10.50

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