10.49

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