4})w = 0`	${\displaystyle{\displaystyle w=\operatorname{ber}_{+\nu}x,w=\operatorname{ber}% _{-\nu}x}}$	(x)^(4)* diff(w, [x$(4)])+ 2(x)^(3) diff(w, [x$(3)])-(1 + 2(nu)^(2))((x)^(2)* diff(w, [x$(2)])- xdiff(w, x))+((nu)^(4)- 4(nu)^(2)+ (x)^(4))*w = 0	(x)^(4)* D[w, {x, 4}]+ 2(x)^(3) D[w, {x, 3}]-(1 + 2\[Nu]^(2))((x)^(2)* D[w, {x, 2}]- xD[w, x])+(\[Nu]^(4)- 4\[Nu]^(2)+ (x)^(4))*w == 0	Error	Failure	-	Skip - No test values generated
10.61#Ex1	${\displaystyle{\displaystyle\operatorname{ber}_{n}\left(-x\right)=(-1)^{n}% \operatorname{ber}_{n}x}}$ `\Kelvinber{n}@{-x} = (-1)^{n}\Kelvinber{n}@@{x}`	${\displaystyle{\displaystyle\Re(n+k+1)>0}}$	KelvinBer(n, - x) = (- 1)^(n)* KelvinBer(n, x)	KelvinBer[n, - x] == (- 1)^(n)* KelvinBer[n, x]	Successful	Failure	-	Successful [Tested: 9]
10.61#Ex2	${\displaystyle{\displaystyle\operatorname{bei}_{n}\left(-x\right)=(-1)^{n}% \operatorname{bei}_{n}x}}$ `\Kelvinbei{n}@{-x} = (-1)^{n}\Kelvinbei{n}@@{x}`		KelvinBei(n, - x) = (- 1)^(n)* KelvinBei(n, x)	KelvinBei[n, - x] == (- 1)^(n)* KelvinBei[n, x]	Successful	Failure	-	Successful [Tested: 9]
10.61#Ex3	${\displaystyle{\displaystyle\operatorname{ber}_{-\nu}x=\cos\left(\nu\pi\right)% \operatorname{ber}_{\nu}x+\sin\left(\nu\pi\right)\operatorname{bei}_{\nu}x+(2/% \pi)\sin\left(\nu\pi\right)\operatorname{ker}_{\nu}x}}$ `\Kelvinber{-\nu}@@{x} = \cos@{\nu\pi}\Kelvinber{\nu}@@{x}+\sin@{\nu\pi}\Kelvinbei{\nu}@@{x}+(2/\pi)\sin@{\nu\pi}\Kelvinker{\nu}@@{x}`	${\displaystyle{\displaystyle\Re((-\nu)+k+1)>0,\Re(\nu+k+1)>0}}$	KelvinBer(- nu, x) = cos(nuPi)KelvinBer(nu, x)+ sin(nuPi)KelvinBei(nu, x)+(2/Pi)sin(nuPi)*KelvinKer(nu, x)	KelvinBer[- \[Nu], x] == Cos[\[Nu]Pi]KelvinBer[\[Nu], x]+ Sin[\[Nu]Pi]KelvinBei[\[Nu], x]+(2/Pi)Sin[\[Nu]Pi]*KelvinKer[\[Nu], x]	Failure	Failure	Successful [Tested: 30]	Successful [Tested: 30]
10.61#Ex4	${\displaystyle{\displaystyle\operatorname{bei}_{-\nu}x=-\sin\left(\nu\pi\right% )\operatorname{ber}_{\nu}x+\cos\left(\nu\pi\right)\operatorname{bei}_{\nu}x+(2% /\pi)\sin\left(\nu\pi\right)\operatorname{kei}_{\nu}x}}$ `\Kelvinbei{-\nu}@@{x} = -\sin@{\nu\pi}\Kelvinber{\nu}@@{x}+\cos@{\nu\pi}\Kelvinbei{\nu}@@{x}+(2/\pi)\sin@{\nu\pi}\Kelvinkei{\nu}@@{x}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	KelvinBei(- nu, x) = - sin(nuPi)KelvinBer(nu, x)+ cos(nuPi)KelvinBei(nu, x)+(2/Pi)sin(nuPi)*KelvinKei(nu, x)	KelvinBei[- \[Nu], x] == - Sin[\[Nu]Pi]KelvinBer[\[Nu], x]+ Cos[\[Nu]Pi]KelvinBei[\[Nu], x]+(2/Pi)Sin[\[Nu]Pi]*KelvinKei[\[Nu], x]	Failure	Failure	Successful [Tested: 30]	Successful [Tested: 30]
10.61#Ex5	${\displaystyle{\displaystyle\operatorname{ker}_{-\nu}x=\cos\left(\nu\pi\right)% \operatorname{ker}_{\nu}x-\sin\left(\nu\pi\right)\operatorname{kei}_{\nu}x}}$ `\Kelvinker{-\nu}@@{x} = \cos@{\nu\pi}\Kelvinker{\nu}@@{x}-\sin@{\nu\pi}\Kelvinkei{\nu}@@{x}`		KelvinKer(- nu, x) = cos(nuPi)KelvinKer(nu, x)- sin(nuPi)KelvinKei(nu, x)	KelvinKer[- \[Nu], x] == Cos[\[Nu]Pi]KelvinKer[\[Nu], x]- Sin[\[Nu]Pi]KelvinKei[\[Nu], x]	Successful	Failure	-	Successful [Tested: 30]
10.61#Ex6	${\displaystyle{\displaystyle\operatorname{kei}_{-\nu}x=\sin\left(\nu\pi\right)% \operatorname{ker}_{\nu}x+\cos\left(\nu\pi\right)\operatorname{kei}_{\nu}x}}$ `\Kelvinkei{-\nu}@@{x} = \sin@{\nu\pi}\Kelvinker{\nu}@@{x}+\cos@{\nu\pi}\Kelvinkei{\nu}@@{x}`		KelvinKei(- nu, x) = sin(nuPi)KelvinKer(nu, x)+ cos(nuPi)KelvinKei(nu, x)	KelvinKei[- \[Nu], x] == Sin[\[Nu]Pi]KelvinKer[\[Nu], x]+ Cos[\[Nu]Pi]KelvinKei[\[Nu], x]	Successful	Failure	-	Successful [Tested: 30]
10.61#Ex7	${\displaystyle{\displaystyle\operatorname{ber}_{-n}x=(-1)^{n}\operatorname{ber% }_{n}x,~{}{}\operatorname{bei}_{-n}x}}$ `\Kelvinber{-n}@@{x} = (-1)^{n}\Kelvinber{n}@@{x},~{}\Kelvinbei{-n}@@{x}`	${\displaystyle{\displaystyle\Re((-n)+k+1)>0,\Re(n+k+1)>0}}$	KelvinBer(- n, x) = (- 1)^(n)* KelvinBer(n, x); *KelvinBei(- n, x)	KelvinBer[- n, x] == (- 1)^(n)* KelvinBer[n, x] *KelvinBei[- n, x]	Error	Failure	-	Error
10.61#Ex7	${\displaystyle{\displaystyle(-1)^{n}\operatorname{ber}_{n}x,~{}{}\operatorname% {bei}_{-n}x=(-1)^{n}\operatorname{bei}_{n}x}}$ `(-1)^{n}\Kelvinber{n}@@{x},~{}\Kelvinbei{-n}@@{x} = (-1)^{n}\Kelvinbei{n}@@{x}`	${\displaystyle{\displaystyle\Re((-n)+k+1)>0,\Re(n+k+1)>0}}$	(- 1)^(n)* KelvinBer(n, x),KelvinBei(- n, x) = (- 1)^(n) KelvinBei(n, x)	(- 1)^(n)* KelvinBer[n, x],KelvinBei[- n, x] == (- 1)^(n) KelvinBei[n, x]	Error	Failure	-	Error
10.61#Ex8	${\displaystyle{\displaystyle\operatorname{ker}_{-n}x=(-1)^{n}\operatorname{ker% }_{n}x,~{}{}\operatorname{kei}_{-n}x}}$ `\Kelvinker{-n}@@{x} = (-1)^{n}\Kelvinker{n}@@{x},~{}\Kelvinkei{-n}@@{x}`		KelvinKer(- n, x) = (- 1)^(n)* KelvinKer(n, x); *KelvinKei(- n, x)	KelvinKer[- n, x] == (- 1)^(n)* KelvinKer[n, x] *KelvinKei[- n, x]	Error	Failure	-	Error
10.61#Ex8	${\displaystyle{\displaystyle(-1)^{n}\operatorname{ker}_{n}x,~{}{}\operatorname% {kei}_{-n}x=(-1)^{n}\operatorname{kei}_{n}x}}$ `(-1)^{n}\Kelvinker{n}@@{x},~{}\Kelvinkei{-n}@@{x} = (-1)^{n}\Kelvinkei{n}@@{x}`		(- 1)^(n)* KelvinKer(n, x),KelvinKei(- n, x) = (- 1)^(n) KelvinKei(n, x)	(- 1)^(n)* KelvinKer[n, x],KelvinKei[- n, x] == (- 1)^(n) KelvinKei[n, x]	Error	Failure	-	Error
10.61#Ex9	${\displaystyle{\displaystyle\operatorname{ber}_{\frac{1}{2}}\left(x\sqrt{2}% \right)=\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\cos\left(x+\frac{\pi}% {8}\right)-e^{-x}\cos\left(x-\frac{\pi}{8}\right)\right)}}$ `\Kelvinber{\frac{1}{2}}@{x\sqrt{2}} = \frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\cos@{x+\frac{\pi}{8}}-e^{-x}\cos@{x-\frac{\pi}{8}}\right)`	${\displaystyle{\displaystyle\Re((\frac{1}{2})+k+1)>0}}$	KelvinBer((1)/(2), xsqrt(2)) = ((2)^(-(3)/(4)))/(sqrt(Pix))(exp(x)cos(x +(Pi)/(8))- exp(- x)*cos(x -(Pi)/(8)))	KelvinBer[Divide[1,2], xSqrt[2]] == Divide[(2)^(-Divide[3,4]),Sqrt[Pix]](Exp[x]Cos[x +Divide[Pi,8]]- Exp[- x]*Cos[x -Divide[Pi,8]])	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
10.61#Ex10	${\displaystyle{\displaystyle\operatorname{bei}_{\frac{1}{2}}\left(x\sqrt{2}% \right)=\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\sin\left(x+\frac{\pi}% {8}\right)+\,e^{-x}\sin\left(x-\frac{\pi}{8}\right)\right)}}$ `\Kelvinbei{\frac{1}{2}}@{x\sqrt{2}} = \frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\sin@{x+\frac{\pi}{8}}+\,e^{-x}\sin@{x-\frac{\pi}{8}}\right)`		KelvinBei((1)/(2), xsqrt(2)) = ((2)^(-(3)/(4)))/(sqrt(Pix))(exp(x)sin(x +(Pi)/(8))+ exp(- x)*sin(x -(Pi)/(8)))	KelvinBei[Divide[1,2], xSqrt[2]] == Divide[(2)^(-Divide[3,4]),Sqrt[Pix]](Exp[x]Sin[x +Divide[Pi,8]]+ Exp[- x]*Sin[x -Divide[Pi,8]])	Failure	Successful	Successful [Tested: 3]	Successful [Tested: 3]
10.61#Ex11	${\displaystyle{\displaystyle\operatorname{ber}_{-\frac{1}{2}}\left(x\sqrt{2}% \right)=\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\sin\left(x+\frac{\pi}% {8}\right)-e^{-x}\sin\left(x-\frac{\pi}{8}\right)\right)}}$ `\Kelvinber{-\frac{1}{2}}@{x\sqrt{2}} = \frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\sin@{x+\frac{\pi}{8}}-e^{-x}\sin@{x-\frac{\pi}{8}}\right)`	${\displaystyle{\displaystyle\Re((-\frac{1}{2})+k+1)>0}}$	KelvinBer(-(1)/(2), xsqrt(2)) = ((2)^(-(3)/(4)))/(sqrt(Pix))(exp(x)sin(x +(Pi)/(8))- exp(- x)*sin(x -(Pi)/(8)))	KelvinBer[-Divide[1,2], xSqrt[2]] == Divide[(2)^(-Divide[3,4]),Sqrt[Pix]](Exp[x]Sin[x +Divide[Pi,8]]- Exp[- x]*Sin[x -Divide[Pi,8]])	Failure	Successful	Successful [Tested: 3]	Successful [Tested: 3]
10.61#Ex12	${\displaystyle{\displaystyle\operatorname{bei}_{-\frac{1}{2}}\left(x\sqrt{2}% \right)=-\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\cos\left(x+\frac{\pi% }{8}\right)+e^{-x}\cos\left(x-\frac{\pi}{8}\right)\right)}}$ `\Kelvinbei{-\frac{1}{2}}@{x\sqrt{2}} = -\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\cos@{x+\frac{\pi}{8}}+e^{-x}\cos@{x-\frac{\pi}{8}}\right)`		KelvinBei(-(1)/(2), xsqrt(2)) = -((2)^(-(3)/(4)))/(sqrt(Pix))(exp(x)cos(x +(Pi)/(8))+ exp(- x)*cos(x -(Pi)/(8)))	KelvinBei[-Divide[1,2], xSqrt[2]] == -Divide[(2)^(-Divide[3,4]),Sqrt[Pix]](Exp[x]Cos[x +Divide[Pi,8]]+ Exp[- x]*Cos[x -Divide[Pi,8]])	Failure	Successful	Successful [Tested: 3]	Successful [Tested: 3]
10.61.E11	${\displaystyle{\displaystyle\operatorname{ker}_{\frac{1}{2}}\left(x\sqrt{2}% \right)=\operatorname{kei}_{-\frac{1}{2}}\left(x\sqrt{2}\right)}}$ `\Kelvinker{\frac{1}{2}}@{x\sqrt{2}} = \Kelvinkei{-\frac{1}{2}}@{x\sqrt{2}}`		KelvinKer((1)/(2), xsqrt(2)) = KelvinKei(-(1)/(2), xsqrt(2))	KelvinKer[Divide[1,2], xSqrt[2]] == KelvinKei[-Divide[1,2], xSqrt[2]]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 3]
10.61.E11	${\displaystyle{\displaystyle\operatorname{kei}_{-\frac{1}{2}}\left(x\sqrt{2}% \right)=-2^{-\frac{3}{4}}\sqrt{\frac{\pi}{x}}e^{-x}\sin\left(x-\frac{\pi}{8}% \right)}}$ `\Kelvinkei{-\frac{1}{2}}@{x\sqrt{2}} = -2^{-\frac{3}{4}}\sqrt{\frac{\pi}{x}}e^{-x}\sin@{x-\frac{\pi}{8}}`		KelvinKei(-(1)/(2), xsqrt(2)) = - (2)^(-(3)/(4))sqrt((Pi)/(x))exp(- x)sin(x -(Pi)/(8))	KelvinKei[-Divide[1,2], xSqrt[2]] == - (2)^(-Divide[3,4])Sqrt[Divide[Pi,x]]Exp[- x]Sin[x -Divide[Pi,8]]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
10.61.E12	${\displaystyle{\displaystyle\operatorname{kei}_{\frac{1}{2}}\left(x\sqrt{2}% \right)=-\operatorname{ker}_{-\frac{1}{2}}\left(x\sqrt{2}\right)}}$ `\Kelvinkei{\frac{1}{2}}@{x\sqrt{2}} = -\Kelvinker{-\frac{1}{2}}@{x\sqrt{2}}`		KelvinKei((1)/(2), xsqrt(2)) = - KelvinKer(-(1)/(2), xsqrt(2))	KelvinKei[Divide[1,2], xSqrt[2]] == - KelvinKer[-Divide[1,2], xSqrt[2]]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 3]
10.61.E12	${\displaystyle{\displaystyle-\operatorname{ker}_{-\frac{1}{2}}\left(x\sqrt{2}% \right)=-2^{-\frac{3}{4}}\sqrt{\frac{\pi}{x}}e^{-x}\cos\left(x-\frac{\pi}{8}% \right)}}$ `-\Kelvinker{-\frac{1}{2}}@{x\sqrt{2}} = -2^{-\frac{3}{4}}\sqrt{\frac{\pi}{x}}e^{-x}\cos@{x-\frac{\pi}{8}}`		- KelvinKer(-(1)/(2), xsqrt(2)) = - (2)^(-(3)/(4))sqrt((Pi)/(x))exp(- x)cos(x -(Pi)/(8))	- KelvinKer[-Divide[1,2], xSqrt[2]] == - (2)^(-Divide[3,4])Sqrt[Divide[Pi,x]]Exp[- x]Cos[x -Divide[Pi,8]]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
10.63#Ex5	${\displaystyle{\displaystyle f_{\nu-1}(x)+f_{\nu+1}(x)=-(\nu\sqrt{2}/x)\left(f% _{\nu}(x)-g_{\nu}(x)\right)}}$ `f_{\nu-1}(x)+f_{\nu+1}(x) = -(\nu\sqrt{2}/x)\left(f_{\nu}(x)-g_{\nu}(x)\right)`		f[nu - 1](x)+ f[nu + 1](x) = -(nusqrt(2)/x)(f[nu](x)- g[nu](x))	Subscript[f, \[Nu]- 1][x]+ Subscript[f, \[Nu]+ 1][x] == -(\[Nu]Sqrt[2]/x)(Subscript[f, \[Nu]][x]- Subscript[g, \[Nu]][x])	Skipped - no semantic math	Skipped - no semantic math	-	-
10.63#Ex9	${\displaystyle{\displaystyle\sqrt{2}\operatorname{ber}'x=\operatorname{ber}_{1% }x+\operatorname{bei}_{1}x}}$ `\sqrt{2}\Kelvinber{}'@@{x} = \Kelvinber{1}@@{x}+\Kelvinbei{1}@@{x}`	${\displaystyle{\displaystyle\Re(1+k+1)>0}}$	sqrt(2)*diff( KelvinBer(, x), x$(1) ) = KelvinBer(1, x)+ KelvinBei(1, x)	Sqrt[2]*D[KelvinBer[, x], {x, 1}] == KelvinBer[1, x]+ KelvinBei[1, x]	Error	Failure	-	Failed [3 / 3] Result: Plus[0.297000428957679, Times[0.35355339059327373, Plus[Times[-1.0, KelvinBei[Plus[-1.0, Null], 1.5]], KelvinBei[Plus[1.0, Null], 1.5], Times[-1.0, KelvinBer[Plus[-1.0, Null], 1.5]], KelvinBer[Plus[1.0, Null], 1.5]]]] Test Values: {Rule[x, 1.5]} Result: Plus[0.011047944038096752, Times[0.35355339059327373, Plus[Times[-1.0, KelvinBei[Plus[-1.0, Null], 0.5]], KelvinBei[Plus[1.0, Null], 0.5], Times[-1.0, KelvinBer[Plus[-1.0, Null], 0.5]], KelvinBer[Plus[1.0, Null], 0.5]]]] Test Values: {Rule[x, 0.5]} ... skip entries to safe data
10.63#Ex10	${\displaystyle{\displaystyle\sqrt{2}\operatorname{bei}'x=-\operatorname{ber}_{% 1}x+\operatorname{bei}_{1}x}}$ `\sqrt{2}\Kelvinbei{}'@@{x} = -\Kelvinber{1}x+\Kelvinbei{1}x`	${\displaystyle{\displaystyle\Re(1+k+1)>0}}$	sqrt(2)*diff( KelvinBei(, x), x$(1) ) = - KelvinBer(1, x)+ KelvinBei(1, x)	Sqrt[2]*D[KelvinBei[, x], {x, 1}] == - KelvinBer[1, x]+ KelvinBei[1, x]	Error	Failure	-	Failed [3 / 3] Result: Plus[-1.0327304069618592, Times[0.35355339059327373, Plus[Times[-1.0, KelvinBei[Plus[-1.0, Null], 1.5]], KelvinBei[Plus[1.0, Null], 1.5], KelvinBer[Plus[-1.0, Null], 1.5], Times[-1.0, KelvinBer[Plus[1.0, Null], 1.5]]]]] Test Values: {Rule[x, 1.5]} Result: Plus[-0.35343830347212746, Times[0.35355339059327373, Plus[Times[-1.0, KelvinBei[Plus[-1.0, Null], 0.5]], KelvinBei[Plus[1.0, Null], 0.5], KelvinBer[Plus[-1.0, Null], 0.5], Times[-1.0, KelvinBer[Plus[1.0, Null], 0.5]]]]] Test Values: {Rule[x, 0.5]} ... skip entries to safe data
10.63#Ex11	${\displaystyle{\displaystyle\sqrt{2}\operatorname{ker}'x=\operatorname{ker}_{1% }x+\operatorname{kei}_{1}x}}$ `\sqrt{2}\Kelvinker{}'@@{x} = \Kelvinker{1}x+\Kelvinkei{1}x`		sqrt(2)*diff( KelvinKer(, x), x$(1) ) = KelvinKer(1, x)+ KelvinKei(1, x)	Sqrt[2]*D[KelvinKer[, x], {x, 1}] == KelvinKer[1, x]+ KelvinKei[1, x]	Error	Failure	-	Failed [3 / 3] Result: Plus[0.4160356041812476, Times[0.35355339059327373, Plus[Times[-1.0, KelvinKei[Plus[-1.0, Null], 1.5]], KelvinKei[Plus[1.0, Null], 1.5], Times[-1.0, KelvinKer[Plus[-1.0, Null], 1.5]], KelvinKer[Plus[1.0, Null], 1.5]]]] Test Values: {Rule[x, 1.5]} Result: Plus[2.5735854919446126, Times[0.35355339059327373, Plus[Times[-1.0, KelvinKei[Plus[-1.0, Null], 0.5]], KelvinKei[Plus[1.0, Null], 0.5], Times[-1.0, KelvinKer[Plus[-1.0, Null], 0.5]], KelvinKer[Plus[1.0, Null], 0.5]]]] Test Values: {Rule[x, 0.5]} ... skip entries to safe data
10.63#Ex12	${\displaystyle{\displaystyle\sqrt{2}\operatorname{kei}'x=-\operatorname{ker}_{% 1}x+\operatorname{kei}_{1}x}}$ `\sqrt{2}\Kelvinkei{}'@@{x} = -\Kelvinker{1}x+\Kelvinkei{1}x`		sqrt(2)*diff( KelvinKei(, x), x$(1) ) = - KelvinKer(1, x)+ KelvinKei(1, x)	Sqrt[2]*D[KelvinKei[, x], {x, 1}] == - KelvinKer[1, x]+ KelvinKei[1, x]	Error	Failure	-	Failed [3 / 3] Result: Plus[-0.418052966151267, Times[0.35355339059327373, Plus[Times[-1.0, KelvinKei[Plus[-1.0, Null], 1.5]], KelvinKei[Plus[1.0, Null], 1.5], KelvinKer[Plus[-1.0, Null], 1.5], Times[-1.0, KelvinKer[Plus[1.0, Null], 1.5]]]]] Test Values: {Rule[x, 1.5]} Result: Plus[-0.47122132111956727, Times[0.35355339059327373, Plus[Times[-1.0, KelvinKei[Plus[-1.0, Null], 0.5]], KelvinKei[Plus[1.0, Null], 0.5], KelvinKer[Plus[-1.0, Null], 0.5], Times[-1.0, KelvinKer[Plus[1.0, Null], 0.5]]]]] Test Values: {Rule[x, 0.5]} ... skip entries to safe data
10.63#Ex17	${\displaystyle{\displaystyle p_{\nu+1}=p_{\nu-1}-(4\nu/x)r_{\nu}}}$ `p_{\nu+1} = p_{\nu-1}-(4\nu/x)r_{\nu}`		p[nu + 1] = p[nu - 1]-(4nu/x)r[nu]	Subscript[p, \[Nu]+ 1] == Subscript[p, \[Nu]- 1]-(4\[Nu]/x)Subscript[r, \[Nu]]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.63#Ex18	${\displaystyle{\displaystyle q_{\nu+1}=-(\nu/x)p_{\nu}+r_{\nu}}}$ `q_{\nu+1} = -(\nu/x)p_{\nu}+r_{\nu}`		q[nu + 1] = -(nu/x)*p[nu]+ r[nu]	Subscript[q, \[Nu]+ 1] == -(\[Nu]/x)*Subscript[p, \[Nu]]+ Subscript[r, \[Nu]]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.63#Ex19	${\displaystyle{\displaystyle r_{\nu+1}=-((\nu+1)/x)p_{\nu+1}+q_{\nu}}}$ `r_{\nu+1} = -((\nu+1)/x)p_{\nu+1}+q_{\nu}`		r[nu + 1] = -((nu + 1)/x)*p[nu + 1]+ q[nu]	Subscript[r, \[Nu]+ 1] == -((\[Nu]+ 1)/x)*Subscript[p, \[Nu]+ 1]+ Subscript[q, \[Nu]]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.63#Ex20	${\displaystyle{\displaystyle s_{\nu}=\tfrac{1}{2}p_{\nu+1}+\tfrac{1}{2}p_{\nu-% 1}-(\nu^{2}/x^{2})p_{\nu}}}$ `s_{\nu} = \tfrac{1}{2}p_{\nu+1}+\tfrac{1}{2}p_{\nu-1}-(\nu^{2}/x^{2})p_{\nu}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	((diff( KelvinBer(nu, x), x$(1) ))^(2)+(diff( KelvinBei(nu, x), x$(1) ))^(2)) = (1)/(2)p[nu + 1]+(1)/(2)p[nu - 1]-((nu)^(2)/(x)^(2))*p[nu]	((D[KelvinBer[\[Nu], x], {x, 1}])^(2)+(D[KelvinBei[\[Nu], x], {x, 1}])^(2)) == Divide[1,2]Subscript[p, \[Nu]+ 1]+Divide[1,2]Subscript[p, \[Nu]- 1]-(\[Nu]^(2)/(x)^(2))*Subscript[p, \[Nu]]	Skipped - no semantic math	Skipped - no semantic math	-	-
10.63.E7	${\displaystyle{\displaystyle p_{\nu}s_{\nu}=r_{\nu}^{2}+q_{\nu}^{2}}}$ `p_{\nu}s_{\nu} = r_{\nu}^{2}+q_{\nu}^{2}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	p[nu]*((diff( KelvinBer(nu, x), x$(1) ))^(2)+(diff( KelvinBei(nu, x), x$(1) ))^(2)) = (r[nu])^(2)+ (q[nu])^(2)	Subscript[p, \[Nu]]*((D[KelvinBer[\[Nu], x], {x, 1}])^(2)+(D[KelvinBei[\[Nu], x], {x, 1}])^(2)) == (Subscript[r, \[Nu]])^(2)+ (Subscript[q, \[Nu]])^(2)	Skipped - no semantic math	Skipped - no semantic math	-	-
10.64.E1	${\displaystyle{\displaystyle\operatorname{ber}_{n}\left(x\sqrt{2}\right)=\frac% {(-1)^{n}}{\pi}\int_{0}^{\pi}\cos\left(x\sin t-nt\right)\cosh\left(x\sin t% \right)\mathrm{d}t}}$ `\Kelvinber{n}@{x\sqrt{2}} = \frac{(-1)^{n}}{\pi}\int_{0}^{\pi}\cos@{x\sin@@{t}-nt}\cosh@{x\sin@@{t}}\diff{t}`	${\displaystyle{\displaystyle\Re(n+k+1)>0}}$	KelvinBer(n, xsqrt(2)) = ((- 1)^(n))/(Pi)int(cos(xsin(t)- nt)cosh(xsin(t)), t = 0..Pi)	KelvinBer[n, xSqrt[2]] == Divide[(- 1)^(n),Pi]Integrate[Cos[xSin[t]- nt]Cosh[xSin[t]], {t, 0, Pi}, GenerateConditions->None]	Failure	Aborted	Successful [Tested: 9]	Skipped - Because timed out
10.64.E2	${\displaystyle{\displaystyle\operatorname{bei}_{n}\left(x\sqrt{2}\right)=\frac% {(-1)^{n}}{\pi}\int_{0}^{\pi}\sin\left(x\sin t-nt\right)\sinh\left(x\sin t% \right)\mathrm{d}t}}$ `\Kelvinbei{n}@{x\sqrt{2}} = \frac{(-1)^{n}}{\pi}\int_{0}^{\pi}\sin@{x\sin@@{t}-nt}\sinh@{x\sin@@{t}}\diff{t}`		KelvinBei(n, xsqrt(2)) = ((- 1)^(n))/(Pi)int(sin(xsin(t)- nt)sinh(xsin(t)), t = 0..Pi)	KelvinBei[n, xSqrt[2]] == Divide[(- 1)^(n),Pi]Integrate[Sin[xSin[t]- nt]Sinh[xSin[t]], {t, 0, Pi}, GenerateConditions->None]	Failure	Aborted	Successful [Tested: 9]	Skipped - Because timed out
10.65#Ex1	${\displaystyle{\displaystyle\operatorname{ber}_{\nu}x=(\tfrac{1}{2}x)^{\nu}% \sum_{k=0}^{\infty}\frac{\cos\left(\frac{3}{4}\nu\pi+\frac{1}{2}k\pi\right)}{k% !\Gamma\left(\nu+k+1\right)}(\tfrac{1}{4}x^{2})^{k}}}$ `\Kelvinber{\nu}@@{x} = (\tfrac{1}{2}x)^{\nu}\sum_{k=0}^{\infty}\frac{\cos@{\frac{3}{4}\nu\pi+\frac{1}{2}k\pi}}{k!\EulerGamma@{\nu+k+1}}(\tfrac{1}{4}x^{2})^{k}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	KelvinBer(nu, x) = ((1)/(2)x)^(nu) sum((cos((3)/(4)nuPi +(1)/(2)kPi))/(factorial(k)GAMMA(nu + k + 1))((1)/(4)*(x)^(2))^(k), k = 0..infinity)	KelvinBer[\[Nu], x] == (Divide[1,2]x)^\[Nu] Sum[Divide[Cos[Divide[3,4]\[Nu]Pi +Divide[1,2]kPi],(k)!Gamma[\[Nu]+ k + 1]](Divide[1,4]*(x)^(2))^(k), {k, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Successful [Tested: 30]	Successful [Tested: 30]
10.65#Ex2	${\displaystyle{\displaystyle\operatorname{bei}_{\nu}x=(\tfrac{1}{2}x)^{\nu}% \sum_{k=0}^{\infty}\frac{\sin\left(\frac{3}{4}\nu\pi+\frac{1}{2}k\pi\right)}{k% !\Gamma\left(\nu+k+1\right)}(\tfrac{1}{4}x^{2})^{k}}}$ `\Kelvinbei{\nu}@@{x} = (\tfrac{1}{2}x)^{\nu}\sum_{k=0}^{\infty}\frac{\sin@{\frac{3}{4}\nu\pi+\frac{1}{2}k\pi}}{k!\EulerGamma@{\nu+k+1}}(\tfrac{1}{4}x^{2})^{k}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	KelvinBei(nu, x) = ((1)/(2)x)^(nu) sum((sin((3)/(4)nuPi +(1)/(2)kPi))/(factorial(k)GAMMA(nu + k + 1))((1)/(4)*(x)^(2))^(k), k = 0..infinity)	KelvinBei[\[Nu], x] == (Divide[1,2]x)^\[Nu] Sum[Divide[Sin[Divide[3,4]\[Nu]Pi +Divide[1,2]kPi],(k)!Gamma[\[Nu]+ k + 1]](Divide[1,4]*(x)^(2))^(k), {k, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Successful [Tested: 30]	Successful [Tested: 30]
10.65#Ex3	${\displaystyle{\displaystyle\operatorname{ber}x=1-\frac{(\frac{1}{4}x^{2})^{2}% }{(2!)^{2}}+\frac{(\frac{1}{4}x^{2})^{4}}{(4!)^{2}}-\cdots}}$ `\Kelvinber{}@@{x} = 1-\frac{(\frac{1}{4}x^{2})^{2}}{(2!)^{2}}+\frac{(\frac{1}{4}x^{2})^{4}}{(4!)^{2}}-\dotsb`		KelvinBer(, x) = 1 -(((1)/(4)(x)^(2))^(2))/((factorial(2))^(2))+(((1)/(4)(x)^(2))^(4))/((factorial(4))^(2))- ..	KelvinBer[, x] == 1 -Divide[(Divide[1,4](x)^(2))^(2),((2)!)^(2)]+Divide[(Divide[1,4](x)^(2))^(4),((4)!)^(2)]- \[Ellipsis]	Error	Failure	-	Failed [3 / 3] Result: Plus[-0.921072244644165, …, KelvinBer[Null, 1.5]] Test Values: {Rule[x, 1.5]} Result: Plus[-0.9990234639909532, …, KelvinBer[Null, 0.5]] Test Values: {Rule[x, 0.5]} ... skip entries to safe data
10.65#Ex4	${\displaystyle{\displaystyle\operatorname{bei}x=\tfrac{1}{4}x^{2}-\frac{(\frac% {1}{4}x^{2})^{3}}{(3!)^{2}}+\frac{(\frac{1}{4}x^{2})^{5}}{(5!)^{2}}-\cdots}}$ `\Kelvinbei{}@@{x} = \tfrac{1}{4}x^{2}-\frac{(\frac{1}{4}x^{2})^{3}}{(3!)^{2}}+\frac{(\frac{1}{4}x^{2})^{5}}{(5!)^{2}}-\dotsi`		KelvinBei(, x) = (1)/(4)(x)^(2)-(((1)/(4)(x)^(2))^(3))/((factorial(3))^(2))+(((1)/(4)*(x)^(2))^(5))/((factorial(5))^(2))- ..	KelvinBei[, x] == Divide[1,4](x)^(2)-Divide[(Divide[1,4](x)^(2))^(3),((3)!)^(2)]+Divide[(Divide[1,4]*(x)^(2))^(5),((5)!)^(2)]- \[Ellipsis]	Error	Failure	-	Failed [3 / 3] Result: Plus[-0.5575600630044937, …, KelvinBei[Null, 1.5]] Test Values: {Rule[x, 1.5]} Result: Plus[-0.06249321838219961, …, KelvinBei[Null, 0.5]] Test Values: {Rule[x, 0.5]} ... skip entries to safe data
10.65.E3	${\displaystyle{\displaystyle\operatorname{ker}_{n}x=\tfrac{1}{2}(\tfrac{1}{2}x% )^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\cos\left(\tfrac{3}{4}n\pi+\tfrac{1}{% 2}k\pi\right)(\tfrac{1}{4}x^{2})^{k}-\ln\left(\tfrac{1}{2}x\right)% \operatorname{ber}_{n}x+\tfrac{1}{4}\pi\operatorname{bei}_{n}x+\tfrac{1}{2}(% \tfrac{1}{2}x)^{n}\sum_{k=0}^{\infty}\frac{\psi\left(k+1\right)+\psi\left(n+k+% 1\right)}{k!(n+k)!}\cos\left(\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi\right)(\tfrac{1% }{4}x^{2})^{k}}}$ `\Kelvinker{n}@@{x} = \tfrac{1}{2}(\tfrac{1}{2}x)^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\cos@{\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi}(\tfrac{1}{4}x^{2})^{k}-\ln@{\tfrac{1}{2}x}\Kelvinber{n}@@{x}+\tfrac{1}{4}\pi\Kelvinbei{n}@@{x}+\tfrac{1}{2}(\tfrac{1}{2}x)^{n}\sum_{k=0}^{\infty}\frac{\digamma@{k+1}+\digamma@{n+k+1}}{k!(n+k)!}\cos@{\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi}(\tfrac{1}{4}x^{2})^{k}`	${\displaystyle{\displaystyle\Re(n+k+1)>0}}$	KelvinKer(n, x) = (1)/(2)((1)/(2)x)^(- n)* sum((factorial(n - k - 1))/(factorial(k))cos((3)/(4)nPi +(1)/(2)kPi)((1)/(4)(x)^(2))^(k), k = 0..n - 1)- ln((1)/(2)x)KelvinBer(n, x)+(1)/(4)PiKelvinBei(n, x)+(1)/(2)((1)/(2)x)^(n) sum((Psi(k + 1)+ Psi(n + k + 1))/(factorial(k)factorial(n + k))cos((3)/(4)nPi +(1)/(2)kPi)((1)/(4)(x)^(2))^(k), k = 0..infinity)	KelvinKer[n, x] == Divide[1,2](Divide[1,2]x)^(- n)* Sum[Divide[(n - k - 1)!,(k)!]Cos[Divide[3,4]nPi +Divide[1,2]kPi](Divide[1,4](x)^(2))^(k), {k, 0, n - 1}, GenerateConditions->None]- Log[Divide[1,2]x]KelvinBer[n, x]+Divide[1,4]PiKelvinBei[n, x]+Divide[1,2](Divide[1,2]x)^(n) Sum[Divide[PolyGamma[k + 1]+ PolyGamma[n + k + 1],(k)!(n + k)!]Cos[Divide[3,4]nPi +Divide[1,2]kPi](Divide[1,4](x)^(2))^(k), {k, 0, Infinity}, GenerateConditions->None]	Aborted	Aborted	Skipped - Because timed out	Failed [9 / 9] Result: Indeterminate Test Values: {Rule[n, 1], Rule[x, 1.5]} Result: Indeterminate Test Values: {Rule[n, 2], Rule[x, 1.5]} ... skip entries to safe data
10.65.E4	${\displaystyle{\displaystyle\operatorname{kei}_{n}x=-\tfrac{1}{2}(\tfrac{1}{2}% x)^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\sin\left(\tfrac{3}{4}n\pi+\tfrac{1}% {2}k\pi\right)(\tfrac{1}{4}x^{2})^{k}-\ln\left(\tfrac{1}{2}x\right)% \operatorname{bei}_{n}x-\tfrac{1}{4}\pi\operatorname{ber}_{n}x+\tfrac{1}{2}(% \tfrac{1}{2}x)^{n}\sum_{k=0}^{\infty}\frac{\psi\left(k+1\right)+\psi\left(n+k+% 1\right)}{k!(n+k)!}\sin\left(\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi\right)(\tfrac{1% }{4}x^{2})^{k}}}$ `\Kelvinkei{n}@@{x} = -\tfrac{1}{2}(\tfrac{1}{2}x)^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\sin@{\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi}(\tfrac{1}{4}x^{2})^{k}-\ln@{\tfrac{1}{2}x}\Kelvinbei{n}@@{x}-\tfrac{1}{4}\pi\Kelvinber{n}@@{x}+\tfrac{1}{2}(\tfrac{1}{2}x)^{n}\sum_{k=0}^{\infty}\frac{\digamma@{k+1}+\digamma@{n+k+1}}{k!(n+k)!}\sin@{\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi}(\tfrac{1}{4}x^{2})^{k}`	${\displaystyle{\displaystyle\Re(n+k+1)>0}}$	KelvinKei(n, x) = -(1)/(2)((1)/(2)x)^(- n)* sum((factorial(n - k - 1))/(factorial(k))sin((3)/(4)nPi +(1)/(2)kPi)((1)/(4)(x)^(2))^(k), k = 0..n - 1)- ln((1)/(2)x)KelvinBei(n, x)-(1)/(4)PiKelvinBer(n, x)+(1)/(2)((1)/(2)x)^(n) sum((Psi(k + 1)+ Psi(n + k + 1))/(factorial(k)factorial(n + k))sin((3)/(4)nPi +(1)/(2)kPi)((1)/(4)(x)^(2))^(k), k = 0..infinity)	KelvinKei[n, x] == -Divide[1,2](Divide[1,2]x)^(- n)* Sum[Divide[(n - k - 1)!,(k)!]Sin[Divide[3,4]nPi +Divide[1,2]kPi](Divide[1,4](x)^(2))^(k), {k, 0, n - 1}, GenerateConditions->None]- Log[Divide[1,2]x]KelvinBei[n, x]-Divide[1,4]PiKelvinBer[n, x]+Divide[1,2](Divide[1,2]x)^(n) Sum[Divide[PolyGamma[k + 1]+ PolyGamma[n + k + 1],(k)!(n + k)!]Sin[Divide[3,4]nPi +Divide[1,2]kPi](Divide[1,4](x)^(2))^(k), {k, 0, Infinity}, GenerateConditions->None]	Aborted	Aborted	Skipped - Because timed out	Failed [9 / 9] Result: Indeterminate Test Values: {Rule[n, 1], Rule[x, 1.5]} Result: Indeterminate Test Values: {Rule[n, 2], Rule[x, 1.5]} ... skip entries to safe data
10.65#Ex5	${\displaystyle{\displaystyle\operatorname{ker}x=-\ln\left(\tfrac{1}{2}x\right)% \operatorname{ber}x+\tfrac{1}{4}\pi\operatorname{bei}x+\sum_{k=0}^{\infty}(-1)% ^{k}\frac{\psi\left(2k+1\right)}{((2k)!)^{2}}(\tfrac{1}{4}x^{2})^{2k}}}$ `\Kelvinker{}@@{x} = -\ln@{\tfrac{1}{2}x}\Kelvinber{}@@{x}+\tfrac{1}{4}\pi\Kelvinbei{}@@{x}+\sum_{k=0}^{\infty}(-1)^{k}\frac{\digamma@{2k+1}}{((2k)!)^{2}}(\tfrac{1}{4}x^{2})^{2k}`		KelvinKer(, x) = - ln((1)/(2)x)KelvinBer(, x)+(1)/(4)PiKelvinBei(, x)+ sum((- 1)^(k)(Psi(2k + 1))/((factorial(2k))^(2))((1)/(4)(x)^(2))^(2k), k = 0..infinity)	KelvinKer[, x] == - Log[Divide[1,2]x]KelvinBer[, x]+Divide[1,4]PiKelvinBei[, x]+ Sum[(- 1)^(k)Divide[PolyGamma[2k + 1],((2k)!)^(2)](Divide[1,4](x)^(2))^(2k), {k, 0, Infinity}, GenerateConditions->None]	Error	Aborted	-	Skipped - Because timed out
10.65#Ex6	${\displaystyle{\displaystyle\operatorname{kei}x=-\ln\left(\tfrac{1}{2}x\right)% \operatorname{bei}x-\tfrac{1}{4}\pi\operatorname{ber}x+\sum_{k=0}^{\infty}(-1)% ^{k}\frac{\psi\left(2k+2\right)}{((2k+1)!)^{2}}(\tfrac{1}{4}x^{2})^{2k+1}}}$ `\Kelvinkei{}@@{x} = -\ln@{\tfrac{1}{2}x}\Kelvinbei{}@@{x}-\tfrac{1}{4}\pi\Kelvinber{}@@{x}+\sum_{k=0}^{\infty}(-1)^{k}\frac{\digamma@{2k+2}}{((2k+1)!)^{2}}(\tfrac{1}{4}x^{2})^{2k+1}`		KelvinKei(, x) = - ln((1)/(2)x)KelvinBei(, x)-(1)/(4)PiKelvinBer(, x)+ sum((- 1)^(k)(Psi(2k + 2))/((factorial(2k + 1))^(2))((1)/(4)(x)^(2))^(2k + 1), k = 0..infinity)	KelvinKei[, x] == - Log[Divide[1,2]x]KelvinBei[, x]-Divide[1,4]PiKelvinBer[, x]+ Sum[(- 1)^(k)Divide[PolyGamma[2k + 2],((2k + 1)!)^(2)](Divide[1,4](x)^(2))^(2k + 1), {k, 0, Infinity}, GenerateConditions->None]	Error	Failure	-	Failed [3 / 3] Result: Plus[-0.23161280473545226, Times[-1.0, KelvinBer[Null, 1.5]], KelvinKei[Null, 1.5]] Test Values: {Rule[x, 1.5]} Result: Plus[-0.02641550246351669, Times[-1.0, KelvinBer[Null, 0.5]], KelvinKei[Null, 0.5]] Test Values: {Rule[x, 0.5]} ... skip entries to safe data
10.65.E6	${\displaystyle{\displaystyle{\operatorname{ber}_{\nu}^{2}}x+{\operatorname{bei% }_{\nu}^{2}}x=(\tfrac{1}{2}x)^{2\nu}\sum_{k=0}^{\infty}\frac{1}{\Gamma\left(% \nu+k+1\right)\Gamma\left(\nu+2k+1\right)}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}}}$ `\Kelvinber{\nu}^{2}@@{x}+\Kelvinbei{\nu}^{2}@@{x} = (\tfrac{1}{2}x)^{2\nu}\sum_{k=0}^{\infty}\frac{1}{\EulerGamma@{\nu+k+1}\EulerGamma@{\nu+2k+1}}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\nu+2k+1)>0}}$	(KelvinBer(nu, x))^(2)+ (KelvinBei(nu, x))^(2) = ((1)/(2)x)^(2nu)* sum((1)/(GAMMA(nu + k + 1)GAMMA(nu + 2k + 1))(((1)/(4)(x)^(2))^(2*k))/(factorial(k)), k = 0..infinity)	(KelvinBer[\[Nu], x])^(2)+ (KelvinBei[\[Nu], x])^(2) == (Divide[1,2]x)^(2\[Nu])* Sum[Divide[1,Gamma[\[Nu]+ k + 1]Gamma[\[Nu]+ 2k + 1]]Divide[(Divide[1,4](x)^(2))^(2*k),(k)!], {k, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 30]
10.65.E7	${\displaystyle{\displaystyle\operatorname{ber}_{\nu}x\operatorname{bei}_{\nu}'% x-\operatorname{ber}_{\nu}'x\operatorname{bei}_{\nu}x=(\tfrac{1}{2}x)^{2\nu+1}% \sum_{k=0}^{\infty}\frac{1}{\Gamma\left(\nu+k+1\right)\Gamma\left(\nu+2k+2% \right)}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}}}$ `\Kelvinber{\nu}@@{x}\Kelvinbei{\nu}'@@{x}-\Kelvinber{\nu}'@@{x}\Kelvinbei{\nu}@@{x} = (\tfrac{1}{2}x)^{2\nu+1}\sum_{k=0}^{\infty}\frac{1}{\EulerGamma@{\nu+k+1}\EulerGamma@{\nu+2k+2}}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\nu+2k+2)>0}}$	KelvinBer(nu, x)diff( KelvinBei(nu, x), x$(1) )- diff( KelvinBer(nu, x), x$(1) )KelvinBei(nu, x) = ((1)/(2)x)^(2nu + 1)* sum((1)/(GAMMA(nu + k + 1)GAMMA(nu + 2k + 2))(((1)/(4)(x)^(2))^(2*k))/(factorial(k)), k = 0..infinity)	KelvinBer[\[Nu], x]D[KelvinBei[\[Nu], x], {x, 1}]- D[KelvinBer[\[Nu], x], {x, 1}]KelvinBei[\[Nu], x] == (Divide[1,2]x)^(2\[Nu]+ 1)* Sum[Divide[1,Gamma[\[Nu]+ k + 1]Gamma[\[Nu]+ 2k + 2]]Divide[(Divide[1,4](x)^(2))^(2*k),(k)!], {k, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Failed [21 / 30] Result: .7271930e-3+.45983036e-2I Test Values: {nu = 1/23^(1/2)+1/2I, x = 3/2} Result: -.41528503e-2+.322695404e-1I Test Values: {nu = 1/23^(1/2)+1/2I, x = 2} ... skip entries to safe data	Failed [3 / 30] Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[ν, -2]} Result: Indeterminate Test Values: {Rule[x, 0.5], Rule[ν, -2]} ... skip entries to safe data
10.65.E8	${\displaystyle{\displaystyle\operatorname{ber}_{\nu}x\operatorname{ber}_{\nu}'% x+\operatorname{bei}_{\nu}x\operatorname{bei}_{\nu}'x=\tfrac{1}{2}(\tfrac{1}{2% }x)^{2\nu-1}\sum_{k=0}^{\infty}\frac{1}{\Gamma\left(\nu+k+1\right)\Gamma\left(% \nu+2k\right)}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}}}$ `\Kelvinber{\nu}@@{x}\Kelvinber{\nu}'@@{x}+\Kelvinbei{\nu}@@{x}\Kelvinbei{\nu}'@@{x} = \tfrac{1}{2}(\tfrac{1}{2}x)^{2\nu-1}\sum_{k=0}^{\infty}\frac{1}{\EulerGamma@{\nu+k+1}\EulerGamma@{\nu+2k}}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\nu+2k)>0}}$	KelvinBer(nu, x)diff( KelvinBer(nu, x), x$(1) )+ KelvinBei(nu, x)diff( KelvinBei(nu, x), x$(1) ) = (1)/(2)((1)/(2)x)^(2nu - 1) sum((1)/(GAMMA(nu + k + 1)GAMMA(nu + 2k))(((1)/(4)(x)^(2))^(2*k))/(factorial(k)), k = 0..infinity)	KelvinBer[\[Nu], x]D[KelvinBer[\[Nu], x], {x, 1}]+ KelvinBei[\[Nu], x]D[KelvinBei[\[Nu], x], {x, 1}] == Divide[1,2](Divide[1,2]x)^(2\[Nu]- 1) Sum[Divide[1,Gamma[\[Nu]+ k + 1]Gamma[\[Nu]+ 2k]]Divide[(Divide[1,4](x)^(2))^(2*k),(k)!], {k, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Failed [25 / 30] Result: .71978298e-2-.3037583875e-1I Test Values: {nu = 1/23^(1/2)+1/2I, x = 3/2} Result: .607273780e-1-.1071579728I Test Values: {nu = 1/23^(1/2)+1/2I, x = 2} ... skip entries to safe data	Failed [3 / 30] Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[ν, -2]} Result: Indeterminate Test Values: {Rule[x, 0.5], Rule[ν, -2]} ... skip entries to safe data
10.65.E9	${\displaystyle{\displaystyle\left(\operatorname{ber}_{\nu}'x\right)^{2}+\left(% \operatorname{bei}_{\nu}'x\right)^{2}=(\tfrac{1}{2}x)^{2\nu-2}\sum_{k=0}^{% \infty}\frac{2k^{2}+2\nu k+\frac{1}{4}\nu^{2}}{\Gamma\left(\nu+k+1\right)% \Gamma\left(\nu+2k+1\right)}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}}}$ `\left(\Kelvinber{\nu}'@@{x}\right)^{2}+\left(\Kelvinbei{\nu}'@@{x}\right)^{2} = (\tfrac{1}{2}x)^{2\nu-2}\sum_{k=0}^{\infty}\frac{2k^{2}+2\nu k+\frac{1}{4}\nu^{2}}{\EulerGamma@{\nu+k+1}\EulerGamma@{\nu+2k+1}}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0,\Re(\nu+2k+1)>0}}$	(diff( KelvinBer(nu, x), x$(1) ))^(2)+(diff( KelvinBei(nu, x), x$(1) ))^(2) = ((1)/(2)x)^(2nu - 2)* sum((2(k)^(2)+ 2nuk +(1)/(4)(nu)^(2))/(GAMMA(nu + k + 1)GAMMA(nu + 2k + 1))(((1)/(4)(x)^(2))^(2*k))/(factorial(k)), k = 0..infinity)	(D[KelvinBer[\[Nu], x], {x, 1}])^(2)+(D[KelvinBei[\[Nu], x], {x, 1}])^(2) == (Divide[1,2]x)^(2\[Nu]- 2)* Sum[Divide[2(k)^(2)+ 2\[Nu]k +Divide[1,4]\[Nu]^(2),Gamma[\[Nu]+ k + 1]Gamma[\[Nu]+ 2k + 1]]Divide[(Divide[1,4](x)^(2))^(2*k),(k)!], {k, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Failed [3 / 30] Result: Float(undefined)+Float(undefined)I Test Values: {nu = -2, x = 3/2} Result: Float(undefined)+Float(undefined)I Test Values: {nu = -2, x = 1/2} ... skip entries to safe data	Failed [3 / 30] Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[ν, -2]} Result: Indeterminate Test Values: {Rule[x, 0.5], Rule[ν, -2]} ... skip entries to safe data
10.66.E1	${\displaystyle{\displaystyle\operatorname{ber}_{\nu}x+i\operatorname{bei}_{\nu% }x=\sum_{k=0}^{\infty}\frac{e^{(3\nu+k)\pi i/4}x^{k}J_{\nu+k}\left(x\right)}{2% ^{k/2}k!}}}$ `\Kelvinber{\nu}@@{x}+i\Kelvinbei{\nu}@@{x} = \sum_{k=0}^{\infty}\frac{e^{(3\nu+k)\pi i/4}x^{k}\BesselJ{\nu+k}@{x}}{2^{k/2}k!}`	${\displaystyle{\displaystyle\Re((\nu+k)+k+1)>0,\Re(\nu+k+1)>0}}$	KelvinBer(nu, x)+ IKelvinBei(nu, x) = sum((exp((3nu + k)PiI/4)(x)^(k) BesselJ(nu + k, x))/((2)^(k/2)* factorial(k)), k = 0..infinity)	KelvinBer[\[Nu], x]+ IKelvinBei[\[Nu], x] == Sum[Divide[Exp[(3\[Nu]+ k)PiI/4](x)^(k) BesselJ[\[Nu]+ k, x],(2)^(k/2)* (k)!], {k, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Skipped - Because timed out	Failed [30 / 30] Result: Plus[Complex[-0.12257968900025018, 0.2735107661041647], Times[-1.0, NSum[Times[Power[1.5, k], Power[2, Times[Rational[-1, 2], k]], Power[E, Times[Complex[0, Rational[1, 4]], Plus[Times[3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], k], Pi]], BesselJ[Plus[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k], 1.5], Power[Factorial[k], -1]] Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[0.3467793075651209, -0.08562995402477025], Times[-1.0, NSum[Times[Power[1.5, k], Power[2, Times[Rational[-1, 2], k]], Power[E, Times[Complex[0, Rational[1, 4]], Plus[Times[3, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], k], Pi]], BesselJ[Plus[Power[E, Times[Complex[0, Rational[2, 3]], Pi]], k], 1.5], Power[Factorial[k], -1]] Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.66.E1	${\displaystyle{\displaystyle\sum_{k=0}^{\infty}\frac{e^{(3\nu+k)\pi i/4}x^{k}J% _{\nu+k}\left(x\right)}{2^{k/2}k!}=\sum_{k=0}^{\infty}\frac{e^{(3\nu+3k)\pi i/% 4}x^{k}I_{\nu+k}\left(x\right)}{2^{k/2}k!}}}$ `\sum_{k=0}^{\infty}\frac{e^{(3\nu+k)\pi i/4}x^{k}\BesselJ{\nu+k}@{x}}{2^{k/2}k!} = \sum_{k=0}^{\infty}\frac{e^{(3\nu+3k)\pi i/4}x^{k}\modBesselI{\nu+k}@{x}}{2^{k/2}k!}`	${\displaystyle{\displaystyle\Re((\nu+k)+k+1)>0,\Re(\nu+k+1)>0}}$	sum((exp((3nu + k)PiI/4)(x)^(k)* BesselJ(nu + k, x))/((2)^(k/2)* factorial(k)), k = 0..infinity) = sum((exp((3nu + 3k)PiI/4)(x)^(k) BesselI(nu + k, x))/((2)^(k/2)* factorial(k)), k = 0..infinity)	Sum[Divide[Exp[(3\[Nu]+ k)PiI/4](x)^(k)* BesselJ[\[Nu]+ k, x],(2)^(k/2)* (k)!], {k, 0, Infinity}, GenerateConditions->None] == Sum[Divide[Exp[(3\[Nu]+ 3k)PiI/4](x)^(k) BesselI[\[Nu]+ k, x],(2)^(k/2)* (k)!], {k, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Skipped - Because timed out	Failed [30 / 30] Result: Plus[Times[-1.0, NSum[Times[Power[1.5, k], Power[2, Times[Rational[-1, 2], k]], Power[E, Times[Complex[0, Rational[1, 4]], Plus[Times[3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[3, k]], Pi]], BesselI[Plus[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k], 1.5], Power[Factorial[k], -1]] Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], NSum[Times[Power[1.5, k], Power[2, Times[Rational[-1, 2], k]], Power[E, Times[Complex[0, Rational[1, 4]], Plus[Times[3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], k], Pi]], BesselJ[Plus[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k], 1.5], Power[Factorial[k], -1]], {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Times[-1.0, NSum[Times[Power[1.5, k], Power[2, Times[Rational[-1, 2], k]], Power[E, Times[Complex[0, Rational[1, 4]], Plus[Times[3, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Times[3, k]], Pi]], BesselI[Plus[Power[E, Times[Complex[0, Rational[2, 3]], Pi]], k], 1.5], Power[Factorial[k], -1]] Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], NSum[Times[Power[1.5, k], Power[2, Times[Rational[-1, 2], k]], Power[E, Times[Complex[0, Rational[1, 4]], Plus[Times[3, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], k], Pi]], BesselJ[Plus[Power[E, Times[Complex[0, Rational[2, 3]], Pi]], k], 1.5], Power[Factorial[k], -1]], {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.66#Ex1	${\displaystyle{\displaystyle\operatorname{ber}_{n}\left(x\sqrt{2}\right)=\sum_% {k=-\infty}^{\infty}(-1)^{n+k}J_{n+2k}\left(x\right)I_{2k}\left(x\right)}}$ `\Kelvinber{n}@{x\sqrt{2}} = \sum_{k=-\infty}^{\infty}(-1)^{n+k}\BesselJ{n+2k}@{x}\modBesselI{2k}@{x}`	${\displaystyle{\displaystyle\Re((n+2k)+k+1)>0,\Re(n+k+1)>0,\Re((2k)+k+1)>0}}$	KelvinBer(n, xsqrt(2)) = sum((- 1)^(n + k) BesselJ(n + 2k, x)BesselI(2*k, x), k = - infinity..infinity)	KelvinBer[n, xSqrt[2]] == Sum[(- 1)^(n + k) BesselJ[n + 2k, x]BesselI[2*k, x], {k, - Infinity, Infinity}, GenerateConditions->None]	Failure	Aborted	Successful [Tested: 9]	Skipped - Because timed out
10.66#Ex2	${\displaystyle{\displaystyle\operatorname{bei}_{n}\left(x\sqrt{2}\right)=\sum_% {k=-\infty}^{\infty}(-1)^{n+k}J_{n+2k+1}\left(x\right)I_{2k+1}\left(x\right)}}$ `\Kelvinbei{n}@{x\sqrt{2}} = \sum_{k=-\infty}^{\infty}(-1)^{n+k}\BesselJ{n+2k+1}@{x}\modBesselI{2k+1}@{x}`	${\displaystyle{\displaystyle\Re((n+2k+1)+k+1)>0,\Re((2k+1)+k+1)>0}}$	KelvinBei(n, xsqrt(2)) = sum((- 1)^(n + k) BesselJ(n + 2k + 1, x)BesselI(2*k + 1, x), k = - infinity..infinity)	KelvinBei[n, xSqrt[2]] == Sum[(- 1)^(n + k) BesselJ[n + 2k + 1, x]BesselI[2*k + 1, x], {k, - Infinity, Infinity}, GenerateConditions->None]	Failure	Aborted	Successful [Tested: 9]	Skipped - Because timed out
10.68#Ex5	${\displaystyle{\displaystyle M_{\nu}\left(x\right)=({\operatorname{ber}_{\nu}^% {2}}x+{\operatorname{bei}_{\nu}^{2}}x)^{\ifrac{1}{2}}}}$ `\HankelmodM{\nu}@{x} = (\Kelvinber{\nu}^{2}@@{x}+\Kelvinbei{\nu}^{2}@@{x})^{\ifrac{1}{2}}`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	Error	Sqrt[KelvinBer[\[Nu], x]^2 + KelvinBei[\[Nu], x]^2] == ((KelvinBer[\[Nu], x])^(2)+ (KelvinBei[\[Nu], x])^(2))^(Divide[1,2])	Missing Macro Error	Successful	-	Successful [Tested: 30]
10.68#Ex6	${\displaystyle{\displaystyle N_{\nu}\left(x\right)=({\operatorname{ker}_{\nu}^% {2}}x+{\operatorname{kei}_{\nu}^{2}}x)^{\ifrac{1}{2}}}}$ `\HankelmodderivN{\nu}@{x} = (\Kelvinker{\nu}^{2}@@{x}+\Kelvinkei{\nu}^{2}@@{x})^{\ifrac{1}{2}}`		Error	Sqrt[KelvinKer[\[Nu], x]^2 + KelvinKei[\[Nu], x]^2] == ((KelvinKer[\[Nu], x])^(2)+ (KelvinKei[\[Nu], x])^(2))^(Divide[1,2])	Missing Macro Error	Successful	-	Successful [Tested: 30]
10.68#Ex9	${\displaystyle{\displaystyle M_{-n}\left(x\right)=M_{n}\left(x\right)}}$ `\HankelmodM{-n}@{x} = \HankelmodM{n}@{x}`		Error	Sqrt[KelvinBer[- n, x]^2 + KelvinBei[- n, x]^2] == Sqrt[KelvinBer[n, x]^2 + KelvinBei[n, x]^2]	Missing Macro Error	Failure	-	Successful [Tested: 9]
10.68#Ex17	${\displaystyle{\displaystyle N_{-\nu}\left(x\right)=N_{\nu}\left(x\right)}}$ `\HankelmodderivN{-\nu}@{x} = \HankelmodderivN{\nu}@{x}`		Error	Sqrt[KelvinKer[- \[Nu], x]^2 + KelvinKei[- \[Nu], x]^2] == Sqrt[KelvinKer[\[Nu], x]^2 + KelvinKei[\[Nu], x]^2]	Missing Macro Error	Failure	-	Successful [Tested: 30]
10.71.E1	${\displaystyle{\displaystyle\int x^{1+\nu}f_{\nu}\mathrm{d}x=-\frac{x^{1+\nu}}% {\sqrt{2}}(f_{\nu+1}-g_{\nu+1})}}$ `\int x^{1+\nu}f_{\nu}\diff{x} = -\frac{x^{1+\nu}}{\sqrt{2}}(f_{\nu+1}-g_{\nu+1})`		int((x)^(1 + nu)* f[nu], x) = -((x)^(1 + nu))/(sqrt(2))*(f[nu + 1]- g[nu + 1])	Integrate[(x)^(1 + \[Nu])* Subscript[f, \[Nu]], x, GenerateConditions->None] == -Divide[(x)^(1 + \[Nu]),Sqrt[2]]*(Subscript[f, \[Nu]+ 1]- Subscript[g, \[Nu]+ 1])	Failure	Failure	Failed [300 / 300] Result: .9346151411+.5776724966I Test Values: {nu = 1/23^(1/2)+1/2I, x = 3/2, f[nu] = 1/23^(1/2)+1/2I, f[1+nu] = 1/23^(1/2)+1/2I, g[1+nu] = 1/23^(1/2)+1/2I} Result: 3.061934630+.4518721345I Test Values: {nu = 1/23^(1/2)+1/2I, x = 3/2, f[nu] = 1/23^(1/2)+1/2I, f[1+nu] = 1/23^(1/2)+1/2I, g[1+nu] = -1/2+1/2I3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.9346151408625077, 0.5776724967688012] Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, ν], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, Plus[1, ν]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, Plus[1, ν]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[3.061934629891139, 0.45187213490403344] Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, ν], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, Plus[1, ν]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, Plus[1, ν]], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.71.E2	${\displaystyle{\displaystyle\int x^{1-\nu}f_{\nu}\mathrm{d}x=\frac{x^{1-\nu}}{% \sqrt{2}}(f_{\nu-1}-g_{\nu-1})}}$ `\int x^{1-\nu}f_{\nu}\diff{x} = \frac{x^{1-\nu}}{\sqrt{2}}(f_{\nu-1}-g_{\nu-1})`		int((x)^(1 - nu)* f[nu], x) = ((x)^(1 - nu))/(sqrt(2))*(f[nu - 1]- g[nu - 1])	Integrate[(x)^(1 - \[Nu])* Subscript[f, \[Nu]], x, GenerateConditions->None] == Divide[(x)^(1 - \[Nu]),Sqrt[2]]*(Subscript[f, \[Nu]- 1]- Subscript[g, \[Nu]- 1])	Failure	Failure	Failed [300 / 300] Result: .9470105611+.8580421171I Test Values: {nu = 1/23^(1/2)+1/2I, x = 3/2, f[nu] = 1/23^(1/2)+1/2I, f[nu-1] = 1/23^(1/2)+1/2I, g[nu-1] = 1/23^(1/2)+1/2I} Result: .30703090e-2+1.331056152I Test Values: {nu = 1/23^(1/2)+1/2I, x = 3/2, f[nu] = 1/23^(1/2)+1/2I, f[nu-1] = 1/23^(1/2)+1/2I, g[nu-1] = -1/2+1/2I3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.9470105613079453, 0.8580421172974921] Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, Plus[-1, ν]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, ν], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, Plus[-1, ν]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.0030703089818392426, 1.3310561520338196] Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, Plus[-1, ν]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, ν], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, Plus[-1, ν]], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.71.E6	${\displaystyle{\displaystyle\int xf_{\nu}g_{\nu}\mathrm{d}x=\tfrac{1}{4}x^{2}% \left(2f_{\nu}g_{\nu}-f_{\nu-1}g_{\nu+1}-f_{\nu+1}g_{\nu-1}\right)}}$ `\int xf_{\nu}g_{\nu}\diff{x} = \tfrac{1}{4}x^{2}\left(2f_{\nu}g_{\nu}-f_{\nu-1}g_{\nu+1}-f_{\nu+1}g_{\nu-1}\right)`		int(xf[nu]g[nu], x) = (1)/(4)(x)^(2)(2f[nu]g[nu]- f[nu - 1]g[nu + 1]- f[nu + 1]g[nu - 1])	Integrate[xSubscript[f, \[Nu]]Subscript[g, \[Nu]], x, GenerateConditions->None] == Divide[1,4](x)^(2)(2Subscript[f, \[Nu]]Subscript[g, \[Nu]]- Subscript[f, \[Nu]- 1]Subscript[g, \[Nu]+ 1]- Subscript[f, \[Nu]+ 1]Subscript[g, \[Nu]- 1])	Failure	Failure	Failed [270 / 300] Result: .5625000004+.9742785795I Test Values: {nu = 1/23^(1/2)+1/2I, x = 3/2, f[nu] = 1/23^(1/2)+1/2I, f[1+nu] = 1/23^(1/2)+1/2I, f[nu-1] = 1/23^(1/2)+1/2I, g[nu] = 1/23^(1/2)+1/2I, g[1+nu] = 1/23^(1/2)+1/2I, g[nu-1] = 1/23^(1/2)+1/2I} Result: -.2058892896+.7683892900I Test Values: {nu = 1/23^(1/2)+1/2I, x = 3/2, f[nu] = 1/23^(1/2)+1/2I, f[1+nu] = 1/23^(1/2)+1/2I, f[nu-1] = 1/23^(1/2)+1/2I, g[nu] = 1/23^(1/2)+1/2I, g[1+nu] = 1/23^(1/2)+1/2I, g[nu-1] = -1/2+1/2I3^(1/2)} ... skip entries to safe data	Skipped - Because timed out
10.71.E7	${\displaystyle{\displaystyle\int x(f_{\nu}^{2}-g_{\nu}^{2})\mathrm{d}x=\tfrac{% 1}{2}x^{2}\left(f_{\nu}^{2}-f_{\nu-1}f_{\nu+1}-g_{\nu}^{2}+g_{\nu-1}g_{\nu+1}% \right)}}$ `\int x(f_{\nu}^{2}-g_{\nu}^{2})\diff{x} = \tfrac{1}{2}x^{2}\left(f_{\nu}^{2}-f_{\nu-1}f_{\nu+1}-g_{\nu}^{2}+g_{\nu-1}g_{\nu+1}\right)`		int(x((f[nu])^(2)- (g[nu])^(2)), x) = (1)/(2)(x)^(2)((f[nu])^(2)- f[nu - 1]f[nu + 1]- (g[nu])^(2)+ g[nu - 1]*g[nu + 1])	Integrate[x((Subscript[f, \[Nu]])^(2)- (Subscript[g, \[Nu]])^(2)), x, GenerateConditions->None] == Divide[1,2](x)^(2)((Subscript[f, \[Nu]])^(2)- Subscript[f, \[Nu]- 1]Subscript[f, \[Nu]+ 1]- (Subscript[g, \[Nu]])^(2)+ Subscript[g, \[Nu]- 1]*Subscript[g, \[Nu]+ 1])	Failure	Failure	Error	Error
10.71#Ex1	${\displaystyle{\displaystyle\int x{M_{\nu}^{2}}\left(x\right)\mathrm{d}x=x(% \operatorname{ber}_{\nu}x\operatorname{bei}_{\nu}'x-\operatorname{ber}_{\nu}'x% \operatorname{bei}_{\nu}x)}}$ `\int x\HankelmodM{\nu}^{2}@{x}\diff{x} = x(\Kelvinber{\nu}@@{x}\Kelvinbei{\nu}'@@{x}-\Kelvinber{\nu}'@@{x}\Kelvinbei{\nu}@@{x})`	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	Error	Integrate[x(Sqrt[KelvinBer[\[Nu], x]^2 + KelvinBei[\[Nu], x]^2])^(2), x, GenerateConditions->None] == x(KelvinBer[\[Nu], x]D[KelvinBei[\[Nu], x], {x, 1}]- D[KelvinBer[\[Nu], x], {x, 1}]KelvinBei[\[Nu], x])	Missing Macro Error	Successful	-	Successful [Tested: 30]
10.71#Ex2	${\displaystyle{\displaystyle\int x{N_{\nu}^{2}}\left(x\right)\mathrm{d}x=x(% \operatorname{ker}_{\nu}x\operatorname{kei}_{\nu}'x-\operatorname{ker}_{\nu}'x% \operatorname{kei}_{\nu}x)}}$ `\int x\HankelmodderivN{\nu}^{2}@{x}\diff{x} = x(\Kelvinker{\nu}@@{x}\Kelvinkei{\nu}'@@{x}-\Kelvinker{\nu}'@@{x}\Kelvinkei{\nu}@@{x})`		Error	Integrate[x(Sqrt[KelvinKer[\[Nu], x]^2 + KelvinKei[\[Nu], x]^2])^(2), x, GenerateConditions->None] == x(KelvinKer[\[Nu], x]D[KelvinKei[\[Nu], x], {x, 1}]- D[KelvinKer[\[Nu], x], {x, 1}]KelvinKei[\[Nu], x])	Missing Macro Error	Successful	-	Successful [Tested: 30]
10.73.E1	${\displaystyle{\displaystyle\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac% {\partial V}{\partial r}\right)+\frac{1}{r^{2}}\frac{{\partial}^{2}V}{{% \partial\phi}^{2}}+\frac{{\partial}^{2}V}{{\partial z}^{2}}=0}}$ `\frac{1}{r}\pderiv{}{r}\left(r\pderiv{V}{r}\right)+\frac{1}{r^{2}}\pderiv[2]{V}{\phi}+\pderiv[2]{V}{z} = 0`		(1)/(r)diff((rdiff(V, r))+(1)/((r)^(2))*diff(V, [phi$(2)]), r)+ diff(V, [z$(2)]) = 0	Divide[1,r]D[(rD[V, r])+Divide[1,(r)^(2)]*D[V, {\[Phi], 2}], r]+ D[V, {z, 2}] == 0	Successful	Successful	-	Successful [Tested: 300]

@@ Line 1: / Line 1: @@
+<div style="width: 100%; height: 75vh; overflow: auto;">
+{| class="wikitable sortable" style="margin: 0;"
+|-
+! scope="col" style="position: sticky; top: 0;" | DLMF
+! scope="col" style="position: sticky; top: 0;" | Formula
+! scope="col" style="position: sticky; top: 0;" | Constraints
+! scope="col" style="position: sticky; top: 0;" | Maple
+! scope="col" style="position: sticky; top: 0;" | Mathematica
+! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
+! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
+! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
+! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
+|-
+| [https://dlmf.nist.gov/10.45.E1 10.45.E1] || [[Item:Q3655|<math>x^{2}\deriv[2]{w}{x}+x\deriv{w}{x}+(\nu^{2}-x^{2})w = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x^{2}\deriv[2]{w}{x}+x\deriv{w}{x}+(\nu^{2}-x^{2})w = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(x)^(2)* diff(w, [x$(2)])+ x*diff(w, x)+((nu)^(2)- (x)^(2))*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>(x)^(2)* D[w, {x, 2}]+ x*D[w, x]+(\[Nu]^(2)- (x)^(2))*w == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [240 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.948557159-.1249999996*I
+Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.2165063507+.8750000006*I
+Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 1/2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [240 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.948557158514987, -0.12499999999999989]
+Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.9485571585149875, -2.125]
+Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.45.E2 10.45.E2] || [[Item:Q3657|<math>\displaystyle\modBesselIimag{\nu}@{x} = \realpart@{\modBesselI{i\nu}@{x}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\displaystyle\modBesselIimag{\nu}@{x} = \realpart@{\modBesselI{i\nu}@{x}}</syntaxhighlight> || <math>\realpart@@{((\iunit \nu)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Re(BesselI(I*(nu), x)) = Re(BesselI(I*nu, x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Re[BesselI[I*\[Nu], x]] == Re[BesselI[I*\[Nu], x]]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 30]
+|-
+| [https://dlmf.nist.gov/10.45.E2 10.45.E2] || [[Item:Q3657|<math>\displaystyle\modBesselKimag{\nu}@{x} = \modBesselK{i\nu}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\displaystyle\modBesselKimag{\nu}@{x} = \modBesselK{i\nu}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>BesselK(I*(nu), x) = BesselK(I*nu, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BesselK[I*\[Nu], x] == BesselK[I*\[Nu], x]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 30]
+|- style="background: #dfe6e9;"
+| [https://dlmf.nist.gov/10.45.E3 10.45.E3] || [[Item:Q3659|<math>\displaystyle\modBesselIimag{-\nu}@{x} = \modBesselIimag{\nu}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\displaystyle\modBesselIimag{-\nu}@{x} = \modBesselIimag{\nu}@{x}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Re(BesselI(I*(- nu), x)) = Re(BesselI(I*(nu), x))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Re[BesselI[I*- \[Nu], x]] == Re[BesselI[I*\[Nu], x]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+|- style="background: #dfe6e9;"
+| [https://dlmf.nist.gov/10.45.E3 10.45.E3] || [[Item:Q3659|<math>\displaystyle\modBesselKimag{-\nu}@{x} = \modBesselKimag{\nu}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\displaystyle\modBesselKimag{-\nu}@{x} = \modBesselKimag{\nu}@{x}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">BesselK(I*(- nu), x) = BesselK(I*(nu), x)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">BesselK[I*- \[Nu], x] == BesselK[I*\[Nu], x]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+|-
+| [https://dlmf.nist.gov/10.45.E4 10.45.E4] || [[Item:Q3660|<math>\Wronskian@{\modBesselKimag{\nu}@{x},\modBesselIimag{\nu}@{x}} = 1/x</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\modBesselKimag{\nu}@{x},\modBesselIimag{\nu}@{x}} = 1/x</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(BesselK(I*(nu), x))*diff(Re(BesselI(I*(nu), x)), x)-diff(BesselK(I*(nu), x), x)*(Re(BesselI(I*(nu), x))) = 1/x</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{BesselK[I*\[Nu], x], Re[BesselI[I*\[Nu], x]]}, x] == 1/x</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-0.6666666666666666, Times[0.5, Plus[Complex[1.0700115379721733, -0.3754447148158467], Times[Complex[0.1636629185333998, 0.09141848176750039], Derivative[1][Re][Complex[2.445786867824693, 0.6492150843755028]]]]]]
+Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-0.6666666666666666, Times[0.5, Plus[Complex[0.8415452902387464, 0.2726729041814867], Times[Complex[0.3412924192180222, 0.19179892830603273], Derivative[1][Re][Complex[1.3137906770541619, -0.7251169608509622]]]]]]
+Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.45.E8 10.45.E8] || [[Item:Q3665|<math>\modBesselKimag{0}@{x} = \modBesselK{0}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modBesselKimag{0}@{x} = \modBesselK{0}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>BesselK(I*(0), x) = BesselK(0, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BesselK[I*0, x] == BesselK[0, x]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
+|-
+| [https://dlmf.nist.gov/10.47.E1 10.47.E1] || [[Item:Q3669|<math>z^{2}\deriv[2]{w}{z}+2z\deriv{w}{z}+\left(z^{2}-n(n+1)\right)w = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z^{2}\deriv[2]{w}{z}+2z\deriv{w}{z}+\left(z^{2}-n(n+1)\right)w = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(z)^(2)* diff(w, [z$(2)])+ 2*z*diff(w, z)+((z)^(2)- n*(n + 1))*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>(z)^(2)* D[w, {z, 2}]+ 2*z*D[w, z]+((z)^(2)- n*(n + 1))*w == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [210 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.732050808+.3733632160e-9*I
+Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -5.196152424-2.000000000*I
+Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [210 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.7320508075688772, 1.1102230246251565*^-16]
+Test Values: {Rule[n, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-5.196152422706633, -1.9999999999999996]
+Test Values: {Rule[n, 2], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.47.E2 10.47.E2] || [[Item:Q3670|<math>z^{2}\deriv[2]{w}{z}+2z\deriv{w}{z}-\left(z^{2}+n(n+1)\right)w = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z^{2}\deriv[2]{w}{z}+2z\deriv{w}{z}-\left(z^{2}+n(n+1)\right)w = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(z)^(2)* diff(w, [z$(2)])+ 2*z*diff(w, z)-((z)^(2)+ n*(n + 1))*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>(z)^(2)* D[w, {z, 2}]+ 2*z*D[w, z]-((z)^(2)+ n*(n + 1))*w == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [210 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.732050808-2.000000000*I
+Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -5.196152424-4.000000000*I
+Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [210 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.7320508075688776, -1.9999999999999998]
+Test Values: {Rule[n, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-5.196152422706632, -3.9999999999999996]
+Test Values: {Rule[n, 2], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.47.E3 10.47.E3] || [[Item:Q3671|<math>\sphBesselJ{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\BesselJ{n+\frac{1}{2}}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselJ{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\BesselJ{n+\frac{1}{2}}@{z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselJ[n, z] == Sqrt[Divide[1,2]*Pi/z]*BesselJ[n +Divide[1,2], z]</syntaxhighlight> || Missing Macro Error || Failure || Skip - symbolical successful subtest || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/10.47.E3 10.47.E3] || [[Item:Q3671|<math>\sqrt{\tfrac{1}{2}\pi/z}\BesselJ{n+\frac{1}{2}}@{z} = (-1)^{n}\sqrt{\tfrac{1}{2}\pi/z}\BesselY{-n-\frac{1}{2}}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{\tfrac{1}{2}\pi/z}\BesselJ{n+\frac{1}{2}}@{z} = (-1)^{n}\sqrt{\tfrac{1}{2}\pi/z}\BesselY{-n-\frac{1}{2}}@{z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>sqrt((1)/(2)*Pi/z)*BesselJ(n +(1)/(2), z) = (- 1)^(n)*sqrt((1)/(2)*Pi/z)*BesselY(- n -(1)/(2), z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[Divide[1,2]*Pi/z]*BesselJ[n +Divide[1,2], z] == (- 1)^(n)*Sqrt[Divide[1,2]*Pi/z]*BesselY[- n -Divide[1,2], z]</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/10.47.E4 10.47.E4] || [[Item:Q3672|<math>\sphBesselY{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\BesselY{n+\frac{1}{2}}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselY{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\BesselY{n+\frac{1}{2}}@{z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-(n+\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselY[n, z] == Sqrt[Divide[1,2]*Pi/z]*BesselY[n +Divide[1,2], z]</syntaxhighlight> || Missing Macro Error || Failure || Skip - symbolical successful subtest || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/10.47.E4 10.47.E4] || [[Item:Q3672|<math>\sqrt{\tfrac{1}{2}\pi/z}\BesselY{n+\frac{1}{2}}@{z} = (-1)^{n+1}\sqrt{\tfrac{1}{2}\pi/z}\BesselJ{-n-\frac{1}{2}}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{\tfrac{1}{2}\pi/z}\BesselY{n+\frac{1}{2}}@{z} = (-1)^{n+1}\sqrt{\tfrac{1}{2}\pi/z}\BesselJ{-n-\frac{1}{2}}@{z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-(n+\frac{1}{2}))+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>sqrt((1)/(2)*Pi/z)*BesselY(n +(1)/(2), z) = (- 1)^(n + 1)*sqrt((1)/(2)*Pi/z)*BesselJ(- n -(1)/(2), z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[Divide[1,2]*Pi/z]*BesselY[n +Divide[1,2], z] == (- 1)^(n + 1)*Sqrt[Divide[1,2]*Pi/z]*BesselJ[- n -Divide[1,2], z]</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/10.47.E5 10.47.E5] || [[Item:Q3673|<math>\sphHankelh{1}{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\HankelH{1}{n+\frac{1}{2}}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphHankelh{1}{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\HankelH{1}{n+\frac{1}{2}}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalHankelH1[n, z] == Sqrt[Divide[1,2]*Pi/z]*HankelH1[n +Divide[1,2], z]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/10.47.E5 10.47.E5] || [[Item:Q3673|<math>\sqrt{\tfrac{1}{2}\pi/z}\HankelH{1}{n+\frac{1}{2}}@{z} = (-1)^{n+1}\iunit\sqrt{\tfrac{1}{2}\pi/z}\HankelH{1}{-n-\frac{1}{2}}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{\tfrac{1}{2}\pi/z}\HankelH{1}{n+\frac{1}{2}}@{z} = (-1)^{n+1}\iunit\sqrt{\tfrac{1}{2}\pi/z}\HankelH{1}{-n-\frac{1}{2}}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sqrt((1)/(2)*Pi/z)*HankelH1(n +(1)/(2), z) = (- 1)^(n + 1)* I*sqrt((1)/(2)*Pi/z)*HankelH1(- n -(1)/(2), z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[Divide[1,2]*Pi/z]*HankelH1[n +Divide[1,2], z] == (- 1)^(n + 1)* I*Sqrt[Divide[1,2]*Pi/z]*HankelH1[- n -Divide[1,2], z]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/10.47.E6 10.47.E6] || [[Item:Q3674|<math>\sphHankelh{2}{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\HankelH{2}{n+\frac{1}{2}}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphHankelh{2}{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\HankelH{2}{n+\frac{1}{2}}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalHankelH2[n, z] == Sqrt[Divide[1,2]*Pi/z]*HankelH2[n +Divide[1,2], z]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/10.47.E6 10.47.E6] || [[Item:Q3674|<math>\sqrt{\tfrac{1}{2}\pi/z}\HankelH{2}{n+\frac{1}{2}}@{z} = (-1)^{n}\iunit\sqrt{\tfrac{1}{2}\pi/z}\HankelH{2}{-n-\frac{1}{2}}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{\tfrac{1}{2}\pi/z}\HankelH{2}{n+\frac{1}{2}}@{z} = (-1)^{n}\iunit\sqrt{\tfrac{1}{2}\pi/z}\HankelH{2}{-n-\frac{1}{2}}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sqrt((1)/(2)*Pi/z)*HankelH2(n +(1)/(2), z) = (- 1)^(n)* I*sqrt((1)/(2)*Pi/z)*HankelH2(- n -(1)/(2), z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[Divide[1,2]*Pi/z]*HankelH2[n +Divide[1,2], z] == (- 1)^(n)* I*Sqrt[Divide[1,2]*Pi/z]*HankelH2[- n -Divide[1,2], z]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/10.47.E7 10.47.E7] || [[Item:Q3675|<math>\modsphBesseli{1}{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\modBesselI{n+\frac{1}{2}}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesseli{1}{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\modBesselI{n+\frac{1}{2}}@{z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n] == Sqrt[Divide[1,2]*Pi/z]*BesselI[n +Divide[1,2], z]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.06771919180965624, -0.29579816936516184]
+Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.4498252419402129, -0.19064547195046921]
+Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.47.E8 10.47.E8] || [[Item:Q3676|<math>\modsphBesseli{2}{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\modBesselI{-n-\frac{1}{2}}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesseli{2}{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\modBesselI{-n-\frac{1}{2}}@{z}</syntaxhighlight> || <math>\realpart@@{((-n-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n] == Sqrt[Divide[1,2]*Pi/z]*BesselI[- n -Divide[1,2], z]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.41419719140728084, -0.8850762711170854]
+Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.1065867555175597, 2.4569570135519543]
+Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.47.E9 10.47.E9] || [[Item:Q3677|<math>\modsphBesselK{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\modBesselK{n+\frac{1}{2}}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesselK{n}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\modBesselK{n+\frac{1}{2}}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[1/2 Pi /z] BesselK[n + 1/2, z] == Sqrt[Divide[1,2]*Pi/z]*BesselK[n +Divide[1,2], z]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/10.47.E9 10.47.E9] || [[Item:Q3677|<math>\sqrt{\tfrac{1}{2}\pi/z}\modBesselK{n+\frac{1}{2}}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\modBesselK{-n-\frac{1}{2}}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{\tfrac{1}{2}\pi/z}\modBesselK{n+\frac{1}{2}}@{z} = \sqrt{\tfrac{1}{2}\pi/z}\modBesselK{-n-\frac{1}{2}}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sqrt((1)/(2)*Pi/z)*BesselK(n +(1)/(2), z) = sqrt((1)/(2)*Pi/z)*BesselK(- n -(1)/(2), z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[Divide[1,2]*Pi/z]*BesselK[n +Divide[1,2], z] == Sqrt[Divide[1,2]*Pi/z]*BesselK[- n -Divide[1,2], z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/10.47#Ex1 10.47#Ex1] || [[Item:Q3678|<math>\sphHankelh{1}{n}@{z} = \sphBesselJ{n}@{z}+i\sphBesselY{n}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphHankelh{1}{n}@{z} = \sphBesselJ{n}@{z}+i\sphBesselY{n}@{z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0, \realpart@@{((-(n+\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalHankelH1[n, z] == SphericalBesselJ[n, z]+ I*SphericalBesselY[n, z]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/10.47#Ex2 10.47#Ex2] || [[Item:Q3679|<math>\sphHankelh{2}{n}@{z} = \sphBesselJ{n}@{z}-i\sphBesselY{n}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphHankelh{2}{n}@{z} = \sphBesselJ{n}@{z}-i\sphBesselY{n}@{z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0, \realpart@@{((-(n+\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalHankelH2[n, z] == SphericalBesselJ[n, z]- I*SphericalBesselY[n, z]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/10.47.E11 10.47.E11] || [[Item:Q3680|<math>\modsphBesselK{n}@{z} = (-1)^{n+1}\tfrac{1}{2}\pi\left(\modsphBesseli{1}{n}@{z}-\modsphBesseli{2}{n}@{z}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesselK{n}@{z} = (-1)^{n+1}\tfrac{1}{2}\pi\left(\modsphBesseli{1}{n}@{z}-\modsphBesseli{2}{n}@{z}\right)</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[1/2 Pi /z] BesselK[n + 1/2, z] == (- 1)^(n + 1)*Divide[1,2]*Pi*(Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n]- Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n])</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.7569924845794465, -0.925635877692591]
+Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.0316385731075524, -4.1588442590402455]
+Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.47#Ex3 10.47#Ex3] || [[Item:Q3681|<math>\modsphBesseli{1}{n}@{z} = i^{-n}\sphBesselJ{n}@{iz}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesseli{1}{n}@{z} = i^{-n}\sphBesselJ{n}@{iz}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n] == (I)^(- n)* SphericalBesselJ[n, I*z]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.06771919180965624, -0.2957981693651618]
+Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.44982524194021284, -0.19064547195046921]
+Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.47#Ex4 10.47#Ex4] || [[Item:Q3682|<math>\modsphBesseli{2}{n}@{z} = i^{-n-1}\sphBesselY{n}@{iz}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesseli{2}{n}@{z} = i^{-n-1}\sphBesselY{n}@{iz}</syntaxhighlight> || <math>\realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-(n+\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n] == (I)^(- n - 1)* SphericalBesselY[n, I*z]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.41419719140728045, -0.8850762711170859]
+Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.1065867555175588, 2.456957013551956]
+Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.47.E13 10.47.E13] || [[Item:Q3683|<math>\modsphBesselK{n}@{z} = -\tfrac{1}{2}\pi i^{n}\sphHankelh{1}{n}@{iz}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesselK{n}@{z} = -\tfrac{1}{2}\pi i^{n}\sphHankelh{1}{n}@{iz}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[1/2 Pi /z] BesselK[n + 1/2, z] == -Divide[1,2]*Pi*(I)^(n)* SphericalHankelH1[n, I*z]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/10.47.E13 10.47.E13] || [[Item:Q3683|<math>-\tfrac{1}{2}\pi i^{n}\sphHankelh{1}{n}@{iz} = -\tfrac{1}{2}\pi i^{-n}\sphHankelh{2}{n}@{-iz}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>-\tfrac{1}{2}\pi i^{n}\sphHankelh{1}{n}@{iz} = -\tfrac{1}{2}\pi i^{-n}\sphHankelh{2}{n}@{-iz}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>-Divide[1,2]*Pi*(I)^(n)* SphericalHankelH1[n, I*z] == -Divide[1,2]*Pi*(I)^(- n)* SphericalHankelH2[n, - I*z]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 21]
+|- style="background: #dfe6e9;"
+| [https://dlmf.nist.gov/10.47.E14 10.47.E14] || [[Item:Q3685|<math>\displaystyle\sphBesselJ{n}@{-z} = (-1)^{n}\sphBesselJ{n}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\displaystyle\sphBesselJ{n}@{-z} = (-1)^{n}\sphBesselJ{n}@{z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Error</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">SphericalBesselJ[n, - z] == (- 1)^(n)* SphericalBesselJ[n, z]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+|- style="background: #dfe6e9;"
+| [https://dlmf.nist.gov/10.47.E14 10.47.E14] || [[Item:Q3685|<math>\displaystyle\sphBesselY{n}@{-z} = (-1)^{n+1}\sphBesselY{n}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\displaystyle\sphBesselY{n}@{-z} = (-1)^{n+1}\sphBesselY{n}@{z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-(n+\frac{1}{2}))+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Error</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">SphericalBesselY[n, - z] == (- 1)^(n + 1)* SphericalBesselY[n, z]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+|- style="background: #dfe6e9;"
+| [https://dlmf.nist.gov/10.47.E15 10.47.E15] || [[Item:Q3687|<math>\displaystyle\sphHankelh{1}{n}@{-z} = (-1)^{n}\sphHankelh{2}{n}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\displaystyle\sphHankelh{1}{n}@{-z} = (-1)^{n}\sphHankelh{2}{n}@{z}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Error</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">SphericalHankelH1[n, - z] == (- 1)^(n)* SphericalHankelH2[n, z]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+|- style="background: #dfe6e9;"
+| [https://dlmf.nist.gov/10.47.E15 10.47.E15] || [[Item:Q3687|<math>\displaystyle\sphHankelh{2}{n}@{-z} = (-1)^{n}\sphHankelh{1}{n}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\displaystyle\sphHankelh{2}{n}@{-z} = (-1)^{n}\sphHankelh{1}{n}@{z}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Error</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">SphericalHankelH2[n, - z] == (- 1)^(n)* SphericalHankelH1[n, z]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+|- style="background: #dfe6e9;"
+| [https://dlmf.nist.gov/10.47.E16 10.47.E16] || [[Item:Q3689|<math>\displaystyle\modsphBesseli{1}{n}@{-z} = (-1)^{n}\modsphBesseli{1}{n}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\displaystyle\modsphBesseli{1}{n}@{-z} = (-1)^{n}\modsphBesseli{1}{n}@{z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Error</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sqrt[Divide[Pi, - z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n] == (- 1)^(n)* Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+|- style="background: #dfe6e9;"
+| [https://dlmf.nist.gov/10.47.E16 10.47.E16] || [[Item:Q3689|<math>\displaystyle\modsphBesseli{2}{n}@{-z} = (-1)^{n+1}\modsphBesseli{2}{n}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\displaystyle\modsphBesseli{2}{n}@{-z} = (-1)^{n+1}\modsphBesseli{2}{n}@{z}</syntaxhighlight> || <math>\realpart@@{((-n-\frac{1}{2})+k+1)} > 0</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Error</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sqrt[Divide[Pi, - z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n] == (- 1)^(n + 1)* Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+|-
+| [https://dlmf.nist.gov/10.47.E17 10.47.E17] || [[Item:Q3690|<math>\modsphBesselK{n}@{-z} = -\tfrac{1}{2}\pi\left(\modsphBesseli{1}{n}@{z}+\modsphBesseli{2}{n}@{z}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesselK{n}@{-z} = -\tfrac{1}{2}\pi\left(\modsphBesseli{1}{n}@{z}+\modsphBesseli{2}{n}@{z}\right)</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[1/2 Pi /- z] BesselK[n + 1/2, - z] == -Divide[1,2]*Pi*(Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n]+ Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n])</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.5442463690831921, -1.8549132335154932]
+Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[2.444806248586177, 3.5599138449204935]
+Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.49.E2 10.49.E2] || [[Item:Q3692|<math>\sphBesselJ{n}@{z} = \sin@{z-\tfrac{1}{2}n\pi}\sum_{k=0}^{\floor{n/2}}(-1)^{k}\frac{a_{2k}(n+\tfrac{1}{2})}{z^{2k+1}}+\cos@{z-\tfrac{1}{2}n\pi}\sum_{k=0}^{\floor{(n-1)/2}}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{2k+2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselJ{n}@{z} = \sin@{z-\tfrac{1}{2}n\pi}\sum_{k=0}^{\floor{n/2}}(-1)^{k}\frac{a_{2k}(n+\tfrac{1}{2})}{z^{2k+1}}+\cos@{z-\tfrac{1}{2}n\pi}\sum_{k=0}^{\floor{(n-1)/2}}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{2k+2}}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0, k \geq 1</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselJ[n, z] == Sin[z -Divide[1,2]*n*Pi]*Sum[(- 1)^(k)*Divide[Subscript[a, 2*k]*(n +Divide[1,2]),(z)^(2*k + 1)], {k, 0, Floor[n/2]}, GenerateConditions->None]+ Cos[z -Divide[1,2]*n*Pi]*Sum[(- 1)^(k)*Divide[Subscript[a, 2*k + 1]*(n +Divide[1,2]),(z)^(2*k + 2)], {k, 0, Floor[(n - 1)/2]}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/10.49#Ex1 10.49#Ex1] || [[Item:Q3693|<math>\sphBesselJ{0}@{z} = \frac{\sin@@{z}}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselJ{0}@{z} = \frac{\sin@@{z}}{z}</syntaxhighlight> || <math>\realpart@@{((0+\frac{1}{2})+k+1)} > 0, \realpart@@{((-0-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-0-\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselJ[0, z] == Divide[Sin[z],z]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 7]
+|-
+| [https://dlmf.nist.gov/10.49#Ex2 10.49#Ex2] || [[Item:Q3694|<math>\sphBesselJ{1}@{z} = \frac{\sin@@{z}}{z^{2}}-\frac{\cos@@{z}}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselJ{1}@{z} = \frac{\sin@@{z}}{z^{2}}-\frac{\cos@@{z}}{z}</syntaxhighlight> || <math>\realpart@@{((1+\frac{1}{2})+k+1)} > 0, \realpart@@{((-1-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-1-\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselJ[1, z] == Divide[Sin[z],(z)^(2)]-Divide[Cos[z],z]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 7]
+|-
+| [https://dlmf.nist.gov/10.49#Ex3 10.49#Ex3] || [[Item:Q3695|<math>\sphBesselJ{2}@{z} = \left(-\frac{1}{z}+\frac{3}{z^{3}}\right)\sin@@{z}-\frac{3}{z^{2}}\cos@@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselJ{2}@{z} = \left(-\frac{1}{z}+\frac{3}{z^{3}}\right)\sin@@{z}-\frac{3}{z^{2}}\cos@@{z}</syntaxhighlight> || <math>\realpart@@{((2+\frac{1}{2})+k+1)} > 0, \realpart@@{((-2-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-2-\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselJ[2, z] == (-Divide[1,z]+Divide[3,(z)^(3)])*Sin[z]-Divide[3,(z)^(2)]*Cos[z]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 7]
+|-
+| [https://dlmf.nist.gov/10.49.E4 10.49.E4] || [[Item:Q3696|<math>\sphBesselY{n}@{z} = -\cos@{z-\tfrac{1}{2}n\pi}\sum_{k=0}^{\floor{n/2}}(-1)^{k}\frac{a_{2k}(n+\tfrac{1}{2})}{z^{2k+1}}+\sin@{z-\tfrac{1}{2}n\pi}\sum_{k=0}^{\floor{(n-1)/2}}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{2k+2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselY{n}@{z} = -\cos@{z-\tfrac{1}{2}n\pi}\sum_{k=0}^{\floor{n/2}}(-1)^{k}\frac{a_{2k}(n+\tfrac{1}{2})}{z^{2k+1}}+\sin@{z-\tfrac{1}{2}n\pi}\sum_{k=0}^{\floor{(n-1)/2}}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{2k+2}}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-(n+\frac{1}{2}))+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, k \geq 1</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselY[n, z] == - Cos[z -Divide[1,2]*n*Pi]*Sum[(- 1)^(k)*Divide[Subscript[a, 2*k]*(n +Divide[1,2]),(z)^(2*k + 1)], {k, 0, Floor[n/2]}, GenerateConditions->None]+ Sin[z -Divide[1,2]*n*Pi]*Sum[(- 1)^(k)*Divide[Subscript[a, 2*k + 1]*(n +Divide[1,2]),(z)^(2*k + 2)], {k, 0, Floor[(n - 1)/2]}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/10.49#Ex4 10.49#Ex4] || [[Item:Q3697|<math>\sphBesselY{0}@{z} = -\frac{\cos@@{z}}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselY{0}@{z} = -\frac{\cos@@{z}}{z}</syntaxhighlight> || <math>\realpart@@{((0+\frac{1}{2})+k+1)} > 0, \realpart@@{((-(0+\frac{1}{2}))+k+1)} > 0, \realpart@@{((-0-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselY[0, z] == -Divide[Cos[z],z]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 7]
+|-
+| [https://dlmf.nist.gov/10.49#Ex5 10.49#Ex5] || [[Item:Q3698|<math>\sphBesselY{1}@{z} = -\frac{\cos@@{z}}{z^{2}}-\frac{\sin@@{z}}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselY{1}@{z} = -\frac{\cos@@{z}}{z^{2}}-\frac{\sin@@{z}}{z}</syntaxhighlight> || <math>\realpart@@{((1+\frac{1}{2})+k+1)} > 0, \realpart@@{((-(1+\frac{1}{2}))+k+1)} > 0, \realpart@@{((-1-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselY[1, z] == -Divide[Cos[z],(z)^(2)]-Divide[Sin[z],z]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 7]
+|-
+| [https://dlmf.nist.gov/10.49#Ex6 10.49#Ex6] || [[Item:Q3699|<math>\sphBesselY{2}@{z} = \left(\frac{1}{z}-\frac{3}{z^{3}}\right)\cos@@{z}-\frac{3}{z^{2}}\sin@@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselY{2}@{z} = \left(\frac{1}{z}-\frac{3}{z^{3}}\right)\cos@@{z}-\frac{3}{z^{2}}\sin@@{z}</syntaxhighlight> || <math>\realpart@@{((2+\frac{1}{2})+k+1)} > 0, \realpart@@{((-(2+\frac{1}{2}))+k+1)} > 0, \realpart@@{((-2-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselY[2, z] == (Divide[1,z]-Divide[3,(z)^(3)])*Cos[z]-Divide[3,(z)^(2)]*Sin[z]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 7]
+|-
+| [https://dlmf.nist.gov/10.49.E6 10.49.E6] || [[Item:Q3700|<math>\sphHankelh{1}{n}@{z} = e^{iz}\sum_{k=0}^{n}i^{k-n-1}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphHankelh{1}{n}@{z} = e^{iz}\sum_{k=0}^{n}i^{k-n-1}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}</syntaxhighlight> || <math>k \geq 1</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalHankelH1[n, z] == Exp[I*z]*Sum[(I)^(k - n - 1)*Divide[Subscript[a, k]*(n +Divide[1,2]),(z)^(k + 1)], {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [210 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.3966692432410339, 0.7497610210111748]
+Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.3157223500929769, 0.5313692545383957]
+Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.49.E7 10.49.E7] || [[Item:Q3701|<math>\sphHankelh{2}{n}@{z} = e^{-iz}\sum_{k=0}^{n}(-i)^{k-n-1}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphHankelh{2}{n}@{z} = e^{-iz}\sum_{k=0}^{n}(-i)^{k-n-1}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}</syntaxhighlight> || <math>k \geq 1</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalHankelH2[n, z] == Exp[- I*z]*Sum[(- I)^(k - n - 1)*Divide[Subscript[a, k]*(n +Divide[1,2]),(z)^(k + 1)], {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/10.49.E8 10.49.E8] || [[Item:Q3702|<math>\modsphBesseli{1}{n}@{z} = \tfrac{1}{2}e^{z}\sum_{k=0}^{n}(-1)^{k}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}+(-1)^{n+1}\*\tfrac{1}{2}e^{-z}\sum_{k=0}^{n}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesseli{1}{n}@{z} = \tfrac{1}{2}e^{z}\sum_{k=0}^{n}(-1)^{k}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}+(-1)^{n+1}\*\tfrac{1}{2}e^{-z}\sum_{k=0}^{n}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, k \geq 1</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n] == Divide[1,2]*Exp[z]*Sum[(- 1)^(k)*Divide[Subscript[a, k]*(n +Divide[1,2]),(z)^(k + 1)], {k, 0, n}, GenerateConditions->None]+(- 1)^(n + 1)*Divide[1,2]*(E)^(- z)* Sum[Divide[Subscript[a, k]*(n +Divide[1,2]),(z)^(k + 1)], {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/10.49#Ex7 10.49#Ex7] || [[Item:Q3703|<math>\modsphBesseli{1}{0}@{z} = \frac{\sinh@@{z}}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesseli{1}{0}@{z} = \frac{\sinh@@{z}}{z}</syntaxhighlight> || <math>\realpart@@{((0+\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(0 + 1/2), 0] == Divide[Sinh[z],z]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.0789668887893185, -0.15155203743332835]
+Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.9126970224666039, 0.13712305377128448]
+Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.49#Ex8 10.49#Ex8] || [[Item:Q3704|<math>\modsphBesseli{1}{1}@{z} = -\frac{\sinh@@{z}}{z^{2}}+\frac{\cosh@@{z}}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesseli{1}{1}@{z} = -\frac{\sinh@@{z}}{z^{2}}+\frac{\cosh@@{z}}{z}</syntaxhighlight> || <math>\realpart@@{((1+\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(1 + 1/2), 1] == -Divide[Sinh[z],(z)^(2)]+Divide[Cosh[z],z]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.06771919180965646, -0.2957981693651617]
+Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.3178790653897484, -0.6062561841669247]
+Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.49#Ex9 10.49#Ex9] || [[Item:Q3705|<math>\modsphBesseli{1}{2}@{z} = \left(\frac{1}{z}+\frac{3}{z^{3}}\right)\sinh@@{z}-\frac{3}{z^{2}}\cosh@@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesseli{1}{2}@{z} = \left(\frac{1}{z}+\frac{3}{z^{3}}\right)\sinh@@{z}-\frac{3}{z^{2}}\cosh@@{z}</syntaxhighlight> || <math>\realpart@@{((2+\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(2 + 1/2), 2] == (Divide[1,z]+Divide[3,(z)^(3)])*Sinh[z]-Divide[3,(z)^(2)]*Cosh[z]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.44982524194021334, -0.19064547195046933]
+Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.2843828483915114, -0.37732112452647515]
+Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.49.E10 10.49.E10] || [[Item:Q3706|<math>\modsphBesseli{2}{n}@{z} = \tfrac{1}{2}e^{z}\sum_{k=0}^{n}(-1)^{k}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}+(-1)^{n}\tfrac{1}{2}e^{-z}\sum_{k=0}^{n}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesseli{2}{n}@{z} = \tfrac{1}{2}e^{z}\sum_{k=0}^{n}(-1)^{k}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}+(-1)^{n}\tfrac{1}{2}e^{-z}\sum_{k=0}^{n}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}</syntaxhighlight> || <math>\realpart@@{((-n-\frac{1}{2})+k+1)} > 0, k \geq 1</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n] == Divide[1,2]*Exp[z]*Sum[(- 1)^(k)*Divide[Subscript[a, k]*(n +Divide[1,2]),(z)^(k + 1)], {k, 0, n}, GenerateConditions->None]+(- 1)^(n)*Divide[1,2]*(E)^(- z)* Sum[Divide[Subscript[a, k]*(n +Divide[1,2]),(z)^(k + 1)], {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/10.49#Ex10 10.49#Ex10] || [[Item:Q3707|<math>\modsphBesseli{2}{0}@{z} = \frac{\cosh@@{z}}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesseli{2}{0}@{z} = \frac{\cosh@@{z}}{z}</syntaxhighlight> || <math>\realpart@@{((-0-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(0 + 1/2), 0] == Divide[Cosh[z],z]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: DirectedInfinity[]
+Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: DirectedInfinity[]
+Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.49#Ex11 10.49#Ex11] || [[Item:Q3708|<math>\modsphBesseli{2}{1}@{z} = -\frac{\cosh@@{z}}{z^{2}}+\frac{\sinh@@{z}}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesseli{2}{1}@{z} = -\frac{\cosh@@{z}}{z^{2}}+\frac{\sinh@@{z}}{z}</syntaxhighlight> || <math>\realpart@@{((-1-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(1 + 1/2), 1] == -Divide[Cosh[z],(z)^(2)]+Divide[Sinh[z],z]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.41419719140728073, -0.8850762711170859]
+Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.1181398580617885, 1.2868595835312289]
+Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.49#Ex12 10.49#Ex12] || [[Item:Q3709|<math>\modsphBesseli{2}{2}@{z} = \left(\frac{1}{z}+\frac{3}{z^{3}}\right)\cosh@@{z}-\frac{3}{z^{2}}\sinh@@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesseli{2}{2}@{z} = \left(\frac{1}{z}+\frac{3}{z^{3}}\right)\cosh@@{z}-\frac{3}{z^{2}}\sinh@@{z}</syntaxhighlight> || <math>\realpart@@{((-2-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(2 + 1/2), 2] == (Divide[1,z]+Divide[3,(z)^(3)])*Cosh[z]-Divide[3,(z)^(2)]*Sinh[z]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.106586755517561, 2.456957013551956]
+Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-2.803584197807803, -1.2408087832280956]
+Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.49.E12 10.49.E12] || [[Item:Q3710|<math>\modsphBesselK{n}@{z} = \tfrac{1}{2}\pi e^{-z}\sum_{k=0}^{n}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesselK{n}@{z} = \tfrac{1}{2}\pi e^{-z}\sum_{k=0}^{n}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}</syntaxhighlight> || <math>k \geq 1</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[1/2 Pi /z] BesselK[n + 1/2, z] == Divide[1,2]*Pi*Exp[- z]*Sum[Divide[Subscript[a, k]*(n +Divide[1,2]),(z)^(k + 1)], {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [210 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.0260307573251746, 0.0967341401667452]
+Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-2.907697530268464, -0.43148595883398677]
+Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.49#Ex13 10.49#Ex13] || [[Item:Q3711|<math>\modsphBesselK{0}@{z} = \tfrac{1}{2}\pi\frac{e^{-z}}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesselK{0}@{z} = \tfrac{1}{2}\pi\frac{e^{-z}}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[1/2 Pi /z] BesselK[0 + 1/2, z] == Divide[1,2]*Pi*Divide[Exp[- z],z]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 7]
+|-
+| [https://dlmf.nist.gov/10.49#Ex14 10.49#Ex14] || [[Item:Q3712|<math>\modsphBesselK{1}@{z} = \tfrac{1}{2}\pi e^{-z}\left(\frac{1}{z}+\frac{1}{z^{2}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesselK{1}@{z} = \tfrac{1}{2}\pi e^{-z}\left(\frac{1}{z}+\frac{1}{z^{2}}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[1/2 Pi /z] BesselK[1 + 1/2, z] == Divide[1,2]*Pi*Exp[- z]*(Divide[1,z]+Divide[1,(z)^(2)])</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 7]
+|-
+| [https://dlmf.nist.gov/10.49#Ex15 10.49#Ex15] || [[Item:Q3713|<math>\modsphBesselK{2}@{z} = \tfrac{1}{2}\pi e^{-z}\left(\frac{1}{z}+\frac{3}{z^{2}}+\frac{3}{z^{3}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesselK{2}@{z} = \tfrac{1}{2}\pi e^{-z}\left(\frac{1}{z}+\frac{3}{z^{2}}+\frac{3}{z^{3}}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[1/2 Pi /z] BesselK[2 + 1/2, z] == Divide[1,2]*Pi*Exp[- z]*(Divide[1,z]+Divide[3,(z)^(2)]+Divide[3,(z)^(3)])</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 7]
+|-
+| [https://dlmf.nist.gov/10.49#Ex16 10.49#Ex16] || [[Item:Q3714|<math>\sphBesselJ{n}@{z} = z^{n}\left(-\frac{1}{z}\deriv{}{z}\right)^{n}\frac{\sin@@{z}}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselJ{n}@{z} = z^{n}\left(-\frac{1}{z}\deriv{}{z}\right)^{n}\frac{\sin@@{z}}{z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>(-Divide[1,z]*D[(z)^(n)*-Divide[1,z], z])^(n)*Divide[Sin[z],z]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.28766324258243325, 0.13393934480402792]
+Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.302013441049254, 0.9125931496973667]
+Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.49#Ex17 10.49#Ex17] || [[Item:Q3715|<math>\sphBesselY{n}@{z} = -z^{n}\left(-\frac{1}{z}\deriv{}{z}\right)^{n}\frac{\cos@@{z}}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselY{n}@{z} = -z^{n}\left(-\frac{1}{z}\deriv{}{z}\right)^{n}\frac{\cos@@{z}}{z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-(n+\frac{1}{2}))+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselY[n, z] (-Divide[1,z]*D[(z)^(n)*-Divide[1,z], z])^(n)*Divide[Cos[z],z]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.9342001374760677, 0.968266641946737]
+Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.14357960272401077, 3.9384338499123404]
+Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.49#Ex18 10.49#Ex18] || [[Item:Q3716|<math>\modsphBesseli{1}{n}@{z} = z^{n}\left(\frac{1}{z}\deriv{}{z}\right)^{n}\frac{\sinh@@{z}}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesseli{1}{n}@{z} = z^{n}\left(\frac{1}{z}\deriv{}{z}\right)^{n}\frac{\sinh@@{z}}{z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n] (Divide[1,z]*D[(z)^(n)*Divide[1,z], z])^(n)*Divide[Sinh[z],z]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.35534425318828616, -0.09521420567684166]
+Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.19008700336701606, 0.7298484499303669]
+Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.49#Ex19 10.49#Ex19] || [[Item:Q3717|<math>\modsphBesseli{2}{n}@{z} = z^{n}\left(\frac{1}{z}\deriv{}{z}\right)^{n}\frac{\cosh@@{z}}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesseli{2}{n}@{z} = z^{n}\left(\frac{1}{z}\deriv{}{z}\right)^{n}\frac{\cosh@@{z}}{z}</syntaxhighlight> || <math>\realpart@@{((-n-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n] (Divide[1,z]*D[(z)^(n)*Divide[1,z], z])^(n)*Divide[Cosh[z],z]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.3553442531882861, 0.09521420567684165]
+Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.31198506093225176, 1.0184810034762684]
+Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.49.E16 10.49.E16] || [[Item:Q3718|<math>\modsphBesselK{n}@{z} = (-1)^{n}\tfrac{1}{2}\pi z^{n}\left(\frac{1}{z}\deriv{}{z}\right)^{n}\frac{e^{-z}}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesselK{n}@{z} = (-1)^{n}\tfrac{1}{2}\pi z^{n}\left(\frac{1}{z}\deriv{}{z}\right)^{n}\frac{e^{-z}}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[1/2 Pi /z] BesselK[n + 1/2, z] == (- 1)^(n)*Divide[1,2]*(Divide[1,z]*D[(z)^(n)*Divide[1,z], z])^(n)*Divide[Exp[- z],z]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.3593544107322247, -1.2247601267643444]
+Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.45891810409859557, -4.100723067341411]
+Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.49.E18 10.49.E18] || [[Item:Q3720|<math>\sphBesselJ{n}^{2}@{z}+\sphBesselY{n}^{2}@{z} = \sum_{k=0}^{n}\frac{s_{k}(n+\frac{1}{2})}{z^{2k+2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselJ{n}^{2}@{z}+\sphBesselY{n}^{2}@{z} = \sum_{k=0}^{n}\frac{s_{k}(n+\frac{1}{2})}{z^{2k+2}}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0, \realpart@@{((-(n+\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>(SphericalBesselJ[n, z])^(2)+ (SphericalBesselY[n, z])^(2) == Sum[Divide[Subscript[s, k]*(n +Divide[1,2]),(z)^(2*k + 2)], {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [210 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.2990381056766571, 0.5179491924311224]
+Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[s, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-9.999999999999996, 1.5358983848622398]
+Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[s, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.49#Ex20 10.49#Ex20] || [[Item:Q3721|<math>\sphBesselJ{0}^{2}@{z}+\sphBesselY{0}^{2}@{z} = z^{-2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselJ{0}^{2}@{z}+\sphBesselY{0}^{2}@{z} = z^{-2}</syntaxhighlight> || <math>\realpart@@{((0+\frac{1}{2})+k+1)} > 0, \realpart@@{((-0-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-0-\frac{1}{2}))+k+1)} > 0, \realpart@@{((-(0+\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>(SphericalBesselJ[0, z])^(2)+ (SphericalBesselY[0, z])^(2) == (z)^(- 2)</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 7]
+|-
+| [https://dlmf.nist.gov/10.49#Ex21 10.49#Ex21] || [[Item:Q3722|<math>\sphBesselJ{1}^{2}@{z}+\sphBesselY{1}^{2}@{z} = z^{-2}+z^{-4}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselJ{1}^{2}@{z}+\sphBesselY{1}^{2}@{z} = z^{-2}+z^{-4}</syntaxhighlight> || <math>\realpart@@{((1+\frac{1}{2})+k+1)} > 0, \realpart@@{((-1-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-1-\frac{1}{2}))+k+1)} > 0, \realpart@@{((-(1+\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>(SphericalBesselJ[1, z])^(2)+ (SphericalBesselY[1, z])^(2) == (z)^(- 2)+ (z)^(- 4)</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 7]
+|-
+| [https://dlmf.nist.gov/10.49#Ex22 10.49#Ex22] || [[Item:Q3723|<math>\sphBesselJ{2}^{2}@{z}+\sphBesselY{2}^{2}@{z} = z^{-2}+3z^{-4}+9z^{-6}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselJ{2}^{2}@{z}+\sphBesselY{2}^{2}@{z} = z^{-2}+3z^{-4}+9z^{-6}</syntaxhighlight> || <math>\realpart@@{((2+\frac{1}{2})+k+1)} > 0, \realpart@@{((-2-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-2-\frac{1}{2}))+k+1)} > 0, \realpart@@{((-(2+\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>(SphericalBesselJ[2, z])^(2)+ (SphericalBesselY[2, z])^(2) == (z)^(- 2)+ 3*(z)^(- 4)+ 9*(z)^(- 6)</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 7]
+|-
+| [https://dlmf.nist.gov/10.49.E20 10.49.E20] || [[Item:Q3724|<math>\left(\modsphBesseli{1}{n}@{z}\right)^{2}-\left(\modsphBesseli{2}{n}@{z}\right)^{2} = (-1)^{n+1}\sum_{k=0}^{n}(-1)^{k}\frac{s_{k}(n+\frac{1}{2})}{z^{2k+2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(\modsphBesseli{1}{n}@{z}\right)^{2}-\left(\modsphBesseli{2}{n}@{z}\right)^{2} = (-1)^{n+1}\sum_{k=0}^{n}(-1)^{k}\frac{s_{k}(n+\frac{1}{2})}{z^{2k+2}}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n])^(2)-(Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n])^(2) == (- 1)^(n + 1)* Sum[(- 1)^(k)*Divide[Subscript[s, k]*(n +Divide[1,2]),(z)^(2*k + 2)], {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [210 / 210]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.299038105676658, -0.7500000000000001]
+Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[s, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.35182282028742856, 0.20312500000000058]
+Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[s, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.50#Ex1 10.50#Ex1] || [[Item:Q3728|<math>\Wronskian@{\sphBesselJ{n}@{z},\sphBesselY{n}@{z}} = z^{-2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\sphBesselJ{n}@{z},\sphBesselY{n}@{z}} = z^{-2}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0, \realpart@@{((-(n+\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{SphericalBesselJ[n, z], SphericalBesselY[n, z]}, z] == (z)^(- 2)</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/10.50#Ex2 10.50#Ex2] || [[Item:Q3729|<math>\Wronskian@{\sphHankelh{1}{n}@{z},\sphHankelh{2}{n}@{z}} = -2iz^{-2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\sphHankelh{1}{n}@{z},\sphHankelh{2}{n}@{z}} = -2iz^{-2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{SphericalHankelH1[n, z], SphericalHankelH2[n, z]}, z] == - 2*I*(z)^(- 2)</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/10.50#Ex3 10.50#Ex3] || [[Item:Q3730|<math>\Wronskian@{\modsphBesseli{1}{n}@{z},\modsphBesseli{2}{n}@{z}} = (-1)^{n+1}z^{-2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\modsphBesseli{1}{n}@{z},\modsphBesseli{2}{n}@{z}} = (-1)^{n+1}z^{-2}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n], Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n]}, z] == (- 1)^(n + 1)* (z)^(- 2)</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.5000000000000001, 0.8660254037844386]
+Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.5000000000000001, -0.8660254037844386]
+Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.50#Ex4 10.50#Ex4] || [[Item:Q3731|<math>\Wronskian@{\modsphBesseli{1}{n}@{z},\modsphBesselK{n}@{z}} = \Wronskian@{\modsphBesseli{2}{n}@{z},\modsphBesselK{n}@{z}}\\</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\modsphBesseli{1}{n}@{z},\modsphBesselK{n}@{z}} = \Wronskian@{\modsphBesseli{2}{n}@{z},\modsphBesselK{n}@{z}}\\</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n], Sqrt[1/2 Pi /z] BesselK[n + 1/2, z]}, z] == Wronskian[{Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n], Sqrt[1/2 Pi /z] BesselK[n + 1/2, z]}, z]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.5384915109869794, 1.7026856201657974]
+Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-2.6544302063904848, -2.4451654315616667]
+Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.50#Ex4 10.50#Ex4] || [[Item:Q3731|<math>\Wronskian@{\modsphBesseli{2}{n}@{z},\modsphBesselK{n}@{z}}\\ = -\tfrac{1}{2}\pi z^{-2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\modsphBesseli{2}{n}@{z},\modsphBesselK{n}@{z}}\\ = -\tfrac{1}{2}\pi z^{-2}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n], Sqrt[1/2 Pi /z] BesselK[n + 1/2, z]}, z] == -Divide[1,2]*Pi*(z)^(- 2)</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.5161524079039588, -2.211692333258562]
+Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[7.686727830477982, 4.996906619076774]
+Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.50#Ex5 10.50#Ex5] || [[Item:Q3732|<math>\sphBesselJ{n+1}@{z}\sphBesselY{n}@{z}-\sphBesselJ{n}@{z}\sphBesselY{n+1}@{z} = z^{-2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselJ{n+1}@{z}\sphBesselY{n}@{z}-\sphBesselJ{n}@{z}\sphBesselY{n+1}@{z} = z^{-2}</syntaxhighlight> || <math>\realpart@@{(((n+1)+\frac{1}{2})+k+1)} > 0, \realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-(n+1)-\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-(n+1)-\frac{1}{2}))+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0, \realpart@@{((-(n+\frac{1}{2}))+k+1)} > 0, \realpart@@{((-((n+1)+\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselJ[n + 1, z]*SphericalBesselY[n, z]- SphericalBesselJ[n, z]*SphericalBesselY[n + 1, z] == (z)^(- 2)</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/10.50#Ex6 10.50#Ex6] || [[Item:Q3733|<math>\sphBesselJ{n+2}@{z}\sphBesselY{n}@{z}-\sphBesselJ{n}@{z}\sphBesselY{n+2}@{z} = (2n+3)z^{-3}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselJ{n+2}@{z}\sphBesselY{n}@{z}-\sphBesselJ{n}@{z}\sphBesselY{n+2}@{z} = (2n+3)z^{-3}</syntaxhighlight> || <math>\realpart@@{(((n+2)+\frac{1}{2})+k+1)} > 0, \realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-(n+2)-\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-(n+2)-\frac{1}{2}))+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0, \realpart@@{((-(n+\frac{1}{2}))+k+1)} > 0, \realpart@@{((-((n+2)+\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselJ[n + 2, z]*SphericalBesselY[n, z]- SphericalBesselJ[n, z]*SphericalBesselY[n + 2, z] == (2*n + 3)*(z)^(- 3)</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/10.50.E4 10.50.E4] || [[Item:Q3734|<math>\sphBesselJ{0}@{z}\sphBesselJ{n}@{z}+\sphBesselY{0}@{z}\sphBesselY{n}@{z} = \cos@{\tfrac{1}{2}n\pi}\sum_{k=0}^{\floor{n/2}}(-1)^{k}\frac{a_{2k}(n+\tfrac{1}{2})}{z^{2k+2}}+\sin@{\tfrac{1}{2}n\pi}\sum_{k=0}^{\floor{(n-1)/2}}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{2k+3}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselJ{0}@{z}\sphBesselJ{n}@{z}+\sphBesselY{0}@{z}\sphBesselY{n}@{z} = \cos@{\tfrac{1}{2}n\pi}\sum_{k=0}^{\floor{n/2}}(-1)^{k}\frac{a_{2k}(n+\tfrac{1}{2})}{z^{2k+2}}+\sin@{\tfrac{1}{2}n\pi}\sum_{k=0}^{\floor{(n-1)/2}}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{2k+3}}</syntaxhighlight> || <math>\realpart@@{((0+\frac{1}{2})+k+1)} > 0, \realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-0-\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-0-\frac{1}{2}))+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0, \realpart@@{((-(0+\frac{1}{2}))+k+1)} > 0, \realpart@@{((-(n+\frac{1}{2}))+k+1)} > 0, k \geq 1</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselJ[0, z]*SphericalBesselJ[n, z]+ SphericalBesselY[0, z]*SphericalBesselY[n, z] == Cos[Divide[1,2]*n*Pi]*Sum[(- 1)^(k)*Divide[Subscript[a, 2*k]*(n +Divide[1,2]),(z)^(2*k + 2)], {k, 0, Floor[n/2]}, GenerateConditions->None]+ Sin[Divide[1,2]*n*Pi]*Sum[(- 1)^(k)*Divide[Subscript[a, 2*k + 1]*(n +Divide[1,2]),(z)^(2*k + 3)], {k, 0, Floor[(n - 1)/2]}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Skipped - Because timed out
+|- style="background: #dfe6e9;"
+| [https://dlmf.nist.gov/10.51#Ex1 10.51#Ex1] || [[Item:Q3735|<math>f_{n-1}(z)+f_{n+1}(z) = ((2n+1)/z)f_{n}(z)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>f_{n-1}(z)+f_{n+1}(z) = ((2n+1)/z)f_{n}(z)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">f[n - 1](z)+ f[n + 1](z) = ((2*n + 1)/z)*f[n](z)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[f, n - 1][z]+ Subscript[f, n + 1][z] == ((2*n + 1)/z)*Subscript[f, n][z]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+|-
+| [https://dlmf.nist.gov/10.51#Ex5 10.51#Ex5] || [[Item:Q3739|<math>\left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{n+1}f_{n}(z)) = z^{n-m+1}f_{n-m}(z)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{n+1}f_{n}(z)) = z^{n-m+1}f_{n-m}(z)</syntaxhighlight> || <math>m = 0</math> || <syntaxhighlight lang=mathematica>(diff((1)/(z), z))^(m)*((z)^(n + 1)* f[n](z)) = (z)^(n - m + 1)* f[n - m](z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[Divide[1,z], z])^(m)*((z)^(n + 1)* Subscript[f, n][z]) == (z)^(n - m + 1)* Subscript[f, n - m][z]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [288 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.49999999999999994, -1.8660254037844388]
+Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, Plus[Times[-1, m], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.49999999999999994, -1.8660254037844388]
+Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, Plus[Times[-1, m], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.51#Ex6 10.51#Ex6] || [[Item:Q3740|<math>\left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{-n}f_{n}(z)) = (-1)^{m}z^{-n-m}f_{n+m}(z)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{-n}f_{n}(z)) = (-1)^{m}z^{-n-m}f_{n+m}(z)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(diff((1)/(z), z))^(m)*((z)^(- n)* f[n](z)) = (- 1)^(m)* (z)^(- n - m)* f[n + m](z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[Divide[1,z], z])^(m)*((z)^(- n)* Subscript[f, n][z]) == (- 1)^(m)* (z)^(- n - m)* Subscript[f, n + m][z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [288 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.366025403-.3660254033*I
+Test Values: {z = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, f[n+m] = 1/2*3^(1/2)+1/2*I, n = 1, m = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .9999999993-.9999999984*I
+Test Values: {z = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, f[n+m] = 1/2*3^(1/2)+1/2*I, n = 2, m = 3}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [288 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.1339745962155613, 0.49999999999999994]
+Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, Plus[m, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.3660254037844386, 0.36602540378443865]
+Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, Plus[m, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|- style="background: #dfe6e9;"
+| [https://dlmf.nist.gov/10.51#Ex7 10.51#Ex7] || [[Item:Q3741|<math>g_{n-1}(z)-g_{n+1}(z) = ((2n+1)/z)g_{n}(z)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>g_{n-1}(z)-g_{n+1}(z) = ((2n+1)/z)g_{n}(z)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">g[n - 1](z)- g[n + 1](z) = ((2*n + 1)/z)*g[n](z)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[g, n - 1][z]- Subscript[g, n + 1][z] == ((2*n + 1)/z)*Subscript[g, n][z]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+|-
+| [https://dlmf.nist.gov/10.51#Ex11 10.51#Ex11] || [[Item:Q3745|<math>\left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{n+1}g_{n}(z)) = z^{n-m+1}g_{n-m}(z)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{n+1}g_{n}(z)) = z^{n-m+1}g_{n-m}(z)</syntaxhighlight> || <math>m = 0</math> || <syntaxhighlight lang=mathematica>(diff((1)/(z), z))^(m)*((z)^(n + 1)* g[n](z)) = (z)^(n - m + 1)* g[n - m](z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[Divide[1,z], z])^(m)*((z)^(n + 1)* Subscript[g, n][z]) == (z)^(n - m + 1)* Subscript[g, n - m][z]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [288 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.49999999999999994, -1.8660254037844388]
+Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, Plus[Times[-1, m], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.49999999999999994, -1.8660254037844388]
+Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, Plus[Times[-1, m], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.51#Ex12 10.51#Ex12] || [[Item:Q3746|<math>\left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{-n}g_{n}(z)) = z^{-n-m}g_{n+m}(z)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{-n}g_{n}(z)) = z^{-n-m}g_{n+m}(z)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(diff((1)/(z), z))^(m)*((z)^(- n)* g[n](z)) = (z)^(- n - m)* g[n + m](z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[Divide[1,z], z])^(m)*((z)^(- n)* Subscript[g, n][z]) == (z)^(- n - m)* Subscript[g, n + m][z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [288 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .3660254028+1.366025403*I
+Test Values: {z = 1/2*3^(1/2)+1/2*I, g[n] = 1/2*3^(1/2)+1/2*I, g[n+m] = 1/2*3^(1/2)+1/2*I, n = 1, m = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .9999999987+.9999999996*I
+Test Values: {z = 1/2*3^(1/2)+1/2*I, g[n] = 1/2*3^(1/2)+1/2*I, g[n+m] = 1/2*3^(1/2)+1/2*I, n = 2, m = 3}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [288 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.8660254037844388, 0.49999999999999994]
+Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, Plus[m, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.3660254037844388, 1.3660254037844386]
+Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, Plus[m, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.53.E1 10.53.E1] || [[Item:Q3755|<math>\sphBesselJ{n}@{z} = z^{n}\sum_{k=0}^{\infty}\frac{(-\frac{1}{2}z^{2})^{k}}{k!(2n+2k+1)!!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselJ{n}@{z} = z^{n}\sum_{k=0}^{\infty}\frac{(-\frac{1}{2}z^{2})^{k}}{k!(2n+2k+1)!!}</syntaxhighlight> || <math>|z| < \infty, \realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselJ[n, z] == (z)^(n)* Sum[Divide[(-Divide[1,2]*(z)^(2))^(k),(k)!*(2*n + 2*k + 1)!!], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/10.53.E2 10.53.E2] || [[Item:Q3756|<math>\sphBesselY{n}@{z} = -\frac{1}{z^{n+1}}\sum_{k=0}^{n}\frac{(2n-2k-1)!!(\frac{1}{2}z^{2})^{k}}{k!}+\frac{(-1)^{n+1}}{z^{n+1}}\sum_{k=n+1}^{\infty}\frac{(-\frac{1}{2}z^{2})^{k}}{k!(2k-2n-1)!!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselY{n}@{z} = -\frac{1}{z^{n+1}}\sum_{k=0}^{n}\frac{(2n-2k-1)!!(\frac{1}{2}z^{2})^{k}}{k!}+\frac{(-1)^{n+1}}{z^{n+1}}\sum_{k=n+1}^{\infty}\frac{(-\frac{1}{2}z^{2})^{k}}{k!(2k-2n-1)!!}</syntaxhighlight> || <math>0 < |z|, |z| < \infty., \realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-(n+\frac{1}{2}))+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselY[n, z] == -Divide[1,(z)^(n + 1)]*Sum[Divide[(2*n - 2*k - 1)!!*(Divide[1,2]*(z)^(2))^(k),(k)!], {k, 0, n}, GenerateConditions->None]+Divide[(- 1)^(n + 1),(z)^(n + 1)]*Sum[Divide[(-Divide[1,2]*(z)^(2))^(k),(k)!*(2*k - 2*n - 1)!!], {k, n + 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/10.53.E3 10.53.E3] || [[Item:Q3757|<math>\modsphBesseli{1}{n}@{z} = z^{n}\sum_{k=0}^{\infty}\frac{(\frac{1}{2}z^{2})^{k}}{k!(2n+2k+1)!!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesseli{1}{n}@{z} = z^{n}\sum_{k=0}^{\infty}\frac{(\frac{1}{2}z^{2})^{k}}{k!(2n+2k+1)!!}</syntaxhighlight> || <math>|z| < \infty, \realpart@@{((n+\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n] == (z)^(n)* Sum[Divide[(Divide[1,2]*(z)^(2))^(k),(k)!*(2*n + 2*k + 1)!!], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.06771919180965624, -0.29579816936516184]
+Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.4498252419402129, -0.19064547195046921]
+Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.53.E4 10.53.E4] || [[Item:Q3758|<math>\modsphBesseli{2}{n}@{z} = \frac{(-1)^{n}}{z^{n+1}}\sum_{k=0}^{n}\frac{(2n-2k-1)!!(-\frac{1}{2}z^{2})^{k}}{k!}+\frac{1}{z^{n+1}}\sum_{k=n+1}^{\infty}\frac{(\frac{1}{2}z^{2})^{k}}{k!(2k-2n-1)!!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesseli{2}{n}@{z} = \frac{(-1)^{n}}{z^{n+1}}\sum_{k=0}^{n}\frac{(2n-2k-1)!!(-\frac{1}{2}z^{2})^{k}}{k!}+\frac{1}{z^{n+1}}\sum_{k=n+1}^{\infty}\frac{(\frac{1}{2}z^{2})^{k}}{k!(2k-2n-1)!!}</syntaxhighlight> || <math>0 < |z|, |z| < \infty., \realpart@@{((-n-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n + 1/2), n] == Divide[(- 1)^(n),(z)^(n + 1)]*Sum[Divide[(2*n - 2*k - 1)!!*(-Divide[1,2]*(z)^(2))^(k),(k)!], {k, 0, n}, GenerateConditions->None]+Divide[1,(z)^(n + 1)]*Sum[Divide[(Divide[1,2]*(z)^(2))^(k),(k)!*(2*k - 2*n - 1)!!], {k, n + 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.4141971914072808, -0.8850762711170854]
+Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.1065867555175597, 2.456957013551954]
+Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.54.E1 10.54.E1] || [[Item:Q3759|<math>\sphBesselJ{n}@{z} = \frac{z^{n}}{2^{n+1}n!}\int_{0}^{\pi}\cos@{z\cos@@{\theta}}(\sin@@{\theta})^{2n+1}\diff{\theta}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselJ{n}@{z} = \frac{z^{n}}{2^{n+1}n!}\int_{0}^{\pi}\cos@{z\cos@@{\theta}}(\sin@@{\theta})^{2n+1}\diff{\theta}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselJ[n, z] == Divide[(z)^(n),(2)^(n + 1)* (n)!]*Integrate[Cos[z*Cos[\[Theta]]]*(Sin[\[Theta]])^(2*n + 1), {\[Theta], 0, Pi}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/10.54.E2 10.54.E2] || [[Item:Q3760|<math>\sphBesselJ{n}@{z} = \frac{(-i)^{n}}{2}\int_{0}^{\pi}e^{iz\cos@@{\theta}}\assLegendreP[]{n}@{\cos@@{\theta}}\sin@@{\theta}\diff{\theta}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselJ{n}@{z} = \frac{(-i)^{n}}{2}\int_{0}^{\pi}e^{iz\cos@@{\theta}}\assLegendreP[]{n}@{\cos@@{\theta}}\sin@@{\theta}\diff{\theta}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselJ[n, z] == Divide[(- I)^(n),2]*Integrate[Exp[I*z*Cos[\[Theta]]]*LegendreP[n, 0, 3, Cos[\[Theta]]]*Sin[\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Aborted || - || Successful [Tested: 21]
+|-
+| [https://dlmf.nist.gov/10.54.E3 10.54.E3] || [[Item:Q3761|<math>\modsphBesselK{n}@{z} = \frac{\pi}{2}\int_{1}^{\infty}e^{-zt}\assLegendreP[]{n}@{t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesselK{n}@{z} = \frac{\pi}{2}\int_{1}^{\infty}e^{-zt}\assLegendreP[]{n}@{t}\diff{t}</syntaxhighlight> || <math>|\phase@@{z}| < \tfrac{1}{2}\pi.</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[1/2 Pi /z] BesselK[n + 1/2, z] == Divide[Pi,2]*Integrate[Exp[- z*t]*LegendreP[n, 0, 3, t], {t, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Aborted || - || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/10.54.E4 10.54.E4] || [[Item:Q3762|<math>\sphBesselJ{n}@{z} = \frac{(-i)^{n+1}}{2\pi}\int_{i\infty}^{(-1+,1+)}e^{izt}\assLegendreQ[]{n}@{t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselJ{n}@{z} = \frac{(-i)^{n+1}}{2\pi}\int_{i\infty}^{(-1+,1+)}e^{izt}\assLegendreQ[]{n}@{t}\diff{t}</syntaxhighlight> || <math>|\phase@@{z}| < \tfrac{1}{2}\pi., \realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselJ[n, z] == Divide[(- I)^(n + 1),2*Pi]*Integrate[Exp[I*z*t]*LegendreQ[n, 0, 3, t], {t, I*Infinity, (- 1 + , 1 +)}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Error
+|-
+| [https://dlmf.nist.gov/10.54#Ex1 10.54#Ex1] || [[Item:Q3763|<math>\sphHankelh{1}{n}@{z} = \frac{(-i)^{n+1}}{\pi}\int_{i\infty}^{(1+)}e^{izt}\assLegendreQ[]{n}@{t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphHankelh{1}{n}@{z} = \frac{(-i)^{n+1}}{\pi}\int_{i\infty}^{(1+)}e^{izt}\assLegendreQ[]{n}@{t}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalHankelH1[n, z] == Divide[(- I)^(n + 1),Pi]*Integrate[Exp[I*z*t]*LegendreQ[n, 0, 3, t], {t, I*Infinity, (1 +)}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Error
+|-
+| [https://dlmf.nist.gov/10.54#Ex2 10.54#Ex2] || [[Item:Q3764|<math>\sphHankelh{2}{n}@{z} = \frac{(-i)^{n+1}}{\pi}\int_{i\infty}^{(-1+)}e^{izt}\assLegendreQ[]{n}@{t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphHankelh{2}{n}@{z} = \frac{(-i)^{n+1}}{\pi}\int_{i\infty}^{(-1+)}e^{izt}\assLegendreQ[]{n}@{t}\diff{t}</syntaxhighlight> || <math>|\phase@@{z}| < \tfrac{1}{2}\pi.</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalHankelH2[n, z] == Divide[(- I)^(n + 1),Pi]*Integrate[Exp[I*z*t]*LegendreQ[n, 0, 3, t], {t, I*Infinity, (- 1 +)}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Error
+|-
+| [https://dlmf.nist.gov/10.56.E1 10.56.E1] || [[Item:Q3765|<math>\frac{\cos@@{\sqrt{z^{2}-2zt}}}{z} = \frac{\cos@@{z}}{z}+\sum_{n=1}^{\infty}\frac{t^{n}}{n!}\sphBesselJ{n-1}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\cos@@{\sqrt{z^{2}-2zt}}}{z} = \frac{\cos@@{z}}{z}+\sum_{n=1}^{\infty}\frac{t^{n}}{n!}\sphBesselJ{n-1}@{z}</syntaxhighlight> || <math>\realpart@@{(((n-1)+\frac{1}{2})+k+1)} > 0, \realpart@@{((-(n-1)-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-(n-1)-\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Cos[Sqrt[(z)^(2)- 2*z*t]],z] == Divide[Cos[z],z]+ Sum[Divide[(t)^(n),(n)!]*SphericalBesselJ[n - 1, z], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-1.0653161526495918, 0.32810386977400907], Times[-1.0, NSum[Times[Power[-1.5, n], Power[Factorial[n], -1], SphericalBesselJ[Plus[-1, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
+Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[-1.8246723112251149, 0.13108435615091096], Times[-1.0, NSum[Times[Power[-1.5, n], Power[Factorial[n], -1], SphericalBesselJ[Plus[-1, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]
+Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.56.E2 10.56.E2] || [[Item:Q3766|<math>\frac{\sin@@{\sqrt{z^{2}-2zt}}}{z} = \frac{\sin@@{z}}{z}+\sum_{n=1}^{\infty}\frac{t^{n}}{n!}\sphBesselY{n-1}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\sin@@{\sqrt{z^{2}-2zt}}}{z} = \frac{\sin@@{z}}{z}+\sum_{n=1}^{\infty}\frac{t^{n}}{n!}\sphBesselY{n-1}@{z}</syntaxhighlight> || <math>\realpart@@{(((n-1)+\frac{1}{2})+k+1)} > 0, \realpart@@{((-((n-1)+\frac{1}{2}))+k+1)} > 0, \realpart@@{((-(n-1)-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Sin[Sqrt[(z)^(2)- 2*z*t]],z] == Divide[Sin[z],z]+ Sum[Divide[(t)^(n),(n)!]*SphericalBesselY[n - 1, z], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Aborted || - || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/10.56.E3 10.56.E3] || [[Item:Q3767|<math>\frac{\cosh@@{\sqrt{z^{2}+2izt}}}{z} = \frac{\cosh@@{z}}{z}+\sum_{n=1}^{\infty}\frac{(it)^{n}}{n!}\modsphBesseli{1}{n-1}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\cosh@@{\sqrt{z^{2}+2izt}}}{z} = \frac{\cosh@@{z}}{z}+\sum_{n=1}^{\infty}\frac{(it)^{n}}{n!}\modsphBesseli{1}{n-1}@{z}</syntaxhighlight> || <math>\realpart@@{(((n-1)+\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Cosh[Sqrt[(z)^(2)+ 2*I*z*t]],z] == Divide[Cosh[z],z]+ Sum[Divide[(I*t)^(n),(n)!]*Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(n - 1 + 1/2), n - 1], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.13108435615091052, -1.8246723112251153], Times[-1.0, NSum[Times[Power[Complex[0.0, -1.5], n], Power[Power[E, Times[Complex[0, Rational[-1, 6]], Pi]], Rational[1, 2]], Power[Times[Rational[1, 2], Pi], Rational[1, 2]], BesselI[Plus[Rational[-1, 2], n], Plus[-1, n]], Power[Factorial[n], -1]]
+Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.022834987510423566, -1.7127448295681993], Times[-1.0, NSum[Times[Power[Complex[0.0, -1.5], n], Power[Power[E, Times[Complex[0, Rational[-2, 3]], Pi]], Rational[1, 2]], Power[Times[Rational[1, 2], Pi], Rational[1, 2]], BesselI[Plus[Rational[-1, 2], n], Plus[-1, n]], Power[Factorial[n], -1]]
+Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.56.E4 10.56.E4] || [[Item:Q3768|<math>\frac{\sinh@@{\sqrt{z^{2}+2izt}}}{z} = \frac{\sinh@@{z}}{z}+\sum_{n=1}^{\infty}\frac{(it)^{n}}{n!}\modsphBesseli{2}{n-1}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\sinh@@{\sqrt{z^{2}+2izt}}}{z} = \frac{\sinh@@{z}}{z}+\sum_{n=1}^{\infty}\frac{(it)^{n}}{n!}\modsphBesseli{2}{n-1}@{z}</syntaxhighlight> || <math>\realpart@@{((-(n-1)-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Sinh[Sqrt[(z)^(2)+ 2*I*z*t]],z] == Divide[Sinh[z],z]+ Sum[Divide[(I*t)^(n),(n)!]*Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n - 1 + 1/2), n - 1], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.12983798012989667, -2.1935922908985273], Times[-1.0, NSum[Times[Power[Complex[0.0, -1.5], n], Power[Power[E, Times[Complex[0, Rational[-1, 6]], Pi]], Rational[1, 2]], Power[Times[Rational[1, 2], Pi], Rational[1, 2]], BesselI[Plus[Rational[1, 2], Times[-1, n]], Plus[-1, n]], Power[Factorial[n], -1]]
+Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[-1.4886830119296848, -1.839102010336905], Times[-1.0, NSum[Times[Power[Complex[0.0, -1.5], n], Power[Power[E, Times[Complex[0, Rational[-2, 3]], Pi]], Rational[1, 2]], Power[Times[Rational[1, 2], Pi], Rational[1, 2]], BesselI[Plus[Rational[1, 2], Times[-1, n]], Plus[-1, n]], Power[Factorial[n], -1]]
+Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.56.E5 10.56.E5] || [[Item:Q3769|<math>\frac{\exp@{-\sqrt{z^{2}+2izt}}}{z} = \frac{e^{-z}}{z}+\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{(-it)^{n}}{n!}\modsphBesselK{n-1}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\exp@{-\sqrt{z^{2}+2izt}}}{z} = \frac{e^{-z}}{z}+\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{(-it)^{n}}{n!}\modsphBesselK{n-1}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Exp[-Sqrt[(z)^(2)+ 2*I*z*t]],z] == Divide[Exp[- z],z]+Divide[2,Pi]*Sum[Divide[(- I*t)^(n),(n)!]*Sqrt[1/2 Pi /z] BesselK[n - 1 + 1/2, z], {n, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Aborted || - || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/10.57.E1 10.57.E1] || [[Item:Q3770|<math>\sphBesselJ{n}'@{(n+\tfrac{1}{2})z} = \frac{\pi^{\frac{1}{2}}}{((2n+1)z)^{\frac{1}{2}}}\BesselJ{n+\frac{1}{2}}'@{(n+\tfrac{1}{2})z}-\frac{\pi^{\frac{1}{2}}}{((2n+1)z)^{\frac{3}{2}}}\BesselJ{n+\frac{1}{2}}@{(n+\tfrac{1}{2})z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselJ{n}'@{(n+\tfrac{1}{2})z} = \frac{\pi^{\frac{1}{2}}}{((2n+1)z)^{\frac{1}{2}}}\BesselJ{n+\frac{1}{2}}'@{(n+\tfrac{1}{2})z}-\frac{\pi^{\frac{1}{2}}}{((2n+1)z)^{\frac{3}{2}}}\BesselJ{n+\frac{1}{2}}@{(n+\tfrac{1}{2})z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[SphericalBesselJ[n, (n +Divide[1,2])*z], {(n +Divide[1,2])*z, 1}] == Divide[(Pi)^(Divide[1,2]),((2*n + 1)*z)^(Divide[1,2])]*D[BesselJ[n +Divide[1,2], (n +Divide[1,2])*z], {(n +Divide[1,2])*z, 1}]-Divide[(Pi)^(Divide[1,2]),((2*n + 1)*z)^(Divide[3,2])]*BesselJ[n +Divide[1,2], (n +Divide[1,2])*z]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.14653389603833195, -0.029869009956249915], Times[Complex[-0.988457695936884, 0.2648564413786163], D[Complex[0.36567703182522004, 0.24184221354059504]
+Test Values: {Complex[1.299038105676658, 0.7499999999999999], 1.0}]], D[Complex[0.425509744388485, 0.14219887983348967], {Complex[1.299038105676658, 0.7499999999999999], 1.0}]], {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.06710374092328811, 0.007963502819859997], Times[Complex[-0.7656560389588212, 0.20515691731902835], D[Complex[0.2637838125883578, 0.3348231997381719]
+Test Values: {Complex[2.165063509461097, 1.2499999999999998], 1.0}]], D[Complex[0.27065896459303473, 0.20224233103375913], {Complex[2.165063509461097, 1.2499999999999998], 1.0}]], {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.60.E1 10.60.E1] || [[Item:Q3776|<math>\frac{\cos@@{w}}{w} = -\sum_{n=0}^{\infty}(2n+1)\sphBesselJ{n}@{v}\sphBesselY{n}@{u}\assLegendreP[]{n}@{\cos@@{\alpha}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\cos@@{w}}{w} = -\sum_{n=0}^{\infty}(2n+1)\sphBesselJ{n}@{v}\sphBesselY{n}@{u}\assLegendreP[]{n}@{\cos@@{\alpha}}</syntaxhighlight> || <math>|ve^{+ i\alpha}| < |u|, |ve^{- i\alpha}| < |u|, \realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0, \realpart@@{((-(n+\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Cos[w],w] == - Sum[(2*n + 1)*SphericalBesselJ[n, v]*SphericalBesselY[n, u]*LegendreP[n, 0, 3, Cos[\[Alpha]]], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.43419403794642014, -0.7090399040477617], NSum[Times[Plus[1, Times[2, n]], LegendreP[n, 0.0707372016677029], SphericalBesselJ[n, -0.5], SphericalBesselY[n, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
+Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, -0.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.43419403794642014, -0.7090399040477617], NSum[Times[Plus[1, Times[2, n]], LegendreP[n, 0.8775825618903728], SphericalBesselJ[n, -0.5], SphericalBesselY[n, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
+Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, -0.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.60.E2 10.60.E2] || [[Item:Q3777|<math>\frac{\sin@@{w}}{w} = \sum_{n=0}^{\infty}(2n+1)\sphBesselJ{n}@{v}\sphBesselJ{n}@{u}\assLegendreP[]{n}@{\cos@@{\alpha}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\sin@@{w}}{w} = \sum_{n=0}^{\infty}(2n+1)\sphBesselJ{n}@{v}\sphBesselJ{n}@{u}\assLegendreP[]{n}@{\cos@@{\alpha}}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Sin[w],w] == Sum[(2*n + 1)*SphericalBesselJ[n, v]*SphericalBesselJ[n, u]*LegendreP[n, 0, 3, Cos[\[Alpha]]], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.912697022466604, -0.13712305377128448], Times[-1.0, NSum[Times[Plus[1, Times[2, n]], LegendreP[n, 0.0707372016677029], Power[SphericalBesselJ[n, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2]]
+Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.912697022466604, -0.13712305377128448], Times[-1.0, NSum[Times[Plus[1, Times[2, n]], LegendreP[n, 0.8775825618903728], Power[SphericalBesselJ[n, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2]]
+Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.60.E3 10.60.E3] || [[Item:Q3778|<math>\frac{e^{-w}}{w} = \frac{2}{\pi}\sum_{n=0}^{\infty}(2n+1)\modsphBesseli{1}{n}@{v}\modsphBesselK{n}@{u}\assLegendreP[]{n}@{\cos@@{\alpha}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{e^{-w}}{w} = \frac{2}{\pi}\sum_{n=0}^{\infty}(2n+1)\modsphBesseli{1}{n}@{v}\modsphBesselK{n}@{u}\assLegendreP[]{n}@{\cos@@{\alpha}}</syntaxhighlight> || <math>|ve^{+ i\alpha}| < |u|, |ve^{- i\alpha}| < |u|, \realpart@@{((n+\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Exp[- w],w] == Divide[2,Pi]*Sum[(2*n + 1)*Sqrt[Divide[Pi, v]/2] BesselI[(-1)^(1-1)*(n + 1/2), n]*Sqrt[1/2 Pi /u] BesselK[n + 1/2, u]*LegendreP[n, 0, 3, Cos[\[Alpha]]], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/10.60.E4 10.60.E4] || [[Item:Q3779|<math>\sphBesselJ{n}@{2z} = -n!z^{n+1}\sum_{k=0}^{n}\frac{2n-2k+1}{k!(2n-k+1)!}\sphBesselJ{n-k}@{z}\sphBesselY{n-k}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselJ{n}@{2z} = -n!z^{n+1}\sum_{k=0}^{n}\frac{2n-2k+1}{k!(2n-k+1)!}\sphBesselJ{n-k}@{z}\sphBesselY{n-k}@{z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{(((n-k)+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(n-k)-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0, \realpart@@{((-(-(n-k)-\frac{1}{2}))+k+1)} > 0, \realpart@@{((-((n-k)+\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselJ[n, 2*z] == - (n)!*(z)^(n + 1)* Sum[Divide[2*n - 2*k + 1,(k)!*(2*n - k + 1)!]*SphericalBesselJ[n - k, z]*SphericalBesselY[n - k, z], {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Aborted || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[0.3456774997623559, Times[2.25, Plus[Times[-2.0, DifferenceRoot[Function[{, }
+Test Values: {Equal[Plus[Times[Plus[-3, Times[-2, ], Times[2, 1]], Plus[-1, Times[-1, ], Times[2, 1]], Plus[Times[-1, ], Times[2, 1]], Plus[1, Times[-1, ], Times[2, 1]], Power[1.5, 2], []], Times[-1, Plus[-1, Times[-1, ], Times[2, 1]], Plus[Times[-1, ], Times[2, 1]], Plus[Times[-3, ], Times[-14, Power[, 2]], Times[-20, Power[, 3]], Times[-8, Power[, 4]], Times[14, , 1], Times[40, Power[, 2], 1], Times[24, Power[, 3], 1], Times[-20, , Power[1, 2]], Times[-24, Power[, 2], Power[1, 2]], Times[8, , Power[1, 3]], Times[-3, Power[1.5, 2]], Times[2, , Power[1.5, 2]], Times[4, Power[, 2], Power[1.5, 2]], Times[-4, 1, Power[1.5, 2]], Times[-8, , 1, Power[1.5, 2]], Times[4, Power[1, 2], Power[1.5, 2]]], [Plus[1, ]]], Times[, Plus[1, , Times[-2, 1]], Plus[3, Times[4, ], Times[4, , 1], Times[-4, Power[1, 2]]], Plus[3, Times[8, ], Times[4, Power[<syntaxhighlight lang=mathematica>Result: Plus[0.2986374970757335, Times[6.75, Plus[Times[-2.0, DifferenceRoot[Function[{, }
+Test Values: {Equal[Plus[Times[Plus[-3, Times[-2, ], Times[2, 2]], Plus[-1, Times[-1, ], Times[2, 2]], Plus[Times[-1, ], Times[2, 2]], Plus[1, Times[-1, ], Times[2, 2]], Power[1.5, 2], []], Times[-1, Plus[-1, Times[-1, ], Times[2, 2]], Plus[Times[-1, ], Times[2, 2]], Plus[Times[-3, ], Times[-14, Power[, 2]], Times[-20, Power[, 3]], Times[-8, Power[, 4]], Times[14, , 2], Times[40, Power[, 2], 2], Times[24, Power[, 3], 2], Times[-20, , Power[2, 2]], Times[-24, Power[, 2], Power[2, 2]], Times[8, , Power[2, 3]], Times[-3, Power[1.5, 2]], Times[2, , Power[1.5, 2]], Times[4, Power[, 2], Power[1.5, 2]], Times[-4, 2, Power[1.5, 2]], Times[-8, , 2, Power[1.5, 2]], Times[4, Power[2, 2], Power[1.5, 2]]], [Plus[1, ]]], Times[, Plus[1, , Times[-2, 2]], Plus[3, Times[4, ], Times[4, , 2], Times[-4, Power[2, 2]]], Plus[3, Times[8, ], Times[4, Power[, 2]], Times[-8, 2], Times[-8, , 2], Times[4, Power[2, 2]], Times[-1, Power[1.5, 2]]], [Plus[2, ]]], Times[-1, , Plus[1, ], Plus[9, Times[39, ], Times[58, Power[, 2]], Times[36, Power[, 3]], Times[8, Power[, 4]], Times[-48, 2], Times[-146, , 2], Times[-136, Power[, 2], 2], Times[-40, Power[, 3], 2], Times[88, Power[2, 2]], Times[164, , Power[2, 2]], Times[72, Power[, 2], Power[2, 2]], Times[-64, Power[2, 3]], Times[-56, , Power[2, 3]], Times[16, Power[2, 4]], Times[-5, Power[1.5, 2]], Times[-10, , Power[1.5, 2]], Times[-4, Power[, 2], Power[1.5, 2]], Times[12, 2, Power[1.5, 2]], Times[8, , 2, Power[1.5, 2]], Times[-4, Power[2, 2], Power[1.5, 2]]], [Plus[3, ]]], Times[-1, , Plus[1, ], Plus[2, ], Plus[1, Times[2, ], Times[-2, 2]], Power[1.5, 2], [Plus[4, ]]]], 0], Equal[[1], 0], Equal[[2], Times[-1, Plus[-1, Times[-2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[Plus[-1, 2], 1.5], SphericalBesselY[Plus[-1, 2], 1.5]]], Equal[[3], Plus[Times[-1, Plus[-1, Times[-2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[Plus[-1, 2], 1.5], SphericalBesselY[Plus[-1, 2], 1.5]], Times[-2, Plus[-1, Times[-2, 2]], 2, Power[1.5, -2], Power[Factorial[Plus[1, Times[2, 2]]], -1], Plus[Times[-1, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselJ[2, 1.5]]], Plus[Times[-1, SphericalBesselY[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselY[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselY[2, 1.5]]]]]], Equal[[4], Plus[Times[-1, Plus[-1, Times[-2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[Plus[-1, 2], 1.5], SphericalBesselY[Plus[-1, 2], 1.5]], Times[-2, Plus[-1, Times[-2, 2]], 2, Power[1.5, -2], Power[Factorial[Plus[1, Times[2, 2]]], -1], Plus[Times[-1, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselJ[2, 1.5]]], Plus[Times[-1, SphericalBesselY[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselY[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselY[2, 1.5]]]], Times[Rational[1, 12], Power[1.5, -2], Plus[Times[12, Plus[-1, Times[-2, 2]], 2, Plus[-1, Times[2, 2]], 1.5, Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[Plus[-1, 2], 1.5]], Times[-12, Plus[-1, Times[-2, 2]], 2, Plus[-3, Times[2, 2]], Plus[-1, Times[2, 2]], Power[1.5, -1], Power[Factorial[Plus[1, Times[2, 2]]], -1], Plus[Times[-1, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselJ[2, 1.5]]]]], Plus[Times[-1, 1.5, SphericalBesselY[Plus[-1, 2], 1.5]], Times[-3, Power[1.5, -1], Plus[Times[-1, SphericalBesselY[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselY[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselY[2, 1.5]]]], Times[2, 2, Power[1.5, -1], Plus[Times[-1, SphericalBesselY[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselY[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselY[2, 1.5]]]]]]]]}]][3.0]], Times[5.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[-3, Times[-2, ], Times[2, 2]], Plus[-1, Times[-1, ], Times[2, 2]], Plus[Times[-1, ], Times[2, 2]], Plus[1, Times[-1, ], Times[2, 2]], Power[1.5, 2], []], Times[-1, Plus[-1, Times[-1, ], Times[2, 2]], Plus[Times[-1, ], Times[2, 2]], Plus[-3, Times[-17, ], Times[-34, Power[, 2]], Times[-28, Power[, 3]], Times[-8, Power[, 4]], Times[14, 2], Times[54, , 2], Times[64, Power[, 2], 2], Times[24, Power[, 3], 2], Times[-20, Power[2, 2]], Times[-44, , Power[2, 2]], Times[-24, Power[, 2], Power[2, 2]], Times[8, Power[2, 3]], Times[8, , Power[2, 3]], Times[-2, Power[1.5, 2]], Times[4, , Power[1.5, 2]], Times[4, Power[, 2], Power[1.5, 2]], Times[-6, 2, Power[1.5, 2]], Times[-8, , 2, Power[1.5, 2]], Times[4, Power[2, 2], Power[1.5, 2]]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[-1, Times[-1, ], 2], Plus[3, Times[2, 2]], Plus[-1, Times[-1, ], Times[2, 2]], Plus[3, Times[8, ], Times[4, Power[, 2]], Times[-8, 2], Times[-8, , 2], Times[4, Power[2, 2]], Times[-1, Power[1.5, 2]]], [Plus[2, ]]], Times[-1, Plus[1, ], Plus[2, ], Plus[9, Times[39, ], Times[58, Power[, 2]], Times[36, Power[, 3]], Times[8, Power[, 4]], Times[-48, 2], Times[-146, , 2], Times[-136, Power[, 2], 2], Times[-40, Power[, 3], 2], Times[88, Power[2, 2]], Times[164, , Power[2, 2]], Times[72, Power[, 2], Power[2, 2]], Times[-64, Power[2, 3]], Times[-56, , Power[2, 3]], Times[16, Power[2, 4]], Times[-6, Power[1.5, 2]], Times[-12, , Power[1.5, 2]], Times[-4, Power[, 2], Power[1.5, 2]], Times[14, 2, Power[1.5, 2]], Times[8, , 2, Power[1.5, 2]], Times[-4, Power[2, 2], Power[1.5, 2]]], [Plus[3, ]]], Times[Plus[1, ], Plus[2, ], Plus[3, ], Plus[-1, Times[-2, ], Times[2, 2]], Power[1.5, 2], [Plus[4, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[2, 1.5], SphericalBesselY[2, 1.5]]], Equal[[2], Plus[Times[-1, Plus[-1, Times[-2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[Plus[-1, 2], 1.5], SphericalBesselY[Plus[-1, 2], 1.5]], Times[Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[2, 1.5], SphericalBesselY[2, 1.5]]]], Equal[[3], Plus[Times[-1, Plus[-1, Times[-2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[Plus[-1, 2], 1.5], SphericalBesselY[Plus[-1, 2], 1.5]], Times[Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[2, 1.5], SphericalBesselY[2, 1.5]], Times[Rational[1, 2], Power[1.5, -2], Plus[Times[2, Plus[-1, Times[-2, 2]], 2, Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[Plus[-1, 2], 1.5]], Times[-4, Plus[-1, Times[-2, 2]], Power[2, 2], Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[Plus[-1, 2], 1.5]], Times[-2, 2, 1.5, Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[2, 1.5]], Times[-4, Power[2, 2], 1.5, Power[Factorial[Plus[1, Times[2, 2]]], -1], SphericalBesselJ[2, 1.5]]], Plus[Times[-1, SphericalBesselY[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselY[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselY[2, 1.5]]]]]]}]][3.0]]]]], {Rule[n, 2], Rule[z, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.60.E5 10.60.E5] || [[Item:Q3780|<math>\sphBesselY{n}@{2z} = n!z^{n+1}\sum_{k=0}^{n}\frac{n-k+\frac{1}{2}}{k!(2n-k+1)!}{\left(\sphBesselJ{n-k}^{2}@{z}-\sphBesselY{n-k}^{2}@{z}\right)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselY{n}@{2z} = n!z^{n+1}\sum_{k=0}^{n}\frac{n-k+\frac{1}{2}}{k!(2n-k+1)!}{\left(\sphBesselJ{n-k}^{2}@{z}-\sphBesselY{n-k}^{2}@{z}\right)}</syntaxhighlight> || <math>\realpart@@{(((n-k)+\frac{1}{2})+k+1)} > 0, \realpart@@{((-(n-k)-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-(n-k)-\frac{1}{2}))+k+1)} > 0, \realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-(n+\frac{1}{2}))+k+1)} > 0, \realpart@@{((-((n-k)+\frac{1}{2}))+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselY[n, 2*z] == (n)!*(z)^(n + 1)* Sum[Divide[n - k +Divide[1,2],(k)!*(2*n - k + 1)!]*((SphericalBesselJ[n - k, z])^(2)- (SphericalBesselY[n - k, z])^(2)), {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Aborted || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[0.06295916360231597, Times[-1.125, Plus[Times[-2.0, DifferenceRoot[Function[{, }
+Test Values: {Equal[Plus[Times[Plus[-3, Times[-2, ], Times[2, 1]], Plus[-1, Times[-1, ], Times[2, 1]], Plus[Times[-1, ], Times[2, 1]], Plus[1, Times[-1, ], Times[2, 1]], Power[1.5, 2], []], Times[-1, Plus[-1, Times[-1, ], Times[2, 1]], Plus[Times[-1, ], Times[2, 1]], Plus[Times[-3, ], Times[-14, Power[, 2]], Times[-20, Power[, 3]], Times[-8, Power[, 4]], Times[14, , 1], Times[40, Power[, 2], 1], Times[24, Power[, 3], 1], Times[-20, , Power[1, 2]], Times[-24, Power[, 2], Power[1, 2]], Times[8, , Power[1, 3]], Times[-3, Power[1.5, 2]], Times[2, , Power[1.5, 2]], Times[4, Power[, 2], Power[1.5, 2]], Times[-4, 1, Power[1.5, 2]], Times[-8, , 1, Power[1.5, 2]], Times[4, Power[1, 2], Power[1.5, 2]]], [Plus[1, ]]], Times[, Plus[1, , Times[-2, 1]], Plus[3, Times[4, ], Times[4, , 1], Times[-4, Power[1, 2]]], Plus[3, Times[8, ], Times[4, Pow<syntaxhighlight lang=mathematica>Result: Plus[-0.26703833526449916, Times[-3.375, Plus[Times[-2.0, DifferenceRoot[Function[{, }
+Test Values: {Equal[Plus[Times[Plus[-3, Times[-2, ], Times[2, 2]], Plus[-1, Times[-1, ], Times[2, 2]], Plus[Times[-1, ], Times[2, 2]], Plus[1, Times[-1, ], Times[2, 2]], Power[1.5, 2], []], Times[-1, Plus[-1, Times[-1, ], Times[2, 2]], Plus[Times[-1, ], Times[2, 2]], Plus[Times[-3, ], Times[-14, Power[, 2]], Times[-20, Power[, 3]], Times[-8, Power[, 4]], Times[14, , 2], Times[40, Power[, 2], 2], Times[24, Power[, 3], 2], Times[-20, , Power[2, 2]], Times[-24, Power[, 2], Power[2, 2]], Times[8, , Power[2, 3]], Times[-3, Power[1.5, 2]], Times[2, , Power[1.5, 2]], Times[4, Power[, 2], Power[1.5, 2]], Times[-4, 2, Power[1.5, 2]], Times[-8, , 2, Power[1.5, 2]], Times[4, Power[2, 2], Power[1.5, 2]]], [Plus[1, ]]], Times[, Plus[1, , Times[-2, 2]], Plus[3, Times[4, ], Times[4, , 2], Times[-4, Power[2, 2]]], Plus[3, Times[8, ], Times[4, Power[, 2]], Times[-8, 2], Times[-8, , 2], Times[4, Power[2, 2]], Times[-1, Power[1.5, 2]]], [Plus[2, ]]], Times[-1, , Plus[1, ], Plus[9, Times[39, ], Times[58, Power[, 2]], Times[36, Power[, 3]], Times[8, Power[, 4]], Times[-48, 2], Times[-146, , 2], Times[-136, Power[, 2], 2], Times[-40, Power[, 3], 2], Times[88, Power[2, 2]], Times[164, , Power[2, 2]], Times[72, Power[, 2], Power[2, 2]], Times[-64, Power[2, 3]], Times[-56, , Power[2, 3]], Times[16, Power[2, 4]], Times[-5, Power[1.5, 2]], Times[-10, , Power[1.5, 2]], Times[-4, Power[, 2], Power[1.5, 2]], Times[12, 2, Power[1.5, 2]], Times[8, , 2, Power[1.5, 2]], Times[-4, Power[2, 2], Power[1.5, 2]]], [Plus[3, ]]], Times[-1, , Plus[1, ], Plus[2, ], Plus[1, Times[2, ], Times[-2, 2]], Power[1.5, 2], [Plus[4, ]]]], 0], Equal[[1], 0], Equal[[2], Times[-1, Plus[-1, Times[-2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselJ[Plus[-1, 2], 1.5], 2]]], Equal[[3], Plus[Times[-1, Plus[-1, Times[-2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselJ[Plus[-1, 2], 1.5], 2]], Times[-2, Plus[-1, Times[-2, 2]], 2, Power[1.5, -2], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[Plus[Times[-1, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselJ[2, 1.5]]], 2]]]], Equal[[4], Plus[Times[-1, Plus[-1, Times[-2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselJ[Plus[-1, 2], 1.5], 2]], Times[-2, Plus[-1, Times[-2, 2]], 2, Power[1.5, -2], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[Plus[Times[-1, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselJ[2, 1.5]]], 2]], Times[-1, Plus[-1, Times[-2, 2]], 2, Power[1.5, -2], Power[Factorial[Plus[1, Times[2, 2]]], -1], Plus[Times[-1, Plus[-1, Times[2, 2]], Plus[3, Times[-8, 2], Times[4, Power[2, 2]], Times[-1, Power[1.5, 2]]], Power[SphericalBesselJ[Plus[-1, 2], 1.5], 2]], Times[Plus[-3, Times[2, 2]], Power[1.5, 2], Power[SphericalBesselJ[2, 1.5], 2]], Times[Plus[-3, Times[2, 2]], Power[1.5, -2], Plus[3, Times[-8, 2], Times[4, Power[2, 2]], Times[-1, Power[1.5, 2]]], Power[Plus[Times[-1, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselJ[2, 1.5]]], 2]]]]]]}]][3.0]], Times[2.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[-3, Times[-2, ], Times[2, 2]], Plus[-1, Times[-1, ], Times[2, 2]], Plus[Times[-1, ], Times[2, 2]], Plus[1, Times[-1, ], Times[2, 2]], Power[1.5, 2], []], Times[-1, Plus[-1, Times[-1, ], Times[2, 2]], Plus[Times[-1, ], Times[2, 2]], Plus[Times[-3, ], Times[-14, Power[, 2]], Times[-20, Power[, 3]], Times[-8, Power[, 4]], Times[14, , 2], Times[40, Power[, 2], 2], Times[24, Power[, 3], 2], Times[-20, , Power[2, 2]], Times[-24, Power[, 2], Power[2, 2]], Times[8, , Power[2, 3]], Times[-3, Power[1.5, 2]], Times[2, , Power[1.5, 2]], Times[4, Power[, 2], Power[1.5, 2]], Times[-4, 2, Power[1.5, 2]], Times[-8, , 2, Power[1.5, 2]], Times[4, Power[2, 2], Power[1.5, 2]]], [Plus[1, ]]], Times[, Plus[1, , Times[-2, 2]], Plus[3, Times[4, ], Times[4, , 2], Times[-4, Power[2, 2]]], Plus[3, Times[8, ], Times[4, Power[, 2]], Times[-8, 2], Times[-8, , 2], Times[4, Power[2, 2]], Times[-1, Power[1.5, 2]]], [Plus[2, ]]], Times[-1, , Plus[1, ], Plus[9, Times[39, ], Times[58, Power[, 2]], Times[36, Power[, 3]], Times[8, Power[, 4]], Times[-48, 2], Times[-146, , 2], Times[-136, Power[, 2], 2], Times[-40, Power[, 3], 2], Times[88, Power[2, 2]], Times[164, , Power[2, 2]], Times[72, Power[, 2], Power[2, 2]], Times[-64, Power[2, 3]], Times[-56, , Power[2, 3]], Times[16, Power[2, 4]], Times[-5, Power[1.5, 2]], Times[-10, , Power[1.5, 2]], Times[-4, Power[, 2], Power[1.5, 2]], Times[12, 2, Power[1.5, 2]], Times[8, , 2, Power[1.5, 2]], Times[-4, Power[2, 2], Power[1.5, 2]]], [Plus[3, ]]], Times[-1, , Plus[1, ], Plus[2, ], Plus[1, Times[2, ], Times[-2, 2]], Power[1.5, 2], [Plus[4, ]]]], 0], Equal[[1], 0], Equal[[2], Times[-1, Plus[-1, Times[-2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselY[Plus[-1, 2], 1.5], 2]]], Equal[[3], Plus[Times[-1, Plus[-1, Times[-2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselY[Plus[-1, 2], 1.5], 2]], Times[-2, Plus[-1, Times[-2, 2]], 2, Power[1.5, -2], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[Plus[Times[-1, SphericalBesselY[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselY[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselY[2, 1.5]]], 2]]]], Equal[[4], Plus[Times[-1, Plus[-1, Times[-2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselY[Plus[-1, 2], 1.5], 2]], Times[-2, Plus[-1, Times[-2, 2]], 2, Power[1.5, -2], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[Plus[Times[-1, SphericalBesselY[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselY[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselY[2, 1.5]]], 2]], Times[-1, Plus[-1, Times[-2, 2]], 2, Power[1.5, -2], Power[Factorial[Plus[1, Times[2, 2]]], -1], Plus[Times[-1, Plus[-1, Times[2, 2]], Plus[3, Times[-8, 2], Times[4, Power[2, 2]], Times[-1, Power[1.5, 2]]], Power[SphericalBesselY[Plus[-1, 2], 1.5], 2]], Times[Plus[-3, Times[2, 2]], Power[1.5, 2], Power[SphericalBesselY[2, 1.5], 2]], Times[Plus[-3, Times[2, 2]], Power[1.5, -2], Plus[3, Times[-8, 2], Times[4, Power[2, 2]], Times[-1, Power[1.5, 2]]], Power[Plus[Times[-1, SphericalBesselY[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselY[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselY[2, 1.5]]], 2]]]]]]}]][3.0]], Times[5.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[-3, Times[-2, ], Times[2, 2]], Plus[-1, Times[-1, ], Times[2, 2]], Plus[Times[-1, ], Times[2, 2]], Plus[1, Times[-1, ], Times[2, 2]], Power[1.5, 2], []], Times[-1, Plus[-1, Times[-1, ], Times[2, 2]], Plus[Times[-1, ], Times[2, 2]], Plus[-3, Times[-17, ], Times[-34, Power[, 2]], Times[-28, Power[, 3]], Times[-8, Power[, 4]], Times[14, 2], Times[54, , 2], Times[64, Power[, 2], 2], Times[24, Power[, 3], 2], Times[-20, Power[2, 2]], Times[-44, , Power[2, 2]], Times[-24, Power[, 2], Power[2, 2]], Times[8, Power[2, 3]], Times[8, , Power[2, 3]], Times[-2, Power[1.5, 2]], Times[4, , Power[1.5, 2]], Times[4, Power[, 2], Power[1.5, 2]], Times[-6, 2, Power[1.5, 2]], Times[-8, , 2, Power[1.5, 2]], Times[4, Power[2, 2], Power[1.5, 2]]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[-1, Times[-1, ], 2], Plus[3, Times[2, 2]], Plus[-1, Times[-1, ], Times[2, 2]], Plus[3, Times[8, ], Times[4, Power[, 2]], Times[-8, 2], Times[-8, , 2], Times[4, Power[2, 2]], Times[-1, Power[1.5, 2]]], [Plus[2, ]]], Times[-1, Plus[1, ], Plus[2, ], Plus[9, Times[39, ], Times[58, Power[, 2]], Times[36, Power[, 3]], Times[8, Power[, 4]], Times[-48, 2], Times[-146, , 2], Times[-136, Power[, 2], 2], Times[-40, Power[, 3], 2], Times[88, Power[2, 2]], Times[164, , Power[2, 2]], Times[72, Power[, 2], Power[2, 2]], Times[-64, Power[2, 3]], Times[-56, , Power[2, 3]], Times[16, Power[2, 4]], Times[-6, Power[1.5, 2]], Times[-12, , Power[1.5, 2]], Times[-4, Power[, 2], Power[1.5, 2]], Times[14, 2, Power[1.5, 2]], Times[8, , 2, Power[1.5, 2]], Times[-4, Power[2, 2], Power[1.5, 2]]], [Plus[3, ]]], Times[Plus[1, ], Plus[2, ], Plus[3, ], Plus[-1, Times[-2, ], Times[2, 2]], Power[1.5, 2], [Plus[4, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselJ[2, 1.5], 2]]], Equal[[2], Plus[Times[Plus[1, Times[2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselJ[Plus[-1, 2], 1.5], 2]], Times[Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselJ[2, 1.5], 2]]]], Equal[[3], Plus[Times[Plus[1, Times[2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselJ[Plus[-1, 2], 1.5], 2]], Times[Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselJ[2, 1.5], 2]], Times[2, Plus[1, Times[2, 2]], Power[1.5, -2], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[Plus[Times[-1, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselJ[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselJ[2, 1.5]]], 2]]]]}]][3.0]], Times[-5.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[-3, Times[-2, ], Times[2, 2]], Plus[-1, Times[-1, ], Times[2, 2]], Plus[Times[-1, ], Times[2, 2]], Plus[1, Times[-1, ], Times[2, 2]], Power[1.5, 2], []], Times[-1, Plus[-1, Times[-1, ], Times[2, 2]], Plus[Times[-1, ], Times[2, 2]], Plus[-3, Times[-17, ], Times[-34, Power[, 2]], Times[-28, Power[, 3]], Times[-8, Power[, 4]], Times[14, 2], Times[54, , 2], Times[64, Power[, 2], 2], Times[24, Power[, 3], 2], Times[-20, Power[2, 2]], Times[-44, , Power[2, 2]], Times[-24, Power[, 2], Power[2, 2]], Times[8, Power[2, 3]], Times[8, , Power[2, 3]], Times[-2, Power[1.5, 2]], Times[4, , Power[1.5, 2]], Times[4, Power[, 2], Power[1.5, 2]], Times[-6, 2, Power[1.5, 2]], Times[-8, , 2, Power[1.5, 2]], Times[4, Power[2, 2], Power[1.5, 2]]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[-1, Times[-1, ], 2], Plus[3, Times[2, 2]], Plus[-1, Times[-1, ], Times[2, 2]], Plus[3, Times[8, ], Times[4, Power[, 2]], Times[-8, 2], Times[-8, , 2], Times[4, Power[2, 2]], Times[-1, Power[1.5, 2]]], [Plus[2, ]]], Times[-1, Plus[1, ], Plus[2, ], Plus[9, Times[39, ], Times[58, Power[, 2]], Times[36, Power[, 3]], Times[8, Power[, 4]], Times[-48, 2], Times[-146, , 2], Times[-136, Power[, 2], 2], Times[-40, Power[, 3], 2], Times[88, Power[2, 2]], Times[164, , Power[2, 2]], Times[72, Power[, 2], Power[2, 2]], Times[-64, Power[2, 3]], Times[-56, , Power[2, 3]], Times[16, Power[2, 4]], Times[-6, Power[1.5, 2]], Times[-12, , Power[1.5, 2]], Times[-4, Power[, 2], Power[1.5, 2]], Times[14, 2, Power[1.5, 2]], Times[8, , 2, Power[1.5, 2]], Times[-4, Power[2, 2], Power[1.5, 2]]], [Plus[3, ]]], Times[Plus[1, ], Plus[2, ], Plus[3, ], Plus[-1, Times[-2, ], Times[2, 2]], Power[1.5, 2], [Plus[4, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselY[2, 1.5], 2]]], Equal[[2], Plus[Times[Plus[1, Times[2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselY[Plus[-1, 2], 1.5], 2]], Times[Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselY[2, 1.5], 2]]]], Equal[[3], Plus[Times[Plus[1, Times[2, 2]], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselY[Plus[-1, 2], 1.5], 2]], Times[Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[SphericalBesselY[2, 1.5], 2]], Times[2, Plus[1, Times[2, 2]], Power[1.5, -2], Power[Factorial[Plus[1, Times[2, 2]]], -1], Power[Plus[Times[-1, SphericalBesselY[Plus[-1, 2], 1.5]], Times[2, 2, SphericalBesselY[Plus[-1, 2], 1.5]], Times[-1, 1.5, SphericalBesselY[2, 1.5]]], 2]]]]}]][3.0]]]]], {Rule[n, 2], Rule[z, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.60.E6 10.60.E6] || [[Item:Q3781|<math>\modsphBesselK{n}@{2z} = \frac{1}{\pi}n!z^{n+1}\sum_{k=0}^{n}(-1)^{k}\frac{2n-2k+1}{k!(2n-k+1)!}\modsphBesselK{n-k}^{2}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesselK{n}@{2z} = \frac{1}{\pi}n!z^{n+1}\sum_{k=0}^{n}(-1)^{k}\frac{2n-2k+1}{k!(2n-k+1)!}\modsphBesselK{n-k}^{2}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[1/2 Pi /2*z] BesselK[n + 1/2, 2*z] == Divide[1,Pi]*(n)!*(z)^(n + 1)* Sum[(- 1)^(k)*Divide[2*n - 2*k + 1,(k)!*(2*n - k + 1)!]*(Sqrt[1/2 Pi /z] BesselK[n - k + 1/2, z])^(2), {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Aborted || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.10365998143807895, 0.01421463603104145]
+Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.21384035370849797, -0.0374061947505589]
+Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.60.E7 10.60.E7] || [[Item:Q3782|<math>e^{iz\cos@@{\alpha}} = \sum_{n=0}^{\infty}(2n+1)i^{n}\sphBesselJ{n}@{z}\assLegendreP[]{n}@{\cos@@{\alpha}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{iz\cos@@{\alpha}} = \sum_{n=0}^{\infty}(2n+1)i^{n}\sphBesselJ{n}@{z}\assLegendreP[]{n}@{\cos@@{\alpha}}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[I*z*Cos[\[Alpha]]] == Sum[(2*n + 1)*(I)^(n)* SphericalBesselJ[n, z]*LegendreP[n, 0, 3, Cos[\[Alpha]]], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.9634389243184156, 0.05909441627762202], Times[-1.0, NSum[Times[Power[Complex[0, 1], n], Plus[1, Times[2, n]], LegendreP[n, 0.0707372016677029], SphericalBesselJ[n, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
+Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.46738148067268087, 0.44423123280344756], Times[-1.0, NSum[Times[Power[Complex[0, 1], n], Plus[1, Times[2, n]], LegendreP[n, 0.8775825618903728], SphericalBesselJ[n, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
+Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.60.E8 10.60.E8] || [[Item:Q3783|<math>e^{z\cos@@{\alpha}} = \sum_{n=0}^{\infty}(2n+1)\modsphBesseli{1}{n}@{z}\assLegendreP[]{n}@{\cos@@{\alpha}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{z\cos@@{\alpha}} = \sum_{n=0}^{\infty}(2n+1)\modsphBesseli{1}{n}@{z}\assLegendreP[]{n}@{\cos@@{\alpha}}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[z*Cos[\[Alpha]]] == Sum[(2*n + 1)*Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n]*LegendreP[n, 0, 3, Cos[\[Alpha]]], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[1.0625106169893304, 0.037595191618525974], Times[-1.0, NSum[Times[Power[Power[E, Times[Complex[0, Rational[-1, 6]], Pi]], Rational[1, 2]], Plus[1, Times[2, n]], Power[Times[Rational[1, 2], Pi], Rational[1, 2]], BesselI[Plus[Rational[1, 2], n], n], LegendreP[n, 0.0707372016677029]]
+Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[1.935725445820811, 0.9084451535292719], Times[-1.0, NSum[Times[Power[Power[E, Times[Complex[0, Rational[-1, 6]], Pi]], Rational[1, 2]], Plus[1, Times[2, n]], Power[Times[Rational[1, 2], Pi], Rational[1, 2]], BesselI[Plus[Rational[1, 2], n], n], LegendreP[n, 0.8775825618903728]]
+Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.60.E9 10.60.E9] || [[Item:Q3784|<math>e^{-z\cos@@{\alpha}} = \sum_{n=0}^{\infty}(-1)^{n}(2n+1)\modsphBesseli{1}{n}@{z}\assLegendreP[]{n}@{\cos@@{\alpha}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{-z\cos@@{\alpha}} = \sum_{n=0}^{\infty}(-1)^{n}(2n+1)\modsphBesseli{1}{n}@{z}\assLegendreP[]{n}@{\cos@@{\alpha}}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[- z*Cos[\[Alpha]]] == Sum[(- 1)^(n)*(2*n + 1)*Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(n + 1/2), n]*LegendreP[n, 0, 3, Cos[\[Alpha]]], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.939990215282077, -0.03326000860415312], Times[-1.0, NSum[Times[Power[-1, n], Power[Power[E, Times[Complex[0, Rational[-1, 6]], Pi]], Rational[1, 2]], Plus[1, Times[2, n]], Power[Times[Rational[1, 2], Pi], Rational[1, 2]], BesselI[Plus[Rational[1, 2], n], n], LegendreP[n, 0.0707372016677029]]
+Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.4233587200353881, -0.19868425982147583], Times[-1.0, NSum[Times[Power[-1, n], Power[Power[E, Times[Complex[0, Rational[-1, 6]], Pi]], Rational[1, 2]], Plus[1, Times[2, n]], Power[Times[Rational[1, 2], Pi], Rational[1, 2]], BesselI[Plus[Rational[1, 2], n], n], LegendreP[n, 0.8775825618903728]]
+Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.60.E10 10.60.E10] || [[Item:Q3785|<math>\BesselJ{0}@{z\sin@@{\alpha}} = \sum_{n=0}^{\infty}(4n+1)\frac{(2n)!}{2^{2n}(n!)^{2}}\sphBesselJ{2n}@{z}\assLegendreP[]{2n}@{\cos@@{\alpha}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BesselJ{0}@{z\sin@@{\alpha}} = \sum_{n=0}^{\infty}(4n+1)\frac{(2n)!}{2^{2n}(n!)^{2}}\sphBesselJ{2n}@{z}\assLegendreP[]{2n}@{\cos@@{\alpha}}</syntaxhighlight> || <math>\realpart@@{(0+k+1)} > 0, \realpart@@{(((2n)+\frac{1}{2})+k+1)} > 0, \realpart@@{((-(2n)-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-(2n)-\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>BesselJ[0, z*Sin[\[Alpha]]] == Sum[(4*n + 1)*Divide[(2*n)!,(2)^(2*n)*((n)!)^(2)]*SphericalBesselJ[2*n, z]*LegendreP[2*n, 0, 3, Cos[\[Alpha]]], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.8683151459050518, -0.20203213835937428], Times[-1.0, NSum[Times[Power[2, Times[-2, n]], Plus[1, Times[4, n]], Power[Factorial[n], -2], Factorial[Times[2, n]], LegendreP[Times[2, n], 0.0707372016677029], SphericalBesselJ[Times[2, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
+Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.9708614168197589, -0.04904886793011446], Times[-1.0, NSum[Times[Power[2, Times[-2, n]], Plus[1, Times[4, n]], Power[Factorial[n], -2], Factorial[Times[2, n]], LegendreP[Times[2, n], 0.8775825618903728], SphericalBesselJ[Times[2, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
+Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.60.E11 10.60.E11] || [[Item:Q3786|<math>\sum_{n=0}^{\infty}\sphBesselJ{n}^{2}@{z} = \frac{\sinint@{2z}}{2z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=0}^{\infty}\sphBesselJ{n}^{2}@{z} = \frac{\sinint@{2z}}{2z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(SphericalBesselJ[n, z])^(2), {n, 0, Infinity}, GenerateConditions->None] == Divide[SinIntegral[2*z],2*z]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 7]
+|-
+| [https://dlmf.nist.gov/10.60.E12 10.60.E12] || [[Item:Q3787|<math>\sum_{n=0}^{\infty}(2n+1)\sphBesselJ{n}^{2}@{z} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=0}^{\infty}(2n+1)\sphBesselJ{n}^{2}@{z} = 1</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(2*n + 1)*(SphericalBesselJ[n, z])^(2), {n, 0, Infinity}, GenerateConditions->None] == 1</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-1.0, NSum[Times[Plus[1, Times[2, n]], Power[SphericalBesselJ[n, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2]]
+Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-1.0, NSum[Times[Plus[1, Times[2, n]], Power[SphericalBesselJ[n, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], 2]]
+Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.60.E13 10.60.E13] || [[Item:Q3788|<math>\sum_{n=0}^{\infty}(-1)^{n}(2n+1)\sphBesselJ{n}^{2}@{z} = \frac{\sin@{2z}}{2z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=0}^{\infty}(-1)^{n}(2n+1)\sphBesselJ{n}^{2}@{z} = \frac{\sin@{2z}}{2z}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(- 1)^(n)*(2*n + 1)*(SphericalBesselJ[n, z])^(2), {n, 0, Infinity}, GenerateConditions->None] == Divide[Sin[2*z],2*z]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.6123335037567501, 0.46246896224791606], NSum[Times[Power[-1, n], Plus[1, Times[2, n]], Power[SphericalBesselJ[n, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2]]
+Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[-1.2536290109103816, -0.6921871649112455], NSum[Times[Power[-1, n], Plus[1, Times[2, n]], Power[SphericalBesselJ[n, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], 2]]
+Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.60.E14 10.60.E14] || [[Item:Q3789|<math>\sum_{n=0}^{\infty}(2n+1)(\sphBesselJ{n}'@{z})^{2} = \tfrac{1}{3}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=0}^{\infty}(2n+1)(\sphBesselJ{n}'@{z})^{2} = \tfrac{1}{3}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(2*n + 1)*(D[SphericalBesselJ[n, z], {z, 1}])^(2), {n, 0, Infinity}, GenerateConditions->None] == Divide[1,3]</syntaxhighlight> || Missing Macro Error || Aborted || - || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/10.61.E1 10.61.E1] || [[Item:Q3790|<math>\Kelvinber{\nu}@@{x}+i\Kelvinbei{\nu}@@{x} = \BesselJ{\nu}@{xe^{3\pi i/4}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinber{\nu}@@{x}+i\Kelvinbei{\nu}@@{x} = \BesselJ{\nu}@{xe^{3\pi i/4}}</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>KelvinBer(nu, x)+ I*KelvinBei(nu, x) = BesselJ(nu, x*exp(3*Pi*I/4))</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinBer[\[Nu], x]+ I*KelvinBei[\[Nu], x] == BesselJ[\[Nu], x*Exp[3*Pi*I/4]]</syntaxhighlight> || Successful || Failure || Skip - symbolical successful subtest || Successful [Tested: 30]
+|-
+| [https://dlmf.nist.gov/10.61.E1 10.61.E1] || [[Item:Q3790|<math>\BesselJ{\nu}@{xe^{3\pi i/4}} = e^{\nu\pi i}\BesselJ{\nu}@{xe^{-\pi i/4}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BesselJ{\nu}@{xe^{3\pi i/4}} = e^{\nu\pi i}\BesselJ{\nu}@{xe^{-\pi i/4}}</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>BesselJ(nu, x*exp(3*Pi*I/4)) = exp(nu*Pi*I)*BesselJ(nu, x*exp(- Pi*I/4))</syntaxhighlight> || <syntaxhighlight lang=mathematica>BesselJ[\[Nu], x*Exp[3*Pi*I/4]] == Exp[\[Nu]*Pi*I]*BesselJ[\[Nu], x*Exp[- Pi*I/4]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 30] || Successful [Tested: 30]
+|-
+| [https://dlmf.nist.gov/10.61.E1 10.61.E1] || [[Item:Q3790|<math>e^{\nu\pi i}\BesselJ{\nu}@{xe^{-\pi i/4}} = e^{\nu\pi i/2}\modBesselI{\nu}@{xe^{\pi i/4}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{\nu\pi i}\BesselJ{\nu}@{xe^{-\pi i/4}} = e^{\nu\pi i/2}\modBesselI{\nu}@{xe^{\pi i/4}}</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>exp(nu*Pi*I)*BesselJ(nu, x*exp(- Pi*I/4)) = exp(nu*Pi*I/2)*BesselI(nu, x*exp(Pi*I/4))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[\[Nu]*Pi*I]*BesselJ[\[Nu], x*Exp[- Pi*I/4]] == Exp[\[Nu]*Pi*I/2]*BesselI[\[Nu], x*Exp[Pi*I/4]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 30] || Successful [Tested: 30]
+|-
+| [https://dlmf.nist.gov/10.61.E1 10.61.E1] || [[Item:Q3790|<math>e^{\nu\pi i/2}\modBesselI{\nu}@{xe^{\pi i/4}} = e^{3\nu\pi i/2}\modBesselI{\nu}@{xe^{-3\pi i/4}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{\nu\pi i/2}\modBesselI{\nu}@{xe^{\pi i/4}} = e^{3\nu\pi i/2}\modBesselI{\nu}@{xe^{-3\pi i/4}}</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>exp(nu*Pi*I/2)*BesselI(nu, x*exp(Pi*I/4)) = exp(3*nu*Pi*I/2)*BesselI(nu, x*exp(- 3*Pi*I/4))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[\[Nu]*Pi*I/2]*BesselI[\[Nu], x*Exp[Pi*I/4]] == Exp[3*\[Nu]*Pi*I/2]*BesselI[\[Nu], x*Exp[- 3*Pi*I/4]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 30] || Successful [Tested: 30]
+|-
+| [https://dlmf.nist.gov/10.61.E2 10.61.E2] || [[Item:Q3791|<math>\Kelvinker{\nu}@@{x}+i\Kelvinkei{\nu}@@{x} = e^{-\nu\pi i/2}\modBesselK{\nu}@{xe^{\pi i/4}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinker{\nu}@@{x}+i\Kelvinkei{\nu}@@{x} = e^{-\nu\pi i/2}\modBesselK{\nu}@{xe^{\pi i/4}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KelvinKer(nu, x)+ I*KelvinKei(nu, x) = exp(- nu*Pi*I/2)*BesselK(nu, x*exp(Pi*I/4))</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinKer[\[Nu], x]+ I*KelvinKei[\[Nu], x] == Exp[- \[Nu]*Pi*I/2]*BesselK[\[Nu], x*Exp[Pi*I/4]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 30] || Successful [Tested: 30]
+|-
+| [https://dlmf.nist.gov/10.61.E2 10.61.E2] || [[Item:Q3791|<math>e^{-\nu\pi i/2}\modBesselK{\nu}@{xe^{\pi i/4}} = \tfrac{1}{2}\pi i\HankelH{1}{\nu}@{xe^{3\pi i/4}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{-\nu\pi i/2}\modBesselK{\nu}@{xe^{\pi i/4}} = \tfrac{1}{2}\pi i\HankelH{1}{\nu}@{xe^{3\pi i/4}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>exp(- nu*Pi*I/2)*BesselK(nu, x*exp(Pi*I/4)) = (1)/(2)*Pi*I*HankelH1(nu, x*exp(3*Pi*I/4))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[- \[Nu]*Pi*I/2]*BesselK[\[Nu], x*Exp[Pi*I/4]] == Divide[1,2]*Pi*I*HankelH1[\[Nu], x*Exp[3*Pi*I/4]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 30] || Successful [Tested: 30]
+|-
+| [https://dlmf.nist.gov/10.61.E2 10.61.E2] || [[Item:Q3791|<math>\tfrac{1}{2}\pi i\HankelH{1}{\nu}@{xe^{3\pi i/4}} = -\tfrac{1}{2}\pi ie^{-\nu\pi i}\HankelH{2}{\nu}@{xe^{-\pi i/4}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\tfrac{1}{2}\pi i\HankelH{1}{\nu}@{xe^{3\pi i/4}} = -\tfrac{1}{2}\pi ie^{-\nu\pi i}\HankelH{2}{\nu}@{xe^{-\pi i/4}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(1)/(2)*Pi*I*HankelH1(nu, x*exp(3*Pi*I/4)) = -(1)/(2)*Pi*I*exp(- nu*Pi*I)*HankelH2(nu, x*exp(- Pi*I/4))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,2]*Pi*I*HankelH1[\[Nu], x*Exp[3*Pi*I/4]] == -Divide[1,2]*Pi*I*Exp[- \[Nu]*Pi*I]*HankelH2[\[Nu], x*Exp[- Pi*I/4]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 30] || Successful [Tested: 30]
+|-
+| [https://dlmf.nist.gov/10.61.E3 10.61.E3] || [[Item:Q3792|<math>x^{2}\deriv[2]{w}{x}+x\deriv{w}{x}-(ix^{2}+\nu^{2})w = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x^{2}\deriv[2]{w}{x}+x\deriv{w}{x}-(ix^{2}+\nu^{2})w = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(x)^(2)* diff(w, [x$(2)])+ x*diff(w, x)-(I*(x)^(2)+ (nu)^(2))*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>(x)^(2)* D[w, {x, 2}]+ x*D[w, x]-(I*(x)^(2)+ \[Nu]^(2))*w == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.125000000-2.948557160*I
+Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .1249999997-1.216506352*I
+Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 1/2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.1249999999999996, -2.948557158514987]
+Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.1249999999999996, -0.9485571585149869]
+Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.61.E4 10.61.E4] || [[Item:Q3793|<math>x^{4}\deriv[4]{w}{x}+2x^{3}\deriv[3]{w}{x}-(1+2\nu^{2})\left(x^{2}\deriv[2]{w}{x}-x\deriv{w}{x}\right)+(\nu^{4}-4\nu^{2}+x^{4})w = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x^{4}\deriv[4]{w}{x}+2x^{3}\deriv[3]{w}{x}-(1+2\nu^{2})\left(x^{2}\deriv[2]{w}{x}-x\deriv{w}{x}\right)+(\nu^{4}-4\nu^{2}+x^{4})w = 0</syntaxhighlight> || <math>w = \Kelvinber{+\nu}@@{x}, w = \Kelvinber{-\nu}@@{x}</math> || <syntaxhighlight lang=mathematica>(x)^(4)* diff(w, [x$(4)])+ 2*(x)^(3)* diff(w, [x$(3)])-(1 + 2*(nu)^(2))*((x)^(2)* diff(w, [x$(2)])- x*diff(w, x))+((nu)^(4)- 4*(nu)^(2)+ (x)^(4))*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>(x)^(4)* D[w, {x, 4}]+ 2*(x)^(3)* D[w, {x, 3}]-(1 + 2*\[Nu]^(2))*((x)^(2)* D[w, {x, 2}]- x*D[w, x])+(\[Nu]^(4)- 4*\[Nu]^(2)+ (x)^(4))*w == 0</syntaxhighlight> || Error || Failure || - || Skip - No test values generated
+|-
+| [https://dlmf.nist.gov/10.61#Ex1 10.61#Ex1] || [[Item:Q3794|<math>\Kelvinber{n}@{-x} = (-1)^{n}\Kelvinber{n}@@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinber{n}@{-x} = (-1)^{n}\Kelvinber{n}@@{x}</syntaxhighlight> || <math>\realpart@@{(n+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>KelvinBer(n, - x) = (- 1)^(n)* KelvinBer(n, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinBer[n, - x] == (- 1)^(n)* KelvinBer[n, x]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 9]
+|-
+| [https://dlmf.nist.gov/10.61#Ex2 10.61#Ex2] || [[Item:Q3795|<math>\Kelvinbei{n}@{-x} = (-1)^{n}\Kelvinbei{n}@@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinbei{n}@{-x} = (-1)^{n}\Kelvinbei{n}@@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KelvinBei(n, - x) = (- 1)^(n)* KelvinBei(n, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinBei[n, - x] == (- 1)^(n)* KelvinBei[n, x]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 9]
+|-
+| [https://dlmf.nist.gov/10.61#Ex3 10.61#Ex3] || [[Item:Q3796|<math>\Kelvinber{-\nu}@@{x} = \cos@{\nu\pi}\Kelvinber{\nu}@@{x}+\sin@{\nu\pi}\Kelvinbei{\nu}@@{x}+(2/\pi)\sin@{\nu\pi}\Kelvinker{\nu}@@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinber{-\nu}@@{x} = \cos@{\nu\pi}\Kelvinber{\nu}@@{x}+\sin@{\nu\pi}\Kelvinbei{\nu}@@{x}+(2/\pi)\sin@{\nu\pi}\Kelvinker{\nu}@@{x}</syntaxhighlight> || <math>\realpart@@{((-\nu)+k+1)} > 0, \realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>KelvinBer(- nu, x) = cos(nu*Pi)*KelvinBer(nu, x)+ sin(nu*Pi)*KelvinBei(nu, x)+(2/Pi)*sin(nu*Pi)*KelvinKer(nu, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinBer[- \[Nu], x] == Cos[\[Nu]*Pi]*KelvinBer[\[Nu], x]+ Sin[\[Nu]*Pi]*KelvinBei[\[Nu], x]+(2/Pi)*Sin[\[Nu]*Pi]*KelvinKer[\[Nu], x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 30] || Successful [Tested: 30]
+|-
+| [https://dlmf.nist.gov/10.61#Ex4 10.61#Ex4] || [[Item:Q3797|<math>\Kelvinbei{-\nu}@@{x} = -\sin@{\nu\pi}\Kelvinber{\nu}@@{x}+\cos@{\nu\pi}\Kelvinbei{\nu}@@{x}+(2/\pi)\sin@{\nu\pi}\Kelvinkei{\nu}@@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinbei{-\nu}@@{x} = -\sin@{\nu\pi}\Kelvinber{\nu}@@{x}+\cos@{\nu\pi}\Kelvinbei{\nu}@@{x}+(2/\pi)\sin@{\nu\pi}\Kelvinkei{\nu}@@{x}</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>KelvinBei(- nu, x) = - sin(nu*Pi)*KelvinBer(nu, x)+ cos(nu*Pi)*KelvinBei(nu, x)+(2/Pi)*sin(nu*Pi)*KelvinKei(nu, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinBei[- \[Nu], x] == - Sin[\[Nu]*Pi]*KelvinBer[\[Nu], x]+ Cos[\[Nu]*Pi]*KelvinBei[\[Nu], x]+(2/Pi)*Sin[\[Nu]*Pi]*KelvinKei[\[Nu], x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 30] || Successful [Tested: 30]
+|-
+| [https://dlmf.nist.gov/10.61#Ex5 10.61#Ex5] || [[Item:Q3798|<math>\Kelvinker{-\nu}@@{x} = \cos@{\nu\pi}\Kelvinker{\nu}@@{x}-\sin@{\nu\pi}\Kelvinkei{\nu}@@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinker{-\nu}@@{x} = \cos@{\nu\pi}\Kelvinker{\nu}@@{x}-\sin@{\nu\pi}\Kelvinkei{\nu}@@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KelvinKer(- nu, x) = cos(nu*Pi)*KelvinKer(nu, x)- sin(nu*Pi)*KelvinKei(nu, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinKer[- \[Nu], x] == Cos[\[Nu]*Pi]*KelvinKer[\[Nu], x]- Sin[\[Nu]*Pi]*KelvinKei[\[Nu], x]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 30]
+|-
+| [https://dlmf.nist.gov/10.61#Ex6 10.61#Ex6] || [[Item:Q3799|<math>\Kelvinkei{-\nu}@@{x} = \sin@{\nu\pi}\Kelvinker{\nu}@@{x}+\cos@{\nu\pi}\Kelvinkei{\nu}@@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinkei{-\nu}@@{x} = \sin@{\nu\pi}\Kelvinker{\nu}@@{x}+\cos@{\nu\pi}\Kelvinkei{\nu}@@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KelvinKei(- nu, x) = sin(nu*Pi)*KelvinKer(nu, x)+ cos(nu*Pi)*KelvinKei(nu, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinKei[- \[Nu], x] == Sin[\[Nu]*Pi]*KelvinKer[\[Nu], x]+ Cos[\[Nu]*Pi]*KelvinKei[\[Nu], x]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 30]
+|-
+| [https://dlmf.nist.gov/10.61#Ex7 10.61#Ex7] || [[Item:Q3800|<math>\Kelvinber{-n}@@{x} = (-1)^{n}\Kelvinber{n}@@{x},~{}\Kelvinbei{-n}@@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinber{-n}@@{x} = (-1)^{n}\Kelvinber{n}@@{x},~{}\Kelvinbei{-n}@@{x}</syntaxhighlight> || <math>\realpart@@{((-n)+k+1)} > 0, \realpart@@{(n+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>KelvinBer(- n, x) = (- 1)^(n)* KelvinBer(n, x); *KelvinBei(- n, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinBer[- n, x] == (- 1)^(n)* KelvinBer[n, x]
+ *KelvinBei[- n, x]</syntaxhighlight> || Error || Failure || - || Error
+|-
+| [https://dlmf.nist.gov/10.61#Ex7 10.61#Ex7] || [[Item:Q3800|<math>(-1)^{n}\Kelvinber{n}@@{x},~{}\Kelvinbei{-n}@@{x} = (-1)^{n}\Kelvinbei{n}@@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(-1)^{n}\Kelvinber{n}@@{x},~{}\Kelvinbei{-n}@@{x} = (-1)^{n}\Kelvinbei{n}@@{x}</syntaxhighlight> || <math>\realpart@@{((-n)+k+1)} > 0, \realpart@@{(n+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>(- 1)^(n)* KelvinBer(n, x),*KelvinBei(- n, x) = (- 1)^(n)* KelvinBei(n, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(- 1)^(n)* KelvinBer[n, x],*KelvinBei[- n, x] == (- 1)^(n)* KelvinBei[n, x]</syntaxhighlight> || Error || Failure || - || Error
+|-
+| [https://dlmf.nist.gov/10.61#Ex8 10.61#Ex8] || [[Item:Q3801|<math>\Kelvinker{-n}@@{x} = (-1)^{n}\Kelvinker{n}@@{x},~{}\Kelvinkei{-n}@@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinker{-n}@@{x} = (-1)^{n}\Kelvinker{n}@@{x},~{}\Kelvinkei{-n}@@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KelvinKer(- n, x) = (- 1)^(n)* KelvinKer(n, x); *KelvinKei(- n, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinKer[- n, x] == (- 1)^(n)* KelvinKer[n, x]
+ *KelvinKei[- n, x]</syntaxhighlight> || Error || Failure || - || Error
+|-
+| [https://dlmf.nist.gov/10.61#Ex8 10.61#Ex8] || [[Item:Q3801|<math>(-1)^{n}\Kelvinker{n}@@{x},~{}\Kelvinkei{-n}@@{x} = (-1)^{n}\Kelvinkei{n}@@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(-1)^{n}\Kelvinker{n}@@{x},~{}\Kelvinkei{-n}@@{x} = (-1)^{n}\Kelvinkei{n}@@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(- 1)^(n)* KelvinKer(n, x),*KelvinKei(- n, x) = (- 1)^(n)* KelvinKei(n, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(- 1)^(n)* KelvinKer[n, x],*KelvinKei[- n, x] == (- 1)^(n)* KelvinKei[n, x]</syntaxhighlight> || Error || Failure || - || Error
+|-
+| [https://dlmf.nist.gov/10.61#Ex9 10.61#Ex9] || [[Item:Q3802|<math>\Kelvinber{\frac{1}{2}}@{x\sqrt{2}} = \frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\cos@{x+\frac{\pi}{8}}-e^{-x}\cos@{x-\frac{\pi}{8}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinber{\frac{1}{2}}@{x\sqrt{2}} = \frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\cos@{x+\frac{\pi}{8}}-e^{-x}\cos@{x-\frac{\pi}{8}}\right)</syntaxhighlight> || <math>\realpart@@{((\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>KelvinBer((1)/(2), x*sqrt(2)) = ((2)^(-(3)/(4)))/(sqrt(Pi*x))*(exp(x)*cos(x +(Pi)/(8))- exp(- x)*cos(x -(Pi)/(8)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinBer[Divide[1,2], x*Sqrt[2]] == Divide[(2)^(-Divide[3,4]),Sqrt[Pi*x]]*(Exp[x]*Cos[x +Divide[Pi,8]]- Exp[- x]*Cos[x -Divide[Pi,8]])</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
+|-
+| [https://dlmf.nist.gov/10.61#Ex10 10.61#Ex10] || [[Item:Q3803|<math>\Kelvinbei{\frac{1}{2}}@{x\sqrt{2}} = \frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\sin@{x+\frac{\pi}{8}}+\,e^{-x}\sin@{x-\frac{\pi}{8}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinbei{\frac{1}{2}}@{x\sqrt{2}} = \frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\sin@{x+\frac{\pi}{8}}+\,e^{-x}\sin@{x-\frac{\pi}{8}}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KelvinBei((1)/(2), x*sqrt(2)) = ((2)^(-(3)/(4)))/(sqrt(Pi*x))*(exp(x)*sin(x +(Pi)/(8))+ exp(- x)*sin(x -(Pi)/(8)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinBei[Divide[1,2], x*Sqrt[2]] == Divide[(2)^(-Divide[3,4]),Sqrt[Pi*x]]*(Exp[x]*Sin[x +Divide[Pi,8]]+ Exp[- x]*Sin[x -Divide[Pi,8]])</syntaxhighlight> || Failure || Successful || Successful [Tested: 3] || Successful [Tested: 3]
+|-
+| [https://dlmf.nist.gov/10.61#Ex11 10.61#Ex11] || [[Item:Q3804|<math>\Kelvinber{-\frac{1}{2}}@{x\sqrt{2}} = \frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\sin@{x+\frac{\pi}{8}}-e^{-x}\sin@{x-\frac{\pi}{8}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinber{-\frac{1}{2}}@{x\sqrt{2}} = \frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\sin@{x+\frac{\pi}{8}}-e^{-x}\sin@{x-\frac{\pi}{8}}\right)</syntaxhighlight> || <math>\realpart@@{((-\frac{1}{2})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>KelvinBer(-(1)/(2), x*sqrt(2)) = ((2)^(-(3)/(4)))/(sqrt(Pi*x))*(exp(x)*sin(x +(Pi)/(8))- exp(- x)*sin(x -(Pi)/(8)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinBer[-Divide[1,2], x*Sqrt[2]] == Divide[(2)^(-Divide[3,4]),Sqrt[Pi*x]]*(Exp[x]*Sin[x +Divide[Pi,8]]- Exp[- x]*Sin[x -Divide[Pi,8]])</syntaxhighlight> || Failure || Successful || Successful [Tested: 3] || Successful [Tested: 3]
+|-
+| [https://dlmf.nist.gov/10.61#Ex12 10.61#Ex12] || [[Item:Q3805|<math>\Kelvinbei{-\frac{1}{2}}@{x\sqrt{2}} = -\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\cos@{x+\frac{\pi}{8}}+e^{-x}\cos@{x-\frac{\pi}{8}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinbei{-\frac{1}{2}}@{x\sqrt{2}} = -\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\cos@{x+\frac{\pi}{8}}+e^{-x}\cos@{x-\frac{\pi}{8}}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KelvinBei(-(1)/(2), x*sqrt(2)) = -((2)^(-(3)/(4)))/(sqrt(Pi*x))*(exp(x)*cos(x +(Pi)/(8))+ exp(- x)*cos(x -(Pi)/(8)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinBei[-Divide[1,2], x*Sqrt[2]] == -Divide[(2)^(-Divide[3,4]),Sqrt[Pi*x]]*(Exp[x]*Cos[x +Divide[Pi,8]]+ Exp[- x]*Cos[x -Divide[Pi,8]])</syntaxhighlight> || Failure || Successful || Successful [Tested: 3] || Successful [Tested: 3]
+|-
+| [https://dlmf.nist.gov/10.61.E11 10.61.E11] || [[Item:Q3806|<math>\Kelvinker{\frac{1}{2}}@{x\sqrt{2}} = \Kelvinkei{-\frac{1}{2}}@{x\sqrt{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinker{\frac{1}{2}}@{x\sqrt{2}} = \Kelvinkei{-\frac{1}{2}}@{x\sqrt{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KelvinKer((1)/(2), x*sqrt(2)) = KelvinKei(-(1)/(2), x*sqrt(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinKer[Divide[1,2], x*Sqrt[2]] == KelvinKei[-Divide[1,2], x*Sqrt[2]]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 3]
+|-
+| [https://dlmf.nist.gov/10.61.E11 10.61.E11] || [[Item:Q3806|<math>\Kelvinkei{-\frac{1}{2}}@{x\sqrt{2}} = -2^{-\frac{3}{4}}\sqrt{\frac{\pi}{x}}e^{-x}\sin@{x-\frac{\pi}{8}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinkei{-\frac{1}{2}}@{x\sqrt{2}} = -2^{-\frac{3}{4}}\sqrt{\frac{\pi}{x}}e^{-x}\sin@{x-\frac{\pi}{8}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KelvinKei(-(1)/(2), x*sqrt(2)) = - (2)^(-(3)/(4))*sqrt((Pi)/(x))*exp(- x)*sin(x -(Pi)/(8))</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinKei[-Divide[1,2], x*Sqrt[2]] == - (2)^(-Divide[3,4])*Sqrt[Divide[Pi,x]]*Exp[- x]*Sin[x -Divide[Pi,8]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
+|-
+| [https://dlmf.nist.gov/10.61.E12 10.61.E12] || [[Item:Q3807|<math>\Kelvinkei{\frac{1}{2}}@{x\sqrt{2}} = -\Kelvinker{-\frac{1}{2}}@{x\sqrt{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinkei{\frac{1}{2}}@{x\sqrt{2}} = -\Kelvinker{-\frac{1}{2}}@{x\sqrt{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KelvinKei((1)/(2), x*sqrt(2)) = - KelvinKer(-(1)/(2), x*sqrt(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinKei[Divide[1,2], x*Sqrt[2]] == - KelvinKer[-Divide[1,2], x*Sqrt[2]]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 3]
+|-
+| [https://dlmf.nist.gov/10.61.E12 10.61.E12] || [[Item:Q3807|<math>-\Kelvinker{-\frac{1}{2}}@{x\sqrt{2}} = -2^{-\frac{3}{4}}\sqrt{\frac{\pi}{x}}e^{-x}\cos@{x-\frac{\pi}{8}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>-\Kelvinker{-\frac{1}{2}}@{x\sqrt{2}} = -2^{-\frac{3}{4}}\sqrt{\frac{\pi}{x}}e^{-x}\cos@{x-\frac{\pi}{8}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>- KelvinKer(-(1)/(2), x*sqrt(2)) = - (2)^(-(3)/(4))*sqrt((Pi)/(x))*exp(- x)*cos(x -(Pi)/(8))</syntaxhighlight> || <syntaxhighlight lang=mathematica>- KelvinKer[-Divide[1,2], x*Sqrt[2]] == - (2)^(-Divide[3,4])*Sqrt[Divide[Pi,x]]*Exp[- x]*Cos[x -Divide[Pi,8]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
+|- style="background: #dfe6e9;"
+| [https://dlmf.nist.gov/10.63#Ex5 10.63#Ex5] || [[Item:Q3812|<math>f_{\nu-1}(x)+f_{\nu+1}(x) = -(\nu\sqrt{2}/x)\left(f_{\nu}(x)-g_{\nu}(x)\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>f_{\nu-1}(x)+f_{\nu+1}(x) = -(\nu\sqrt{2}/x)\left(f_{\nu}(x)-g_{\nu}(x)\right)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">f[nu - 1](x)+ f[nu + 1](x) = -(nu*sqrt(2)/x)*(f[nu](x)- g[nu](x))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[f, \[Nu]- 1][x]+ Subscript[f, \[Nu]+ 1][x] == -(\[Nu]*Sqrt[2]/x)*(Subscript[f, \[Nu]][x]- Subscript[g, \[Nu]][x])</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+|-
+| [https://dlmf.nist.gov/10.63#Ex9 10.63#Ex9] || [[Item:Q3816|<math>\sqrt{2}\Kelvinber{}'@@{x} = \Kelvinber{1}@@{x}+\Kelvinbei{1}@@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{2}\Kelvinber{}'@@{x} = \Kelvinber{1}@@{x}+\Kelvinbei{1}@@{x}</syntaxhighlight> || <math>\realpart@@{(1+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>sqrt(2)*diff( KelvinBer(, x), x$(1) ) = KelvinBer(1, x)+ KelvinBei(1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[2]*D[KelvinBer[, x], {x, 1}] == KelvinBer[1, x]+ KelvinBei[1, x]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[0.297000428957679, Times[0.35355339059327373, Plus[Times[-1.0, KelvinBei[Plus[-1.0, Null], 1.5]], KelvinBei[Plus[1.0, Null], 1.5], Times[-1.0, KelvinBer[Plus[-1.0, Null], 1.5]], KelvinBer[Plus[1.0, Null], 1.5]]]]
+Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[0.011047944038096752, Times[0.35355339059327373, Plus[Times[-1.0, KelvinBei[Plus[-1.0, Null], 0.5]], KelvinBei[Plus[1.0, Null], 0.5], Times[-1.0, KelvinBer[Plus[-1.0, Null], 0.5]], KelvinBer[Plus[1.0, Null], 0.5]]]]
+Test Values: {Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.63#Ex10 10.63#Ex10] || [[Item:Q3817|<math>\sqrt{2}\Kelvinbei{}'@@{x} = -\Kelvinber{1}x+\Kelvinbei{1}x</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{2}\Kelvinbei{}'@@{x} = -\Kelvinber{1}x+\Kelvinbei{1}x</syntaxhighlight> || <math>\realpart@@{(1+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>sqrt(2)*diff( KelvinBei(, x), x$(1) ) = - KelvinBer(1, x)+ KelvinBei(1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[2]*D[KelvinBei[, x], {x, 1}] == - KelvinBer[1, x]+ KelvinBei[1, x]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-1.0327304069618592, Times[0.35355339059327373, Plus[Times[-1.0, KelvinBei[Plus[-1.0, Null], 1.5]], KelvinBei[Plus[1.0, Null], 1.5], KelvinBer[Plus[-1.0, Null], 1.5], Times[-1.0, KelvinBer[Plus[1.0, Null], 1.5]]]]]
+Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-0.35343830347212746, Times[0.35355339059327373, Plus[Times[-1.0, KelvinBei[Plus[-1.0, Null], 0.5]], KelvinBei[Plus[1.0, Null], 0.5], KelvinBer[Plus[-1.0, Null], 0.5], Times[-1.0, KelvinBer[Plus[1.0, Null], 0.5]]]]]
+Test Values: {Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.63#Ex11 10.63#Ex11] || [[Item:Q3818|<math>\sqrt{2}\Kelvinker{}'@@{x} = \Kelvinker{1}x+\Kelvinkei{1}x</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{2}\Kelvinker{}'@@{x} = \Kelvinker{1}x+\Kelvinkei{1}x</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sqrt(2)*diff( KelvinKer(, x), x$(1) ) = KelvinKer(1, x)+ KelvinKei(1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[2]*D[KelvinKer[, x], {x, 1}] == KelvinKer[1, x]+ KelvinKei[1, x]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[0.4160356041812476, Times[0.35355339059327373, Plus[Times[-1.0, KelvinKei[Plus[-1.0, Null], 1.5]], KelvinKei[Plus[1.0, Null], 1.5], Times[-1.0, KelvinKer[Plus[-1.0, Null], 1.5]], KelvinKer[Plus[1.0, Null], 1.5]]]]
+Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[2.5735854919446126, Times[0.35355339059327373, Plus[Times[-1.0, KelvinKei[Plus[-1.0, Null], 0.5]], KelvinKei[Plus[1.0, Null], 0.5], Times[-1.0, KelvinKer[Plus[-1.0, Null], 0.5]], KelvinKer[Plus[1.0, Null], 0.5]]]]
+Test Values: {Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.63#Ex12 10.63#Ex12] || [[Item:Q3819|<math>\sqrt{2}\Kelvinkei{}'@@{x} = -\Kelvinker{1}x+\Kelvinkei{1}x</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{2}\Kelvinkei{}'@@{x} = -\Kelvinker{1}x+\Kelvinkei{1}x</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sqrt(2)*diff( KelvinKei(, x), x$(1) ) = - KelvinKer(1, x)+ KelvinKei(1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[2]*D[KelvinKei[, x], {x, 1}] == - KelvinKer[1, x]+ KelvinKei[1, x]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-0.418052966151267, Times[0.35355339059327373, Plus[Times[-1.0, KelvinKei[Plus[-1.0, Null], 1.5]], KelvinKei[Plus[1.0, Null], 1.5], KelvinKer[Plus[-1.0, Null], 1.5], Times[-1.0, KelvinKer[Plus[1.0, Null], 1.5]]]]]
+Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-0.47122132111956727, Times[0.35355339059327373, Plus[Times[-1.0, KelvinKei[Plus[-1.0, Null], 0.5]], KelvinKei[Plus[1.0, Null], 0.5], KelvinKer[Plus[-1.0, Null], 0.5], Times[-1.0, KelvinKer[Plus[1.0, Null], 0.5]]]]]
+Test Values: {Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|- style="background: #dfe6e9;"
+| [https://dlmf.nist.gov/10.63#Ex17 10.63#Ex17] || [[Item:Q3824|<math>p_{\nu+1} = p_{\nu-1}-(4\nu/x)r_{\nu}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>p_{\nu+1} = p_{\nu-1}-(4\nu/x)r_{\nu}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">p[nu + 1] = p[nu - 1]-(4*nu/x)*r[nu]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[p, \[Nu]+ 1] == Subscript[p, \[Nu]- 1]-(4*\[Nu]/x)*Subscript[r, \[Nu]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+|- style="background: #dfe6e9;"
+| [https://dlmf.nist.gov/10.63#Ex18 10.63#Ex18] || [[Item:Q3825|<math>q_{\nu+1} = -(\nu/x)p_{\nu}+r_{\nu}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>q_{\nu+1} = -(\nu/x)p_{\nu}+r_{\nu}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">q[nu + 1] = -(nu/x)*p[nu]+ r[nu]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[q, \[Nu]+ 1] == -(\[Nu]/x)*Subscript[p, \[Nu]]+ Subscript[r, \[Nu]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+|- style="background: #dfe6e9;"
+| [https://dlmf.nist.gov/10.63#Ex19 10.63#Ex19] || [[Item:Q3826|<math>r_{\nu+1} = -((\nu+1)/x)p_{\nu+1}+q_{\nu}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>r_{\nu+1} = -((\nu+1)/x)p_{\nu+1}+q_{\nu}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">r[nu + 1] = -((nu + 1)/x)*p[nu + 1]+ q[nu]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[r, \[Nu]+ 1] == -((\[Nu]+ 1)/x)*Subscript[p, \[Nu]+ 1]+ Subscript[q, \[Nu]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+|- style="background: #dfe6e9;"
+| [https://dlmf.nist.gov/10.63#Ex20 10.63#Ex20] || [[Item:Q3827|<math>s_{\nu} = \tfrac{1}{2}p_{\nu+1}+\tfrac{1}{2}p_{\nu-1}-(\nu^{2}/x^{2})p_{\nu}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>s_{\nu} = \tfrac{1}{2}p_{\nu+1}+\tfrac{1}{2}p_{\nu-1}-(\nu^{2}/x^{2})p_{\nu}</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((diff( KelvinBer(nu, x), x$(1) ))^(2)+(diff( KelvinBei(nu, x), x$(1) ))^(2)) = (1)/(2)*p[nu + 1]+(1)/(2)*p[nu - 1]-((nu)^(2)/(x)^(2))*p[nu]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((D[KelvinBer[\[Nu], x], {x, 1}])^(2)+(D[KelvinBei[\[Nu], x], {x, 1}])^(2)) == Divide[1,2]*Subscript[p, \[Nu]+ 1]+Divide[1,2]*Subscript[p, \[Nu]- 1]-(\[Nu]^(2)/(x)^(2))*Subscript[p, \[Nu]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+|- style="background: #dfe6e9;"
+| [https://dlmf.nist.gov/10.63.E7 10.63.E7] || [[Item:Q3828|<math>p_{\nu}s_{\nu} = r_{\nu}^{2}+q_{\nu}^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>p_{\nu}s_{\nu} = r_{\nu}^{2}+q_{\nu}^{2}</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">p[nu]*((diff( KelvinBer(nu, x), x$(1) ))^(2)+(diff( KelvinBei(nu, x), x$(1) ))^(2)) = (r[nu])^(2)+ (q[nu])^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[p, \[Nu]]*((D[KelvinBer[\[Nu], x], {x, 1}])^(2)+(D[KelvinBei[\[Nu], x], {x, 1}])^(2)) == (Subscript[r, \[Nu]])^(2)+ (Subscript[q, \[Nu]])^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
+|-
+| [https://dlmf.nist.gov/10.64.E1 10.64.E1] || [[Item:Q3829|<math>\Kelvinber{n}@{x\sqrt{2}} = \frac{(-1)^{n}}{\pi}\int_{0}^{\pi}\cos@{x\sin@@{t}-nt}\cosh@{x\sin@@{t}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinber{n}@{x\sqrt{2}} = \frac{(-1)^{n}}{\pi}\int_{0}^{\pi}\cos@{x\sin@@{t}-nt}\cosh@{x\sin@@{t}}\diff{t}</syntaxhighlight> || <math>\realpart@@{(n+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>KelvinBer(n, x*sqrt(2)) = ((- 1)^(n))/(Pi)*int(cos(x*sin(t)- n*t)*cosh(x*sin(t)), t = 0..Pi)</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinBer[n, x*Sqrt[2]] == Divide[(- 1)^(n),Pi]*Integrate[Cos[x*Sin[t]- n*t]*Cosh[x*Sin[t]], {t, 0, Pi}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Successful [Tested: 9] || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/10.64.E2 10.64.E2] || [[Item:Q3830|<math>\Kelvinbei{n}@{x\sqrt{2}} = \frac{(-1)^{n}}{\pi}\int_{0}^{\pi}\sin@{x\sin@@{t}-nt}\sinh@{x\sin@@{t}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinbei{n}@{x\sqrt{2}} = \frac{(-1)^{n}}{\pi}\int_{0}^{\pi}\sin@{x\sin@@{t}-nt}\sinh@{x\sin@@{t}}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KelvinBei(n, x*sqrt(2)) = ((- 1)^(n))/(Pi)*int(sin(x*sin(t)- n*t)*sinh(x*sin(t)), t = 0..Pi)</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinBei[n, x*Sqrt[2]] == Divide[(- 1)^(n),Pi]*Integrate[Sin[x*Sin[t]- n*t]*Sinh[x*Sin[t]], {t, 0, Pi}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Successful [Tested: 9] || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/10.65#Ex1 10.65#Ex1] || [[Item:Q3831|<math>\Kelvinber{\nu}@@{x} = (\tfrac{1}{2}x)^{\nu}\sum_{k=0}^{\infty}\frac{\cos@{\frac{3}{4}\nu\pi+\frac{1}{2}k\pi}}{k!\EulerGamma@{\nu+k+1}}(\tfrac{1}{4}x^{2})^{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinber{\nu}@@{x} = (\tfrac{1}{2}x)^{\nu}\sum_{k=0}^{\infty}\frac{\cos@{\frac{3}{4}\nu\pi+\frac{1}{2}k\pi}}{k!\EulerGamma@{\nu+k+1}}(\tfrac{1}{4}x^{2})^{k}</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>KelvinBer(nu, x) = ((1)/(2)*x)^(nu)* sum((cos((3)/(4)*nu*Pi +(1)/(2)*k*Pi))/(factorial(k)*GAMMA(nu + k + 1))*((1)/(4)*(x)^(2))^(k), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinBer[\[Nu], x] == (Divide[1,2]*x)^\[Nu]* Sum[Divide[Cos[Divide[3,4]*\[Nu]*Pi +Divide[1,2]*k*Pi],(k)!*Gamma[\[Nu]+ k + 1]]*(Divide[1,4]*(x)^(2))^(k), {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 30] || Successful [Tested: 30]
+|-
+| [https://dlmf.nist.gov/10.65#Ex2 10.65#Ex2] || [[Item:Q3832|<math>\Kelvinbei{\nu}@@{x} = (\tfrac{1}{2}x)^{\nu}\sum_{k=0}^{\infty}\frac{\sin@{\frac{3}{4}\nu\pi+\frac{1}{2}k\pi}}{k!\EulerGamma@{\nu+k+1}}(\tfrac{1}{4}x^{2})^{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinbei{\nu}@@{x} = (\tfrac{1}{2}x)^{\nu}\sum_{k=0}^{\infty}\frac{\sin@{\frac{3}{4}\nu\pi+\frac{1}{2}k\pi}}{k!\EulerGamma@{\nu+k+1}}(\tfrac{1}{4}x^{2})^{k}</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>KelvinBei(nu, x) = ((1)/(2)*x)^(nu)* sum((sin((3)/(4)*nu*Pi +(1)/(2)*k*Pi))/(factorial(k)*GAMMA(nu + k + 1))*((1)/(4)*(x)^(2))^(k), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinBei[\[Nu], x] == (Divide[1,2]*x)^\[Nu]* Sum[Divide[Sin[Divide[3,4]*\[Nu]*Pi +Divide[1,2]*k*Pi],(k)!*Gamma[\[Nu]+ k + 1]]*(Divide[1,4]*(x)^(2))^(k), {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 30] || Successful [Tested: 30]
+|-
+| [https://dlmf.nist.gov/10.65#Ex3 10.65#Ex3] || [[Item:Q3833|<math>\Kelvinber{}@@{x} = 1-\frac{(\frac{1}{4}x^{2})^{2}}{(2!)^{2}}+\frac{(\frac{1}{4}x^{2})^{4}}{(4!)^{2}}-\dotsb</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinber{}@@{x} = 1-\frac{(\frac{1}{4}x^{2})^{2}}{(2!)^{2}}+\frac{(\frac{1}{4}x^{2})^{4}}{(4!)^{2}}-\dotsb</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KelvinBer(, x) = 1 -(((1)/(4)*(x)^(2))^(2))/((factorial(2))^(2))+(((1)/(4)*(x)^(2))^(4))/((factorial(4))^(2))- ..</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinBer[, x] == 1 -Divide[(Divide[1,4]*(x)^(2))^(2),((2)!)^(2)]+Divide[(Divide[1,4]*(x)^(2))^(4),((4)!)^(2)]- \[Ellipsis]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-0.921072244644165, …, KelvinBer[Null, 1.5]]
+Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-0.9990234639909532, …, KelvinBer[Null, 0.5]]
+Test Values: {Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.65#Ex4 10.65#Ex4] || [[Item:Q3834|<math>\Kelvinbei{}@@{x} = \tfrac{1}{4}x^{2}-\frac{(\frac{1}{4}x^{2})^{3}}{(3!)^{2}}+\frac{(\frac{1}{4}x^{2})^{5}}{(5!)^{2}}-\dotsi</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinbei{}@@{x} = \tfrac{1}{4}x^{2}-\frac{(\frac{1}{4}x^{2})^{3}}{(3!)^{2}}+\frac{(\frac{1}{4}x^{2})^{5}}{(5!)^{2}}-\dotsi</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KelvinBei(, x) = (1)/(4)*(x)^(2)-(((1)/(4)*(x)^(2))^(3))/((factorial(3))^(2))+(((1)/(4)*(x)^(2))^(5))/((factorial(5))^(2))- ..</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinBei[, x] == Divide[1,4]*(x)^(2)-Divide[(Divide[1,4]*(x)^(2))^(3),((3)!)^(2)]+Divide[(Divide[1,4]*(x)^(2))^(5),((5)!)^(2)]- \[Ellipsis]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-0.5575600630044937, …, KelvinBei[Null, 1.5]]
+Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-0.06249321838219961, …, KelvinBei[Null, 0.5]]
+Test Values: {Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.65.E3 10.65.E3] || [[Item:Q3835|<math>\Kelvinker{n}@@{x} = \tfrac{1}{2}(\tfrac{1}{2}x)^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\cos@{\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi}(\tfrac{1}{4}x^{2})^{k}-\ln@{\tfrac{1}{2}x}\Kelvinber{n}@@{x}+\tfrac{1}{4}\pi\Kelvinbei{n}@@{x}+\tfrac{1}{2}(\tfrac{1}{2}x)^{n}\sum_{k=0}^{\infty}\frac{\digamma@{k+1}+\digamma@{n+k+1}}{k!(n+k)!}\cos@{\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi}(\tfrac{1}{4}x^{2})^{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinker{n}@@{x} = \tfrac{1}{2}(\tfrac{1}{2}x)^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\cos@{\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi}(\tfrac{1}{4}x^{2})^{k}-\ln@{\tfrac{1}{2}x}\Kelvinber{n}@@{x}+\tfrac{1}{4}\pi\Kelvinbei{n}@@{x}+\tfrac{1}{2}(\tfrac{1}{2}x)^{n}\sum_{k=0}^{\infty}\frac{\digamma@{k+1}+\digamma@{n+k+1}}{k!(n+k)!}\cos@{\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi}(\tfrac{1}{4}x^{2})^{k}</syntaxhighlight> || <math>\realpart@@{(n+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>KelvinKer(n, x) = (1)/(2)*((1)/(2)*x)^(- n)* sum((factorial(n - k - 1))/(factorial(k))*cos((3)/(4)*n*Pi +(1)/(2)*k*Pi)*((1)/(4)*(x)^(2))^(k), k = 0..n - 1)- ln((1)/(2)*x)*KelvinBer(n, x)+(1)/(4)*Pi*KelvinBei(n, x)+(1)/(2)*((1)/(2)*x)^(n)* sum((Psi(k + 1)+ Psi(n + k + 1))/(factorial(k)*factorial(n + k))*cos((3)/(4)*n*Pi +(1)/(2)*k*Pi)*((1)/(4)*(x)^(2))^(k), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinKer[n, x] == Divide[1,2]*(Divide[1,2]*x)^(- n)* Sum[Divide[(n - k - 1)!,(k)!]*Cos[Divide[3,4]*n*Pi +Divide[1,2]*k*Pi]*(Divide[1,4]*(x)^(2))^(k), {k, 0, n - 1}, GenerateConditions->None]- Log[Divide[1,2]*x]*KelvinBer[n, x]+Divide[1,4]*Pi*KelvinBei[n, x]+Divide[1,2]*(Divide[1,2]*x)^(n)* Sum[Divide[PolyGamma[k + 1]+ PolyGamma[n + k + 1],(k)!*(n + k)!]*Cos[Divide[3,4]*n*Pi +Divide[1,2]*k*Pi]*(Divide[1,4]*(x)^(2))^(k), {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Aborted || Aborted || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 9]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[n, 1], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[n, 2], Rule[x, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.65.E4 10.65.E4] || [[Item:Q3836|<math>\Kelvinkei{n}@@{x} = -\tfrac{1}{2}(\tfrac{1}{2}x)^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\sin@{\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi}(\tfrac{1}{4}x^{2})^{k}-\ln@{\tfrac{1}{2}x}\Kelvinbei{n}@@{x}-\tfrac{1}{4}\pi\Kelvinber{n}@@{x}+\tfrac{1}{2}(\tfrac{1}{2}x)^{n}\sum_{k=0}^{\infty}\frac{\digamma@{k+1}+\digamma@{n+k+1}}{k!(n+k)!}\sin@{\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi}(\tfrac{1}{4}x^{2})^{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinkei{n}@@{x} = -\tfrac{1}{2}(\tfrac{1}{2}x)^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\sin@{\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi}(\tfrac{1}{4}x^{2})^{k}-\ln@{\tfrac{1}{2}x}\Kelvinbei{n}@@{x}-\tfrac{1}{4}\pi\Kelvinber{n}@@{x}+\tfrac{1}{2}(\tfrac{1}{2}x)^{n}\sum_{k=0}^{\infty}\frac{\digamma@{k+1}+\digamma@{n+k+1}}{k!(n+k)!}\sin@{\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi}(\tfrac{1}{4}x^{2})^{k}</syntaxhighlight> || <math>\realpart@@{(n+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>KelvinKei(n, x) = -(1)/(2)*((1)/(2)*x)^(- n)* sum((factorial(n - k - 1))/(factorial(k))*sin((3)/(4)*n*Pi +(1)/(2)*k*Pi)*((1)/(4)*(x)^(2))^(k), k = 0..n - 1)- ln((1)/(2)*x)*KelvinBei(n, x)-(1)/(4)*Pi*KelvinBer(n, x)+(1)/(2)*((1)/(2)*x)^(n)* sum((Psi(k + 1)+ Psi(n + k + 1))/(factorial(k)*factorial(n + k))*sin((3)/(4)*n*Pi +(1)/(2)*k*Pi)*((1)/(4)*(x)^(2))^(k), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinKei[n, x] == -Divide[1,2]*(Divide[1,2]*x)^(- n)* Sum[Divide[(n - k - 1)!,(k)!]*Sin[Divide[3,4]*n*Pi +Divide[1,2]*k*Pi]*(Divide[1,4]*(x)^(2))^(k), {k, 0, n - 1}, GenerateConditions->None]- Log[Divide[1,2]*x]*KelvinBei[n, x]-Divide[1,4]*Pi*KelvinBer[n, x]+Divide[1,2]*(Divide[1,2]*x)^(n)* Sum[Divide[PolyGamma[k + 1]+ PolyGamma[n + k + 1],(k)!*(n + k)!]*Sin[Divide[3,4]*n*Pi +Divide[1,2]*k*Pi]*(Divide[1,4]*(x)^(2))^(k), {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Aborted || Aborted || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 9]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[n, 1], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[n, 2], Rule[x, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.65#Ex5 10.65#Ex5] || [[Item:Q3837|<math>\Kelvinker{}@@{x} = -\ln@{\tfrac{1}{2}x}\Kelvinber{}@@{x}+\tfrac{1}{4}\pi\Kelvinbei{}@@{x}+\sum_{k=0}^{\infty}(-1)^{k}\frac{\digamma@{2k+1}}{((2k)!)^{2}}(\tfrac{1}{4}x^{2})^{2k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinker{}@@{x} = -\ln@{\tfrac{1}{2}x}\Kelvinber{}@@{x}+\tfrac{1}{4}\pi\Kelvinbei{}@@{x}+\sum_{k=0}^{\infty}(-1)^{k}\frac{\digamma@{2k+1}}{((2k)!)^{2}}(\tfrac{1}{4}x^{2})^{2k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KelvinKer(, x) = - ln((1)/(2)*x)*KelvinBer(, x)+(1)/(4)*Pi*KelvinBei(, x)+ sum((- 1)^(k)*(Psi(2*k + 1))/((factorial(2*k))^(2))*((1)/(4)*(x)^(2))^(2*k), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinKer[, x] == - Log[Divide[1,2]*x]*KelvinBer[, x]+Divide[1,4]*Pi*KelvinBei[, x]+ Sum[(- 1)^(k)*Divide[PolyGamma[2*k + 1],((2*k)!)^(2)]*(Divide[1,4]*(x)^(2))^(2*k), {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Error || Aborted || - || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/10.65#Ex6 10.65#Ex6] || [[Item:Q3838|<math>\Kelvinkei{}@@{x} = -\ln@{\tfrac{1}{2}x}\Kelvinbei{}@@{x}-\tfrac{1}{4}\pi\Kelvinber{}@@{x}+\sum_{k=0}^{\infty}(-1)^{k}\frac{\digamma@{2k+2}}{((2k+1)!)^{2}}(\tfrac{1}{4}x^{2})^{2k+1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinkei{}@@{x} = -\ln@{\tfrac{1}{2}x}\Kelvinbei{}@@{x}-\tfrac{1}{4}\pi\Kelvinber{}@@{x}+\sum_{k=0}^{\infty}(-1)^{k}\frac{\digamma@{2k+2}}{((2k+1)!)^{2}}(\tfrac{1}{4}x^{2})^{2k+1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>KelvinKei(, x) = - ln((1)/(2)*x)*KelvinBei(, x)-(1)/(4)*Pi*KelvinBer(, x)+ sum((- 1)^(k)*(Psi(2*k + 2))/((factorial(2*k + 1))^(2))*((1)/(4)*(x)^(2))^(2*k + 1), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinKei[, x] == - Log[Divide[1,2]*x]*KelvinBei[, x]-Divide[1,4]*Pi*KelvinBer[, x]+ Sum[(- 1)^(k)*Divide[PolyGamma[2*k + 2],((2*k + 1)!)^(2)]*(Divide[1,4]*(x)^(2))^(2*k + 1), {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-0.23161280473545226, Times[-1.0, KelvinBer[Null, 1.5]], KelvinKei[Null, 1.5]]
+Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-0.02641550246351669, Times[-1.0, KelvinBer[Null, 0.5]], KelvinKei[Null, 0.5]]
+Test Values: {Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.65.E6 10.65.E6] || [[Item:Q3839|<math>\Kelvinber{\nu}^{2}@@{x}+\Kelvinbei{\nu}^{2}@@{x} = (\tfrac{1}{2}x)^{2\nu}\sum_{k=0}^{\infty}\frac{1}{\EulerGamma@{\nu+k+1}\EulerGamma@{\nu+2k+1}}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinber{\nu}^{2}@@{x}+\Kelvinbei{\nu}^{2}@@{x} = (\tfrac{1}{2}x)^{2\nu}\sum_{k=0}^{\infty}\frac{1}{\EulerGamma@{\nu+k+1}\EulerGamma@{\nu+2k+1}}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0, \realpart@@{(\nu+2k+1)} > 0</math> || <syntaxhighlight lang=mathematica>(KelvinBer(nu, x))^(2)+ (KelvinBei(nu, x))^(2) = ((1)/(2)*x)^(2*nu)* sum((1)/(GAMMA(nu + k + 1)*GAMMA(nu + 2*k + 1))*(((1)/(4)*(x)^(2))^(2*k))/(factorial(k)), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(KelvinBer[\[Nu], x])^(2)+ (KelvinBei[\[Nu], x])^(2) == (Divide[1,2]*x)^(2*\[Nu])* Sum[Divide[1,Gamma[\[Nu]+ k + 1]*Gamma[\[Nu]+ 2*k + 1]]*Divide[(Divide[1,4]*(x)^(2))^(2*k),(k)!], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 30]
+|-
+| [https://dlmf.nist.gov/10.65.E7 10.65.E7] || [[Item:Q3840|<math>\Kelvinber{\nu}@@{x}\Kelvinbei{\nu}'@@{x}-\Kelvinber{\nu}'@@{x}\Kelvinbei{\nu}@@{x} = (\tfrac{1}{2}x)^{2\nu+1}\sum_{k=0}^{\infty}\frac{1}{\EulerGamma@{\nu+k+1}\EulerGamma@{\nu+2k+2}}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinber{\nu}@@{x}\Kelvinbei{\nu}'@@{x}-\Kelvinber{\nu}'@@{x}\Kelvinbei{\nu}@@{x} = (\tfrac{1}{2}x)^{2\nu+1}\sum_{k=0}^{\infty}\frac{1}{\EulerGamma@{\nu+k+1}\EulerGamma@{\nu+2k+2}}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0, \realpart@@{(\nu+2k+2)} > 0</math> || <syntaxhighlight lang=mathematica>KelvinBer(nu, x)*diff( KelvinBei(nu, x), x$(1) )- diff( KelvinBer(nu, x), x$(1) )*KelvinBei(nu, x) = ((1)/(2)*x)^(2*nu + 1)* sum((1)/(GAMMA(nu + k + 1)*GAMMA(nu + 2*k + 2))*(((1)/(4)*(x)^(2))^(2*k))/(factorial(k)), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinBer[\[Nu], x]*D[KelvinBei[\[Nu], x], {x, 1}]- D[KelvinBer[\[Nu], x], {x, 1}]*KelvinBei[\[Nu], x] == (Divide[1,2]*x)^(2*\[Nu]+ 1)* Sum[Divide[1,Gamma[\[Nu]+ k + 1]*Gamma[\[Nu]+ 2*k + 2]]*Divide[(Divide[1,4]*(x)^(2))^(2*k),(k)!], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .7271930e-3+.45983036e-2*I
+Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.41528503e-2+.322695404e-1*I
+Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[x, 1.5], Rule[ν, -2]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[x, 0.5], Rule[ν, -2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.65.E8 10.65.E8] || [[Item:Q3841|<math>\Kelvinber{\nu}@@{x}\Kelvinber{\nu}'@@{x}+\Kelvinbei{\nu}@@{x}\Kelvinbei{\nu}'@@{x} = \tfrac{1}{2}(\tfrac{1}{2}x)^{2\nu-1}\sum_{k=0}^{\infty}\frac{1}{\EulerGamma@{\nu+k+1}\EulerGamma@{\nu+2k}}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinber{\nu}@@{x}\Kelvinber{\nu}'@@{x}+\Kelvinbei{\nu}@@{x}\Kelvinbei{\nu}'@@{x} = \tfrac{1}{2}(\tfrac{1}{2}x)^{2\nu-1}\sum_{k=0}^{\infty}\frac{1}{\EulerGamma@{\nu+k+1}\EulerGamma@{\nu+2k}}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0, \realpart@@{(\nu+2k)} > 0</math> || <syntaxhighlight lang=mathematica>KelvinBer(nu, x)*diff( KelvinBer(nu, x), x$(1) )+ KelvinBei(nu, x)*diff( KelvinBei(nu, x), x$(1) ) = (1)/(2)*((1)/(2)*x)^(2*nu - 1)* sum((1)/(GAMMA(nu + k + 1)*GAMMA(nu + 2*k))*(((1)/(4)*(x)^(2))^(2*k))/(factorial(k)), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinBer[\[Nu], x]*D[KelvinBer[\[Nu], x], {x, 1}]+ KelvinBei[\[Nu], x]*D[KelvinBei[\[Nu], x], {x, 1}] == Divide[1,2]*(Divide[1,2]*x)^(2*\[Nu]- 1)* Sum[Divide[1,Gamma[\[Nu]+ k + 1]*Gamma[\[Nu]+ 2*k]]*Divide[(Divide[1,4]*(x)^(2))^(2*k),(k)!], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [25 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .71978298e-2-.3037583875e-1*I
+Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .607273780e-1-.1071579728*I
+Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[x, 1.5], Rule[ν, -2]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[x, 0.5], Rule[ν, -2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.65.E9 10.65.E9] || [[Item:Q3842|<math>\left(\Kelvinber{\nu}'@@{x}\right)^{2}+\left(\Kelvinbei{\nu}'@@{x}\right)^{2} = (\tfrac{1}{2}x)^{2\nu-2}\sum_{k=0}^{\infty}\frac{2k^{2}+2\nu k+\frac{1}{4}\nu^{2}}{\EulerGamma@{\nu+k+1}\EulerGamma@{\nu+2k+1}}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(\Kelvinber{\nu}'@@{x}\right)^{2}+\left(\Kelvinbei{\nu}'@@{x}\right)^{2} = (\tfrac{1}{2}x)^{2\nu-2}\sum_{k=0}^{\infty}\frac{2k^{2}+2\nu k+\frac{1}{4}\nu^{2}}{\EulerGamma@{\nu+k+1}\EulerGamma@{\nu+2k+1}}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0, \realpart@@{(\nu+2k+1)} > 0</math> || <syntaxhighlight lang=mathematica>(diff( KelvinBer(nu, x), x$(1) ))^(2)+(diff( KelvinBei(nu, x), x$(1) ))^(2) = ((1)/(2)*x)^(2*nu - 2)* sum((2*(k)^(2)+ 2*nu*k +(1)/(4)*(nu)^(2))/(GAMMA(nu + k + 1)*GAMMA(nu + 2*k + 1))*(((1)/(4)*(x)^(2))^(2*k))/(factorial(k)), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[KelvinBer[\[Nu], x], {x, 1}])^(2)+(D[KelvinBei[\[Nu], x], {x, 1}])^(2) == (Divide[1,2]*x)^(2*\[Nu]- 2)* Sum[Divide[2*(k)^(2)+ 2*\[Nu]*k +Divide[1,4]*\[Nu]^(2),Gamma[\[Nu]+ k + 1]*Gamma[\[Nu]+ 2*k + 1]]*Divide[(Divide[1,4]*(x)^(2))^(2*k),(k)!], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(undefined)+Float(undefined)*I
+Test Values: {nu = -2, x = 3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(undefined)+Float(undefined)*I
+Test Values: {nu = -2, x = 1/2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[x, 1.5], Rule[ν, -2]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
+Test Values: {Rule[x, 0.5], Rule[ν, -2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.66.E1 10.66.E1] || [[Item:Q3843|<math>\Kelvinber{\nu}@@{x}+i\Kelvinbei{\nu}@@{x} = \sum_{k=0}^{\infty}\frac{e^{(3\nu+k)\pi i/4}x^{k}\BesselJ{\nu+k}@{x}}{2^{k/2}k!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinber{\nu}@@{x}+i\Kelvinbei{\nu}@@{x} = \sum_{k=0}^{\infty}\frac{e^{(3\nu+k)\pi i/4}x^{k}\BesselJ{\nu+k}@{x}}{2^{k/2}k!}</syntaxhighlight> || <math>\realpart@@{((\nu+k)+k+1)} > 0, \realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>KelvinBer(nu, x)+ I*KelvinBei(nu, x) = sum((exp((3*nu + k)*Pi*I/4)*(x)^(k)* BesselJ(nu + k, x))/((2)^(k/2)* factorial(k)), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinBer[\[Nu], x]+ I*KelvinBei[\[Nu], x] == Sum[Divide[Exp[(3*\[Nu]+ k)*Pi*I/4]*(x)^(k)* BesselJ[\[Nu]+ k, x],(2)^(k/2)* (k)!], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.12257968900025018, 0.2735107661041647], Times[-1.0, NSum[Times[Power[1.5, k], Power[2, Times[Rational[-1, 2], k]], Power[E, Times[Complex[0, Rational[1, 4]], Plus[Times[3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], k], Pi]], BesselJ[Plus[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k], 1.5], Power[Factorial[k], -1]]
+Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.3467793075651209, -0.08562995402477025], Times[-1.0, NSum[Times[Power[1.5, k], Power[2, Times[Rational[-1, 2], k]], Power[E, Times[Complex[0, Rational[1, 4]], Plus[Times[3, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], k], Pi]], BesselJ[Plus[Power[E, Times[Complex[0, Rational[2, 3]], Pi]], k], 1.5], Power[Factorial[k], -1]]
+Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.66.E1 10.66.E1] || [[Item:Q3843|<math>\sum_{k=0}^{\infty}\frac{e^{(3\nu+k)\pi i/4}x^{k}\BesselJ{\nu+k}@{x}}{2^{k/2}k!} = \sum_{k=0}^{\infty}\frac{e^{(3\nu+3k)\pi i/4}x^{k}\modBesselI{\nu+k}@{x}}{2^{k/2}k!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=0}^{\infty}\frac{e^{(3\nu+k)\pi i/4}x^{k}\BesselJ{\nu+k}@{x}}{2^{k/2}k!} = \sum_{k=0}^{\infty}\frac{e^{(3\nu+3k)\pi i/4}x^{k}\modBesselI{\nu+k}@{x}}{2^{k/2}k!}</syntaxhighlight> || <math>\realpart@@{((\nu+k)+k+1)} > 0, \realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>sum((exp((3*nu + k)*Pi*I/4)*(x)^(k)* BesselJ(nu + k, x))/((2)^(k/2)* factorial(k)), k = 0..infinity) = sum((exp((3*nu + 3*k)*Pi*I/4)*(x)^(k)* BesselI(nu + k, x))/((2)^(k/2)* factorial(k)), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[Divide[Exp[(3*\[Nu]+ k)*Pi*I/4]*(x)^(k)* BesselJ[\[Nu]+ k, x],(2)^(k/2)* (k)!], {k, 0, Infinity}, GenerateConditions->None] == Sum[Divide[Exp[(3*\[Nu]+ 3*k)*Pi*I/4]*(x)^(k)* BesselI[\[Nu]+ k, x],(2)^(k/2)* (k)!], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Times[-1.0, NSum[Times[Power[1.5, k], Power[2, Times[Rational[-1, 2], k]], Power[E, Times[Complex[0, Rational[1, 4]], Plus[Times[3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[3, k]], Pi]], BesselI[Plus[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k], 1.5], Power[Factorial[k], -1]]
+Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], NSum[Times[Power[1.5, k], Power[2, Times[Rational[-1, 2], k]], Power[E, Times[Complex[0, Rational[1, 4]], Plus[Times[3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], k], Pi]], BesselJ[Plus[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k], 1.5], Power[Factorial[k], -1]], {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Times[-1.0, NSum[Times[Power[1.5, k], Power[2, Times[Rational[-1, 2], k]], Power[E, Times[Complex[0, Rational[1, 4]], Plus[Times[3, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Times[3, k]], Pi]], BesselI[Plus[Power[E, Times[Complex[0, Rational[2, 3]], Pi]], k], 1.5], Power[Factorial[k], -1]]
+Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], NSum[Times[Power[1.5, k], Power[2, Times[Rational[-1, 2], k]], Power[E, Times[Complex[0, Rational[1, 4]], Plus[Times[3, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], k], Pi]], BesselJ[Plus[Power[E, Times[Complex[0, Rational[2, 3]], Pi]], k], 1.5], Power[Factorial[k], -1]], {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.66#Ex1 10.66#Ex1] || [[Item:Q3844|<math>\Kelvinber{n}@{x\sqrt{2}} = \sum_{k=-\infty}^{\infty}(-1)^{n+k}\BesselJ{n+2k}@{x}\modBesselI{2k}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinber{n}@{x\sqrt{2}} = \sum_{k=-\infty}^{\infty}(-1)^{n+k}\BesselJ{n+2k}@{x}\modBesselI{2k}@{x}</syntaxhighlight> || <math>\realpart@@{((n+2k)+k+1)} > 0, \realpart@@{(n+k+1)} > 0, \realpart@@{((2k)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>KelvinBer(n, x*sqrt(2)) = sum((- 1)^(n + k)* BesselJ(n + 2*k, x)*BesselI(2*k, x), k = - infinity..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinBer[n, x*Sqrt[2]] == Sum[(- 1)^(n + k)* BesselJ[n + 2*k, x]*BesselI[2*k, x], {k, - Infinity, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Successful [Tested: 9] || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/10.66#Ex2 10.66#Ex2] || [[Item:Q3845|<math>\Kelvinbei{n}@{x\sqrt{2}} = \sum_{k=-\infty}^{\infty}(-1)^{n+k}\BesselJ{n+2k+1}@{x}\modBesselI{2k+1}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Kelvinbei{n}@{x\sqrt{2}} = \sum_{k=-\infty}^{\infty}(-1)^{n+k}\BesselJ{n+2k+1}@{x}\modBesselI{2k+1}@{x}</syntaxhighlight> || <math>\realpart@@{((n+2k+1)+k+1)} > 0, \realpart@@{((2k+1)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>KelvinBei(n, x*sqrt(2)) = sum((- 1)^(n + k)* BesselJ(n + 2*k + 1, x)*BesselI(2*k + 1, x), k = - infinity..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>KelvinBei[n, x*Sqrt[2]] == Sum[(- 1)^(n + k)* BesselJ[n + 2*k + 1, x]*BesselI[2*k + 1, x], {k, - Infinity, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Successful [Tested: 9] || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/10.68#Ex5 10.68#Ex5] || [[Item:Q3868|<math>\HankelmodM{\nu}@{x} = (\Kelvinber{\nu}^{2}@@{x}+\Kelvinbei{\nu}^{2}@@{x})^{\ifrac{1}{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\HankelmodM{\nu}@{x} = (\Kelvinber{\nu}^{2}@@{x}+\Kelvinbei{\nu}^{2}@@{x})^{\ifrac{1}{2}}</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[KelvinBer[\[Nu], x]^2 + KelvinBei[\[Nu], x]^2] == ((KelvinBer[\[Nu], x])^(2)+ (KelvinBei[\[Nu], x])^(2))^(Divide[1,2])</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 30]
+|-
+| [https://dlmf.nist.gov/10.68#Ex6 10.68#Ex6] || [[Item:Q3869|<math>\HankelmodderivN{\nu}@{x} = (\Kelvinker{\nu}^{2}@@{x}+\Kelvinkei{\nu}^{2}@@{x})^{\ifrac{1}{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\HankelmodderivN{\nu}@{x} = (\Kelvinker{\nu}^{2}@@{x}+\Kelvinkei{\nu}^{2}@@{x})^{\ifrac{1}{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[KelvinKer[\[Nu], x]^2 + KelvinKei[\[Nu], x]^2] == ((KelvinKer[\[Nu], x])^(2)+ (KelvinKei[\[Nu], x])^(2))^(Divide[1,2])</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 30]
+|-
+| [https://dlmf.nist.gov/10.68#Ex9 10.68#Ex9] || [[Item:Q3872|<math>\HankelmodM{-n}@{x} = \HankelmodM{n}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\HankelmodM{-n}@{x} = \HankelmodM{n}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[KelvinBer[- n, x]^2 + KelvinBei[- n, x]^2] == Sqrt[KelvinBer[n, x]^2 + KelvinBei[n, x]^2]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 9]
+|-
+| [https://dlmf.nist.gov/10.68#Ex17 10.68#Ex17] || [[Item:Q3884|<math>\HankelmodderivN{-\nu}@{x} = \HankelmodderivN{\nu}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\HankelmodderivN{-\nu}@{x} = \HankelmodderivN{\nu}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[KelvinKer[- \[Nu], x]^2 + KelvinKei[- \[Nu], x]^2] == Sqrt[KelvinKer[\[Nu], x]^2 + KelvinKei[\[Nu], x]^2]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 30]
+|-
+| [https://dlmf.nist.gov/10.71.E1 10.71.E1] || [[Item:Q3902|<math>\int x^{1+\nu}f_{\nu}\diff{x} = -\frac{x^{1+\nu}}{\sqrt{2}}(f_{\nu+1}-g_{\nu+1})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int x^{1+\nu}f_{\nu}\diff{x} = -\frac{x^{1+\nu}}{\sqrt{2}}(f_{\nu+1}-g_{\nu+1})</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int((x)^(1 + nu)* f[nu], x) = -((x)^(1 + nu))/(sqrt(2))*(f[nu + 1]- g[nu + 1])</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(x)^(1 + \[Nu])* Subscript[f, \[Nu]], x, GenerateConditions->None] == -Divide[(x)^(1 + \[Nu]),Sqrt[2]]*(Subscript[f, \[Nu]+ 1]- Subscript[g, \[Nu]+ 1])</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .9346151411+.5776724966*I
+Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2, f[nu] = 1/2*3^(1/2)+1/2*I, f[1+nu] = 1/2*3^(1/2)+1/2*I, g[1+nu] = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 3.061934630+.4518721345*I
+Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2, f[nu] = 1/2*3^(1/2)+1/2*I, f[1+nu] = 1/2*3^(1/2)+1/2*I, g[1+nu] = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.9346151408625077, 0.5776724967688012]
+Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, ν], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, Plus[1, ν]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, Plus[1, ν]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[3.061934629891139, 0.45187213490403344]
+Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, ν], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, Plus[1, ν]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, Plus[1, ν]], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.71.E2 10.71.E2] || [[Item:Q3903|<math>\int x^{1-\nu}f_{\nu}\diff{x} = \frac{x^{1-\nu}}{\sqrt{2}}(f_{\nu-1}-g_{\nu-1})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int x^{1-\nu}f_{\nu}\diff{x} = \frac{x^{1-\nu}}{\sqrt{2}}(f_{\nu-1}-g_{\nu-1})</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int((x)^(1 - nu)* f[nu], x) = ((x)^(1 - nu))/(sqrt(2))*(f[nu - 1]- g[nu - 1])</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(x)^(1 - \[Nu])* Subscript[f, \[Nu]], x, GenerateConditions->None] == Divide[(x)^(1 - \[Nu]),Sqrt[2]]*(Subscript[f, \[Nu]- 1]- Subscript[g, \[Nu]- 1])</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .9470105611+.8580421171*I
+Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2, f[nu] = 1/2*3^(1/2)+1/2*I, f[nu-1] = 1/2*3^(1/2)+1/2*I, g[nu-1] = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .30703090e-2+1.331056152*I
+Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2, f[nu] = 1/2*3^(1/2)+1/2*I, f[nu-1] = 1/2*3^(1/2)+1/2*I, g[nu-1] = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.9470105613079453, 0.8580421172974921]
+Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, Plus[-1, ν]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, ν], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, Plus[-1, ν]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.0030703089818392426, 1.3310561520338196]
+Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, Plus[-1, ν]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, ν], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, Plus[-1, ν]], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
+|-
+| [https://dlmf.nist.gov/10.71.E6 10.71.E6] || [[Item:Q3907|<math>\int xf_{\nu}g_{\nu}\diff{x} = \tfrac{1}{4}x^{2}\left(2f_{\nu}g_{\nu}-f_{\nu-1}g_{\nu+1}-f_{\nu+1}g_{\nu-1}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int xf_{\nu}g_{\nu}\diff{x} = \tfrac{1}{4}x^{2}\left(2f_{\nu}g_{\nu}-f_{\nu-1}g_{\nu+1}-f_{\nu+1}g_{\nu-1}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(x*f[nu]*g[nu], x) = (1)/(4)*(x)^(2)*(2*f[nu]*g[nu]- f[nu - 1]*g[nu + 1]- f[nu + 1]*g[nu - 1])</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[x*Subscript[f, \[Nu]]*Subscript[g, \[Nu]], x, GenerateConditions->None] == Divide[1,4]*(x)^(2)*(2*Subscript[f, \[Nu]]*Subscript[g, \[Nu]]- Subscript[f, \[Nu]- 1]*Subscript[g, \[Nu]+ 1]- Subscript[f, \[Nu]+ 1]*Subscript[g, \[Nu]- 1])</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [270 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .5625000004+.9742785795*I
+Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2, f[nu] = 1/2*3^(1/2)+1/2*I, f[1+nu] = 1/2*3^(1/2)+1/2*I, f[nu-1] = 1/2*3^(1/2)+1/2*I, g[nu] = 1/2*3^(1/2)+1/2*I, g[1+nu] = 1/2*3^(1/2)+1/2*I, g[nu-1] = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.2058892896+.7683892900*I
+Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2, f[nu] = 1/2*3^(1/2)+1/2*I, f[1+nu] = 1/2*3^(1/2)+1/2*I, f[nu-1] = 1/2*3^(1/2)+1/2*I, g[nu] = 1/2*3^(1/2)+1/2*I, g[1+nu] = 1/2*3^(1/2)+1/2*I, g[nu-1] = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
+|-
+| [https://dlmf.nist.gov/10.71.E7 10.71.E7] || [[Item:Q3908|<math>\int x(f_{\nu}^{2}-g_{\nu}^{2})\diff{x} = \tfrac{1}{2}x^{2}\left(f_{\nu}^{2}-f_{\nu-1}f_{\nu+1}-g_{\nu}^{2}+g_{\nu-1}g_{\nu+1}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int x(f_{\nu}^{2}-g_{\nu}^{2})\diff{x} = \tfrac{1}{2}x^{2}\left(f_{\nu}^{2}-f_{\nu-1}f_{\nu+1}-g_{\nu}^{2}+g_{\nu-1}g_{\nu+1}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(x*((f[nu])^(2)- (g[nu])^(2)), x) = (1)/(2)*(x)^(2)*((f[nu])^(2)- f[nu - 1]*f[nu + 1]- (g[nu])^(2)+ g[nu - 1]*g[nu + 1])</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[x*((Subscript[f, \[Nu]])^(2)- (Subscript[g, \[Nu]])^(2)), x, GenerateConditions->None] == Divide[1,2]*(x)^(2)*((Subscript[f, \[Nu]])^(2)- Subscript[f, \[Nu]- 1]*Subscript[f, \[Nu]+ 1]- (Subscript[g, \[Nu]])^(2)+ Subscript[g, \[Nu]- 1]*Subscript[g, \[Nu]+ 1])</syntaxhighlight> || Failure || Failure || Error || Error
+|-
+| [https://dlmf.nist.gov/10.71#Ex1 10.71#Ex1] || [[Item:Q3909|<math>\int x\HankelmodM{\nu}^{2}@{x}\diff{x} = x(\Kelvinber{\nu}@@{x}\Kelvinbei{\nu}'@@{x}-\Kelvinber{\nu}'@@{x}\Kelvinbei{\nu}@@{x})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int x\HankelmodM{\nu}^{2}@{x}\diff{x} = x(\Kelvinber{\nu}@@{x}\Kelvinbei{\nu}'@@{x}-\Kelvinber{\nu}'@@{x}\Kelvinbei{\nu}@@{x})</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[x*(Sqrt[KelvinBer[\[Nu], x]^2 + KelvinBei[\[Nu], x]^2])^(2), x, GenerateConditions->None] == x*(KelvinBer[\[Nu], x]*D[KelvinBei[\[Nu], x], {x, 1}]- D[KelvinBer[\[Nu], x], {x, 1}]*KelvinBei[\[Nu], x])</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 30]
+|-
+| [https://dlmf.nist.gov/10.71#Ex2 10.71#Ex2] || [[Item:Q3910|<math>\int x\HankelmodderivN{\nu}^{2}@{x}\diff{x} = x(\Kelvinker{\nu}@@{x}\Kelvinkei{\nu}'@@{x}-\Kelvinker{\nu}'@@{x}\Kelvinkei{\nu}@@{x})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int x\HankelmodderivN{\nu}^{2}@{x}\diff{x} = x(\Kelvinker{\nu}@@{x}\Kelvinkei{\nu}'@@{x}-\Kelvinker{\nu}'@@{x}\Kelvinkei{\nu}@@{x})</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[x*(Sqrt[KelvinKer[\[Nu], x]^2 + KelvinKei[\[Nu], x]^2])^(2), x, GenerateConditions->None] == x*(KelvinKer[\[Nu], x]*D[KelvinKei[\[Nu], x], {x, 1}]- D[KelvinKer[\[Nu], x], {x, 1}]*KelvinKei[\[Nu], x])</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 30]
+|-
+| [https://dlmf.nist.gov/10.73.E1 10.73.E1] || [[Item:Q3912|<math>\frac{1}{r}\pderiv{}{r}\left(r\pderiv{V}{r}\right)+\frac{1}{r^{2}}\pderiv[2]{V}{\phi}+\pderiv[2]{V}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{r}\pderiv{}{r}\left(r\pderiv{V}{r}\right)+\frac{1}{r^{2}}\pderiv[2]{V}{\phi}+\pderiv[2]{V}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(1)/(r)*diff((r*diff(V, r))+(1)/((r)^(2))*diff(V, [phi$(2)]), r)+ diff(V, [z$(2)]) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,r]*D[(r*D[V, r])+Divide[1,(r)^(2)]*D[V, {\[Phi], 2}], r]+ D[V, {z, 2}] == 0</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 300]
+|}
+</div>

Results of Bessel Functions III: Difference between revisions

Latest revision as of 07:04, 25 May 2021

Navigation menu

Search