Bessel Functions - 10.49 Explicit Formulas

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10.49.E2 𝗃 n ⁑ ( z ) = sin ⁑ ( z - 1 2 ⁒ n ⁒ Ο€ ) ⁒ βˆ‘ k = 0 ⌊ n / 2 βŒ‹ ( - 1 ) k ⁒ a 2 ⁒ k ⁒ ( n + 1 2 ) z 2 ⁒ k + 1 + cos ⁑ ( z - 1 2 ⁒ n ⁒ Ο€ ) ⁒ βˆ‘ k = 0 ⌊ ( n - 1 ) / 2 βŒ‹ ( - 1 ) k ⁒ a 2 ⁒ k + 1 ⁒ ( n + 1 2 ) z 2 ⁒ k + 2 spherical-Bessel-J 𝑛 𝑧 𝑧 1 2 𝑛 πœ‹ superscript subscript π‘˜ 0 𝑛 2 superscript 1 π‘˜ subscript π‘Ž 2 π‘˜ 𝑛 1 2 superscript 𝑧 2 π‘˜ 1 𝑧 1 2 𝑛 πœ‹ superscript subscript π‘˜ 0 𝑛 1 2 superscript 1 π‘˜ subscript π‘Ž 2 π‘˜ 1 𝑛 1 2 superscript 𝑧 2 π‘˜ 2 {\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}}		 β„œ ⁑ ( ( n + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - n - 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( - n - 1 2 ) ) + k + 1 ) > 0 , k β‰₯ 1 formulae-sequence 𝑛 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 π‘˜ 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,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 𝗃 0 ⁑ ( z ) = sin ⁑ z z spherical-Bessel-J 0 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\mathsf{j}_{0}\left(z\right)=\frac{\sin z}{z}}}
\sphBesselJ{0}@{z} = \frac{\sin@@{z}}{z}		 β„œ ⁑ ( ( 0 + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - 0 - 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( - 0 - 1 2 ) ) + k + 1 ) > 0 formulae-sequence 0 1 2 π‘˜ 1 0 formulae-sequence 0 1 2 π‘˜ 1 0 0 1 2 π‘˜ 1 0 {\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 𝗃 1 ⁑ ( z ) = sin ⁑ z z 2 - cos ⁑ z z spherical-Bessel-J 1 𝑧 𝑧 superscript 𝑧 2 𝑧 𝑧 {\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}		 β„œ ⁑ ( ( 1 + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - 1 - 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( - 1 - 1 2 ) ) + k + 1 ) > 0 formulae-sequence 1 1 2 π‘˜ 1 0 formulae-sequence 1 1 2 π‘˜ 1 0 1 1 2 π‘˜ 1 0 {\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 𝗃 2 ⁑ ( z ) = ( - 1 z + 3 z 3 ) ⁒ sin ⁑ z - 3 z 2 ⁒ cos ⁑ z spherical-Bessel-J 2 𝑧 1 𝑧 3 superscript 𝑧 3 𝑧 3 superscript 𝑧 2 𝑧 {\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}		 β„œ ⁑ ( ( 2 + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - 2 - 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( - 2 - 1 2 ) ) + k + 1 ) > 0 formulae-sequence 2 1 2 π‘˜ 1 0 formulae-sequence 2 1 2 π‘˜ 1 0 2 1 2 π‘˜ 1 0 {\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 𝗒 n ⁑ ( z ) = - cos ⁑ ( z - 1 2 ⁒ n ⁒ Ο€ ) ⁒ βˆ‘ k = 0 ⌊ n / 2 βŒ‹ ( - 1 ) k ⁒ a 2 ⁒ k ⁒ ( n + 1 2 ) z 2 ⁒ k + 1 + sin ⁑ ( z - 1 2 ⁒ n ⁒ Ο€ ) ⁒ βˆ‘ k = 0 ⌊ ( n - 1 ) / 2 βŒ‹ ( - 1 ) k ⁒ a 2 ⁒ k + 1 ⁒ ( n + 1 2 ) z 2 ⁒ k + 2 spherical-Bessel-Y 𝑛 𝑧 𝑧 1 2 𝑛 πœ‹ superscript subscript π‘˜ 0 𝑛 2 superscript 1 π‘˜ subscript π‘Ž 2 π‘˜ 𝑛 1 2 superscript 𝑧 2 π‘˜ 1 𝑧 1 2 𝑛 πœ‹ superscript subscript π‘˜ 0 𝑛 1 2 superscript 1 π‘˜ subscript π‘Ž 2 π‘˜ 1 𝑛 1 2 superscript 𝑧 2 π‘˜ 2 {\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}}		 β„œ ⁑ ( ( n + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( n + 1 2 ) ) + k + 1 ) > 0 , β„œ ⁑ ( ( - n - 1 2 ) + k + 1 ) > 0 , k β‰₯ 1 formulae-sequence 𝑛 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 π‘˜ 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,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 𝗒 0 ⁑ ( z ) = - cos ⁑ z z spherical-Bessel-Y 0 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle\mathsf{y}_{0}\left(z\right)=-\frac{\cos z}{z}}}
\sphBesselY{0}@{z} = -\frac{\cos@@{z}}{z}		 β„œ ⁑ ( ( 0 + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( 0 + 1 2 ) ) + k + 1 ) > 0 , β„œ ⁑ ( ( - 0 - 1 2 ) + k + 1 ) > 0 formulae-sequence 0 1 2 π‘˜ 1 0 formulae-sequence 0 1 2 π‘˜ 1 0 0 1 2 π‘˜ 1 0 {\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 𝗒 1 ⁑ ( z ) = - cos ⁑ z z 2 - sin ⁑ z z spherical-Bessel-Y 1 𝑧 𝑧 superscript 𝑧 2 𝑧 𝑧 {\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}		 β„œ ⁑ ( ( 1 + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( 1 + 1 2 ) ) + k + 1 ) > 0 , β„œ ⁑ ( ( - 1 - 1 2 ) + k + 1 ) > 0 formulae-sequence 1 1 2 π‘˜ 1 0 formulae-sequence 1 1 2 π‘˜ 1 0 1 1 2 π‘˜ 1 0 {\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 𝗒 2 ⁑ ( z ) = ( 1 z - 3 z 3 ) ⁒ cos ⁑ z - 3 z 2 ⁒ sin ⁑ z spherical-Bessel-Y 2 𝑧 1 𝑧 3 superscript 𝑧 3 𝑧 3 superscript 𝑧 2 𝑧 {\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}		 β„œ ⁑ ( ( 2 + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( 2 + 1 2 ) ) + k + 1 ) > 0 , β„œ ⁑ ( ( - 2 - 1 2 ) + k + 1 ) > 0 formulae-sequence 2 1 2 π‘˜ 1 0 formulae-sequence 2 1 2 π‘˜ 1 0 2 1 2 π‘˜ 1 0 {\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 𝗁 n ( 1 ) ⁑ ( z ) = e i ⁒ z ⁒ βˆ‘ k = 0 n i k - n - 1 ⁒ a k ⁒ ( n + 1 2 ) z k + 1 spherical-Hankel-H-1-Bessel-third-kind 𝑛 𝑧 superscript 𝑒 𝑖 𝑧 superscript subscript π‘˜ 0 𝑛 superscript 𝑖 π‘˜ 𝑛 1 subscript π‘Ž π‘˜ 𝑛 1 2 superscript 𝑧 π‘˜ 1 {\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}}		 k β‰₯ 1 π‘˜ 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 𝗁 n ( 2 ) ⁑ ( z ) = e - i ⁒ z ⁒ βˆ‘ k = 0 n ( - i ) k - n - 1 ⁒ a k ⁒ ( n + 1 2 ) z k + 1 spherical-Hankel-H-2-Bessel-third-kind 𝑛 𝑧 superscript 𝑒 𝑖 𝑧 superscript subscript π‘˜ 0 𝑛 superscript 𝑖 π‘˜ 𝑛 1 subscript π‘Ž π‘˜ 𝑛 1 2 superscript 𝑧 π‘˜ 1 {\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}}		 k β‰₯ 1 π‘˜ 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 𝗂 n ( 1 ) ⁑ ( z ) = 1 2 ⁒ e z ⁒ βˆ‘ k = 0 n ( - 1 ) k ⁒ a k ⁒ ( n + 1 2 ) z k + 1 + ( - 1 ) n + 1 ⁒ 1 2 ⁒ e - z ⁒ βˆ‘ k = 0 n a k ⁒ ( n + 1 2 ) z k + 1 spherical-Bessel-I-1 𝑛 𝑧 1 2 superscript 𝑒 𝑧 superscript subscript π‘˜ 0 𝑛 superscript 1 π‘˜ subscript π‘Ž π‘˜ 𝑛 1 2 superscript 𝑧 π‘˜ 1 superscript 1 𝑛 1 1 2 superscript 𝑒 𝑧 superscript subscript π‘˜ 0 𝑛 subscript π‘Ž π‘˜ 𝑛 1 2 superscript 𝑧 π‘˜ 1 {\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}}		 β„œ ⁑ ( ( n + 1 2 ) + k + 1 ) > 0 , k β‰₯ 1 formulae-sequence 𝑛 1 2 π‘˜ 1 0 π‘˜ 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 𝗂 0 ( 1 ) ⁑ ( z ) = sinh ⁑ z z spherical-Bessel-I-1 0 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle{\mathsf{i}^{(1)}_{0}}\left(z\right)=\frac{\sinh z% }{z}}}
\modsphBesseli{1}{0}@{z} = \frac{\sinh@@{z}}{z}		 β„œ ⁑ ( ( 0 + 1 2 ) + k + 1 ) > 0 0 1 2 π‘˜ 1 0 {\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 𝗂 1 ( 1 ) ⁑ ( z ) = - sinh ⁑ z z 2 + cosh ⁑ z z spherical-Bessel-I-1 1 𝑧 𝑧 superscript 𝑧 2 𝑧 𝑧 {\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}		 β„œ ⁑ ( ( 1 + 1 2 ) + k + 1 ) > 0 1 1 2 π‘˜ 1 0 {\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 𝗂 2 ( 1 ) ⁑ ( z ) = ( 1 z + 3 z 3 ) ⁒ sinh ⁑ z - 3 z 2 ⁒ cosh ⁑ z spherical-Bessel-I-1 2 𝑧 1 𝑧 3 superscript 𝑧 3 𝑧 3 superscript 𝑧 2 𝑧 {\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}		 β„œ ⁑ ( ( 2 + 1 2 ) + k + 1 ) > 0 2 1 2 π‘˜ 1 0 {\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 𝗂 n ( 2 ) ⁑ ( z ) = 1 2 ⁒ e z ⁒ βˆ‘ k = 0 n ( - 1 ) k ⁒ a k ⁒ ( n + 1 2 ) z k + 1 + ( - 1 ) n ⁒ 1 2 ⁒ e - z ⁒ βˆ‘ k = 0 n a k ⁒ ( n + 1 2 ) z k + 1 spherical-Bessel-I-2 𝑛 𝑧 1 2 superscript 𝑒 𝑧 superscript subscript π‘˜ 0 𝑛 superscript 1 π‘˜ subscript π‘Ž π‘˜ 𝑛 1 2 superscript 𝑧 π‘˜ 1 superscript 1 𝑛 1 2 superscript 𝑒 𝑧 superscript subscript π‘˜ 0 𝑛 subscript π‘Ž π‘˜ 𝑛 1 2 superscript 𝑧 π‘˜ 1 {\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}}		 β„œ ⁑ ( ( - n - 1 2 ) + k + 1 ) > 0 , k β‰₯ 1 formulae-sequence 𝑛 1 2 π‘˜ 1 0 π‘˜ 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 𝗂 0 ( 2 ) ⁑ ( z ) = cosh ⁑ z z spherical-Bessel-I-2 0 𝑧 𝑧 𝑧 {\displaystyle{\displaystyle{\mathsf{i}^{(2)}_{0}}\left(z\right)=\frac{\cosh z% }{z}}}
\modsphBesseli{2}{0}@{z} = \frac{\cosh@@{z}}{z}		 β„œ ⁑ ( ( - 0 - 1 2 ) + k + 1 ) > 0 0 1 2 π‘˜ 1 0 {\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 𝗂 1 ( 2 ) ⁑ ( z ) = - cosh ⁑ z z 2 + sinh ⁑ z z spherical-Bessel-I-2 1 𝑧 𝑧 superscript 𝑧 2 𝑧 𝑧 {\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}		 β„œ ⁑ ( ( - 1 - 1 2 ) + k + 1 ) > 0 1 1 2 π‘˜ 1 0 {\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 𝗂 2 ( 2 ) ⁑ ( z ) = ( 1 z + 3 z 3 ) ⁒ cosh ⁑ z - 3 z 2 ⁒ sinh ⁑ z spherical-Bessel-I-2 2 𝑧 1 𝑧 3 superscript 𝑧 3 𝑧 3 superscript 𝑧 2 𝑧 {\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}		 β„œ ⁑ ( ( - 2 - 1 2 ) + k + 1 ) > 0 2 1 2 π‘˜ 1 0 {\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 𝗄 n ⁑ ( z ) = 1 2 ⁒ Ο€ ⁒ e - z ⁒ βˆ‘ k = 0 n a k ⁒ ( n + 1 2 ) z k + 1 spherical-Bessel-K 𝑛 𝑧 1 2 πœ‹ superscript 𝑒 𝑧 superscript subscript π‘˜ 0 𝑛 subscript π‘Ž π‘˜ 𝑛 1 2 superscript 𝑧 π‘˜ 1 {\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}}		 k β‰₯ 1 π‘˜ 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 𝗄 0 ⁑ ( z ) = 1 2 ⁒ Ο€ ⁒ e - z z spherical-Bessel-K 0 𝑧 1 2 πœ‹ superscript 𝑒 𝑧 𝑧 {\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 𝗄 1 ⁑ ( z ) = 1 2 ⁒ Ο€ ⁒ e - z ⁒ ( 1 z + 1 z 2 ) spherical-Bessel-K 1 𝑧 1 2 πœ‹ superscript 𝑒 𝑧 1 𝑧 1 superscript 𝑧 2 {\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 𝗄 2 ⁑ ( z ) = 1 2 ⁒ Ο€ ⁒ e - z ⁒ ( 1 z + 3 z 2 + 3 z 3 ) spherical-Bessel-K 2 𝑧 1 2 πœ‹ superscript 𝑒 𝑧 1 𝑧 3 superscript 𝑧 2 3 superscript 𝑧 3 {\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 𝗃 n ⁑ ( z ) = z n ⁒ ( - 1 z ⁒ d d z ) n ⁒ sin ⁑ z z spherical-Bessel-J 𝑛 𝑧 superscript 𝑧 𝑛 superscript 1 𝑧 derivative 𝑧 𝑛 𝑧 𝑧 {\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}		 β„œ ⁑ ( ( n + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - n - 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( - n - 1 2 ) ) + k + 1 ) > 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 𝑛 1 2 π‘˜ 1 0 {\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 𝗒 n ⁑ ( z ) = - z n ⁒ ( - 1 z ⁒ d d z ) n ⁒ cos ⁑ z z spherical-Bessel-Y 𝑛 𝑧 superscript 𝑧 𝑛 superscript 1 𝑧 derivative 𝑧 𝑛 𝑧 𝑧 {\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}		 β„œ ⁑ ( ( n + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( n + 1 2 ) ) + k + 1 ) > 0 , β„œ ⁑ ( ( - n - 1 2 ) + k + 1 ) > 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 𝑛 1 2 π‘˜ 1 0 {\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 𝗂 n ( 1 ) ⁑ ( z ) = z n ⁒ ( 1 z ⁒ d d z ) n ⁑ sinh ⁑ z z spherical-Bessel-I-1 𝑛 𝑧 superscript 𝑧 𝑛 superscript 1 𝑧 derivative 𝑧 𝑛 𝑧 𝑧 {\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}		 β„œ ⁑ ( ( n + 1 2 ) + k + 1 ) > 0 𝑛 1 2 π‘˜ 1 0 {\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 𝗂 n ( 2 ) ⁑ ( z ) = z n ⁒ ( 1 z ⁒ d d z ) n ⁑ cosh ⁑ z z spherical-Bessel-I-2 𝑛 𝑧 superscript 𝑧 𝑛 superscript 1 𝑧 derivative 𝑧 𝑛 𝑧 𝑧 {\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}		 β„œ ⁑ ( ( - n - 1 2 ) + k + 1 ) > 0 𝑛 1 2 π‘˜ 1 0 {\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 𝗄 n ⁑ ( z ) = ( - 1 ) n ⁒ 1 2 ⁒ Ο€ ⁒ z n ⁒ ( 1 z ⁒ d d z ) n ⁑ e - z z spherical-Bessel-K 𝑛 𝑧 superscript 1 𝑛 1 2 πœ‹ superscript 𝑧 𝑛 superscript 1 𝑧 derivative 𝑧 𝑛 superscript 𝑒 𝑧 𝑧 {\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 𝗃 n 2 ⁑ ( z ) + 𝗒 n 2 ⁑ ( z ) = βˆ‘ k = 0 n s k ⁒ ( n + 1 2 ) z 2 ⁒ k + 2 spherical-Bessel-J 𝑛 2 𝑧 spherical-Bessel-Y 𝑛 2 𝑧 superscript subscript π‘˜ 0 𝑛 subscript 𝑠 π‘˜ 𝑛 1 2 superscript 𝑧 2 π‘˜ 2 {\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}}		 β„œ ⁑ ( ( n + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - n - 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( - n - 1 2 ) ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( n + 1 2 ) ) + k + 1 ) > 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 𝑛 1 2 π‘˜ 1 0 {\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 𝗃 0 2 ⁑ ( z ) + 𝗒 0 2 ⁑ ( z ) = z - 2 spherical-Bessel-J 0 2 𝑧 spherical-Bessel-Y 0 2 𝑧 superscript 𝑧 2 {\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}		 β„œ ⁑ ( ( 0 + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - 0 - 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( - 0 - 1 2 ) ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( 0 + 1 2 ) ) + k + 1 ) > 0 formulae-sequence 0 1 2 π‘˜ 1 0 formulae-sequence 0 1 2 π‘˜ 1 0 formulae-sequence 0 1 2 π‘˜ 1 0 0 1 2 π‘˜ 1 0 {\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 𝗃 1 2 ⁑ ( z ) + 𝗒 1 2 ⁑ ( z ) = z - 2 + z - 4 spherical-Bessel-J 1 2 𝑧 spherical-Bessel-Y 1 2 𝑧 superscript 𝑧 2 superscript 𝑧 4 {\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}		 β„œ ⁑ ( ( 1 + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - 1 - 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( - 1 - 1 2 ) ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( 1 + 1 2 ) ) + k + 1 ) > 0 formulae-sequence 1 1 2 π‘˜ 1 0 formulae-sequence 1 1 2 π‘˜ 1 0 formulae-sequence 1 1 2 π‘˜ 1 0 1 1 2 π‘˜ 1 0 {\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 𝗃 2 2 ⁑ ( z ) + 𝗒 2 2 ⁑ ( z ) = z - 2 + 3 ⁒ z - 4 + 9 ⁒ z - 6 spherical-Bessel-J 2 2 𝑧 spherical-Bessel-Y 2 2 𝑧 superscript 𝑧 2 3 superscript 𝑧 4 9 superscript 𝑧 6 {\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}		 β„œ ⁑ ( ( 2 + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - 2 - 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( - 2 - 1 2 ) ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( 2 + 1 2 ) ) + k + 1 ) > 0 formulae-sequence 2 1 2 π‘˜ 1 0 formulae-sequence 2 1 2 π‘˜ 1 0 formulae-sequence 2 1 2 π‘˜ 1 0 2 1 2 π‘˜ 1 0 {\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 ( 𝗂 n ( 1 ) ⁑ ( z ) ) 2 - ( 𝗂 n ( 2 ) ⁑ ( z ) ) 2 = ( - 1 ) n + 1 ⁒ βˆ‘ k = 0 n ( - 1 ) k ⁒ s k ⁒ ( n + 1 2 ) z 2 ⁒ k + 2 superscript spherical-Bessel-I-1 𝑛 𝑧 2 superscript spherical-Bessel-I-2 𝑛 𝑧 2 superscript 1 𝑛 1 superscript subscript π‘˜ 0 𝑛 superscript 1 π‘˜ subscript 𝑠 π‘˜ 𝑛 1 2 superscript 𝑧 2 π‘˜ 2 {\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}}		 β„œ ⁑ ( ( n + 1 2 ) + k + 1 ) > 0 , β„œ ⁑ ( ( - n - 1 2 ) + k + 1 ) > 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 𝑛 1 2 π‘˜ 1 0 {\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
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