Bessel Functions - 10.53 Power Series

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
10.53.E1 𝗃 n ⁑ ( z ) = z n ⁒ βˆ‘ k = 0 ∞ ( - 1 2 ⁒ z 2 ) k k ! ⁒ ( 2 ⁒ n + 2 ⁒ k + 1 ) !! spherical-Bessel-J 𝑛 𝑧 superscript 𝑧 𝑛 superscript subscript π‘˜ 0 superscript 1 2 superscript 𝑧 2 π‘˜ π‘˜ double-factorial 2 𝑛 2 π‘˜ 1 {\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)!!}		 | 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 𝑧 formulae-sequence 𝑛 1 2 π‘˜ 1 0 formulae-sequence 𝑛 1 2 π‘˜ 1 0 𝑛 1 2 π‘˜ 1 0 {\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 𝗒 n ⁑ ( z ) = - 1 z n + 1 ⁒ βˆ‘ k = 0 n ( 2 ⁒ n - 2 ⁒ k - 1 ) !! ⁒ ( 1 2 ⁒ z 2 ) k k ! + ( - 1 ) n + 1 z n + 1 ⁒ βˆ‘ k = n + 1 ∞ ( - 1 2 ⁒ z 2 ) k k ! ⁒ ( 2 ⁒ k - 2 ⁒ n - 1 ) !! spherical-Bessel-Y 𝑛 𝑧 1 superscript 𝑧 𝑛 1 superscript subscript π‘˜ 0 𝑛 double-factorial 2 𝑛 2 π‘˜ 1 superscript 1 2 superscript 𝑧 2 π‘˜ π‘˜ superscript 1 𝑛 1 superscript 𝑧 𝑛 1 superscript subscript π‘˜ 𝑛 1 superscript 1 2 superscript 𝑧 2 π‘˜ π‘˜ double-factorial 2 π‘˜ 2 𝑛 1 {\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)!!}		 0 < | 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 fragments 0 | z | , | z | . , 𝑛 1 2 π‘˜ 1 0 , 𝑛 1 2 π‘˜ 1 0 , 𝑛 1 2 π‘˜ 1 0 {\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 𝗂 n ( 1 ) ⁑ ( z ) = z n ⁒ βˆ‘ k = 0 ∞ ( 1 2 ⁒ z 2 ) k k ! ⁒ ( 2 ⁒ n + 2 ⁒ k + 1 ) !! spherical-Bessel-I-1 𝑛 𝑧 superscript 𝑧 𝑛 superscript subscript π‘˜ 0 superscript 1 2 superscript 𝑧 2 π‘˜ π‘˜ double-factorial 2 𝑛 2 π‘˜ 1 {\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)!!}		 | z | < ∞ , β„œ ⁑ ( ( n + 1 2 ) + k + 1 ) > 0 formulae-sequence 𝑧 𝑛 1 2 π‘˜ 1 0 {\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 𝗂 n ( 2 ) ⁑ ( z ) = ( - 1 ) n z n + 1 ⁒ βˆ‘ k = 0 n ( 2 ⁒ n - 2 ⁒ k - 1 ) !! ⁒ ( - 1 2 ⁒ z 2 ) k k ! + 1 z n + 1 ⁒ βˆ‘ k = n + 1 ∞ ( 1 2 ⁒ z 2 ) k k ! ⁒ ( 2 ⁒ k - 2 ⁒ n - 1 ) !! spherical-Bessel-I-2 𝑛 𝑧 superscript 1 𝑛 superscript 𝑧 𝑛 1 superscript subscript π‘˜ 0 𝑛 double-factorial 2 𝑛 2 π‘˜ 1 superscript 1 2 superscript 𝑧 2 π‘˜ π‘˜ 1 superscript 𝑧 𝑛 1 superscript subscript π‘˜ 𝑛 1 superscript 1 2 superscript 𝑧 2 π‘˜ π‘˜ double-factorial 2 π‘˜ 2 𝑛 1 {\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)!!}		 0 < | z | , | z | < ∞ . , β„œ ⁑ ( ( - n - 1 2 ) + k + 1 ) > 0 fragments 0 | z | , | z | . , 𝑛 1 2 π‘˜ 1 0 {\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
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