10.47

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
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