Results of Bessel Functions III

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)])+ x*diff(w, x)+((nu)^(2)- (x)^(2))*w = 0	(x)^(2)* D[w, {x, 2}]+ x*D[w, x]+(\[Nu]^(2)- (x)^(2))*w == 0	Failure	Failure	Failed [240 / 300] 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} 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} ... 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(I*nu, 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(I*nu, 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)])+ 2*z*diff(w, z)+((z)^(2)- n*(n + 1))*w = 0	(z)^(2)* D[w, {z, 2}]+ 2*z*D[w, z]+((z)^(2)- n*(n + 1))*w == 0	Failure	Failure	Failed [210 / 210] 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} 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} ... 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)])+ 2*z*diff(w, z)-((z)^(2)+ n*(n + 1))*w = 0	(z)^(2)* D[w, {z, 2}]+ 2*z*D[w, z]-((z)^(2)+ n*(n + 1))*w == 0	Failure	Failure	Failed [210 / 210] 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} 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} ... 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)* I*sqrt((1)/(2)*Pi/z)*HankelH1(- n -(1)/(2), z)	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]	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)* I*sqrt((1)/(2)*Pi/z)*HankelH2(- n -(1)/(2), z)	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]	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, I*z] == -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]*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]	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]*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]	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[I*z]*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[- I*z]*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]*Pi*Exp[- 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]*Pi*Divide[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]*Pi*Exp[- 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]*Pi*Exp[- 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)^(2*k + 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] == - 2*I*(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] == (2*n + 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]*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]	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) = ((2*n + 1)/z)*f[n](z)	Subscript[f, n - 1][z]+ Subscript[f, n + 1][z] == ((2*n + 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-.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} 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} ... 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) = ((2*n + 1)/z)*g[n](z)	Subscript[g, n - 1][z]- Subscript[g, n + 1][z] == ((2*n + 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+1.366025403*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} 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} ... 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)!*(2*n + 2*k + 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[(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]	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)!*(2*n + 2*k + 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[(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]	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[z*Cos[\[Theta]]]*(Sin[\[Theta]])^(2*n + 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[I*z*Cos[\[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[- z*t]*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),2*Pi]*Integrate[Exp[I*z*t]*LegendreQ[n, 0, 3, t], {t, I*Infinity, (- 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[I*z*t]*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[I*z*t]*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)- 2*z*t]],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)- 2*z*t]],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)+ 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]	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)+ 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]	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)+ 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]	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]),((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]	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[(2*n + 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[(2*n + 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[(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]	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, 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]	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, 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]	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 /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]	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[I*z*Cos[\[Alpha]]] == Sum[(2*n + 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[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]	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[- 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]	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, 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]	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[2*z],2*z]	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[(2*n + 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)*(2*n + 1)*(SphericalBesselJ[n, z])^(2), {n, 0, Infinity}, GenerateConditions->None] == Divide[Sin[2*z],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[(2*n + 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)+ I*KelvinBei(nu, x) = BesselJ(nu, x*exp(3*Pi*I/4))	KelvinBer[\[Nu], x]+ I*KelvinBei[\[Nu], x] == BesselJ[\[Nu], x*Exp[3*Pi*I/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, x*exp(3*Pi*I/4)) = exp(nu*Pi*I)*BesselJ(nu, x*exp(- Pi*I/4))	BesselJ[\[Nu], x*Exp[3*Pi*I/4]] == Exp[\[Nu]*Pi*I]*BesselJ[\[Nu], x*Exp[- Pi*I/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(nu*Pi*I)*BesselJ(nu, x*exp(- Pi*I/4)) = exp(nu*Pi*I/2)*BesselI(nu, x*exp(Pi*I/4))	Exp[\[Nu]*Pi*I]*BesselJ[\[Nu], x*Exp[- Pi*I/4]] == Exp[\[Nu]*Pi*I/2]*BesselI[\[Nu], x*Exp[Pi*I/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(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))	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]]	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)+ I*KelvinKei(nu, x) = exp(- nu*Pi*I/2)*BesselK(nu, x*exp(Pi*I/4))	KelvinKer[\[Nu], x]+ I*KelvinKei[\[Nu], x] == Exp[- \[Nu]*Pi*I/2]*BesselK[\[Nu], x*Exp[Pi*I/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(- nu*Pi*I/2)*BesselK(nu, x*exp(Pi*I/4)) = (1)/(2)*Pi*I*HankelH1(nu, x*exp(3*Pi*I/4))	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]]	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)*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))	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]]	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)])+ x*diff(w, x)-(I*(x)^(2)+ (nu)^(2))*w = 0	(x)^(2)* D[w, {x, 2}]+ x*D[w, x]-(I*(x)^(2)+ \[Nu]^(2))*w == 0	Failure	Failure	Failed [300 / 300] 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} 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} ... 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)])- x*diff(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}]- x*D[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(nu*Pi)*KelvinBer(nu, x)+ sin(nu*Pi)*KelvinBei(nu, x)+(2/Pi)*sin(nu*Pi)*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(nu*Pi)*KelvinBer(nu, x)+ cos(nu*Pi)*KelvinBei(nu, x)+(2/Pi)*sin(nu*Pi)*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(nu*Pi)*KelvinKer(nu, x)- sin(nu*Pi)*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(nu*Pi)*KelvinKer(nu, x)+ cos(nu*Pi)*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), x*sqrt(2)) = ((2)^(-(3)/(4)))/(sqrt(Pi*x))*(exp(x)*cos(x +(Pi)/(8))- exp(- x)*cos(x -(Pi)/(8)))	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]])	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), x*sqrt(2)) = ((2)^(-(3)/(4)))/(sqrt(Pi*x))*(exp(x)*sin(x +(Pi)/(8))+ exp(- x)*sin(x -(Pi)/(8)))	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]])	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), x*sqrt(2)) = ((2)^(-(3)/(4)))/(sqrt(Pi*x))*(exp(x)*sin(x +(Pi)/(8))- exp(- x)*sin(x -(Pi)/(8)))	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]])	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), x*sqrt(2)) = -((2)^(-(3)/(4)))/(sqrt(Pi*x))*(exp(x)*cos(x +(Pi)/(8))+ exp(- x)*cos(x -(Pi)/(8)))	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]])	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), x*sqrt(2)) = KelvinKei(-(1)/(2), x*sqrt(2))	KelvinKer[Divide[1,2], x*Sqrt[2]] == KelvinKei[-Divide[1,2], x*Sqrt[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), x*sqrt(2)) = - (2)^(-(3)/(4))*sqrt((Pi)/(x))*exp(- x)*sin(x -(Pi)/(8))	KelvinKei[-Divide[1,2], x*Sqrt[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), x*sqrt(2)) = - KelvinKer(-(1)/(2), x*sqrt(2))	KelvinKei[Divide[1,2], x*Sqrt[2]] == - KelvinKer[-Divide[1,2], x*Sqrt[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), x*sqrt(2)) = - (2)^(-(3)/(4))*sqrt((Pi)/(x))*exp(- x)*cos(x -(Pi)/(8))	- KelvinKer[-Divide[1,2], x*Sqrt[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) = -(nu*sqrt(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]-(4*nu/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, x*sqrt(2)) = ((- 1)^(n))/(Pi)*int(cos(x*sin(t)- n*t)*cosh(x*sin(t)), t = 0..Pi)	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]	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, x*sqrt(2)) = ((- 1)^(n))/(Pi)*int(sin(x*sin(t)- n*t)*sinh(x*sin(t)), t = 0..Pi)	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]	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)*nu*Pi +(1)/(2)*k*Pi))/(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]*k*Pi],(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)*nu*Pi +(1)/(2)*k*Pi))/(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]*k*Pi],(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)*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)	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]	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)*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)	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]	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)*Pi*KelvinBei(, x)+ sum((- 1)^(k)*(Psi(2*k + 1))/((factorial(2*k))^(2))*((1)/(4)*(x)^(2))^(2*k), k = 0..infinity)	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]	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)*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)	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]	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)^(2*nu)* sum((1)/(GAMMA(nu + k + 1)*GAMMA(nu + 2*k + 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]+ 2*k + 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)^(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)	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]	Failure	Successful	Failed [21 / 30] Result: .7271930e-3+.45983036e-2*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2} Result: -.41528503e-2+.322695404e-1*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, 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)^(2*nu - 1)* sum((1)/(GAMMA(nu + k + 1)*GAMMA(nu + 2*k))*(((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]+ 2*k]]*Divide[(Divide[1,4]*(x)^(2))^(2*k),(k)!], {k, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Failed [25 / 30] Result: .71978298e-2-.3037583875e-1*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2} Result: .607273780e-1-.1071579728*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, 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)^(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)	(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]	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)+ 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)	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]	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((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)	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]	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, x*sqrt(2)) = sum((- 1)^(n + k)* BesselJ(n + 2*k, x)*BesselI(2*k, x), k = - infinity..infinity)	KelvinBer[n, x*Sqrt[2]] == Sum[(- 1)^(n + k)* BesselJ[n + 2*k, 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, x*sqrt(2)) = sum((- 1)^(n + k)* BesselJ(n + 2*k + 1, x)*BesselI(2*k + 1, x), k = - infinity..infinity)	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]	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+.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} 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)} ... 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+.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} 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)} ... 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(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])	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])	Failure	Failure	Failed [270 / 300] 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} 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)} ... 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((r*diff(V, r))+(1)/((r)^(2))*diff(V, [phi$(2)]), r)+ diff(V, [z$(2)]) = 0	Divide[1,r]*D[(r*D[V, r])+Divide[1,(r)^(2)]*D[V, {\[Phi], 2}], r]+ D[V, {z, 2}] == 0	Successful	Successful	-	Successful [Tested: 300]