Coulomb Functions - 33.20 Expansions for Small

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DLMF Formula Constraints Maple Mathematica Symbolic
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33.20.E4 π–₯ k ⁒ ( β„“ ; r ) = βˆ‘ p = 2 ⁒ k 3 ⁒ k ( 2 ⁒ r ) ( p + 1 ) / 2 ⁒ C k , p ⁒ J 2 ⁒ β„“ + 1 + p ⁑ ( 8 ⁒ r ) subscript π–₯ π‘˜ β„“ π‘Ÿ superscript subscript 𝑝 2 π‘˜ 3 π‘˜ superscript 2 π‘Ÿ 𝑝 1 2 subscript 𝐢 π‘˜ 𝑝 Bessel-J 2 β„“ 1 𝑝 8 π‘Ÿ {\displaystyle{\displaystyle{\sf F}_{k}(\ell;r)=\sum_{p=2k}^{3k}(2r)^{(p+1)/2}% C_{k,p}J_{2\ell+1+p}\left(\sqrt{8r}\right)}}
{\sf F}_{k}(\ell;r) = \sum_{p=2k}^{3k}(2r)^{(p+1)/2}C_{k,p}\BesselJ{2\ell+1+p}@{\sqrt{8r}}		 r > 0 , β„œ ⁑ ( ( 2 ⁒ β„“ + 1 + p ) + k + 1 ) > 0 formulae-sequence π‘Ÿ 0 2 β„“ 1 𝑝 π‘˜ 1 0 {\displaystyle{\displaystyle r>0,\Re((2\ell+1+p)+k+1)>0}} 
F[k](ell ; r) = sum((2*r)^((p + 1)/2)* C[k , p]*BesselJ(2*ell + 1 + p, sqrt(8*r)), p = 2*k..3*k)

Subscript[F, k][\[ScriptL]; r] == Sum[(2*r)^((p + 1)/2)* Subscript[C, k , p]*BesselJ[2*\[ScriptL]+ 1 + p, Sqrt[8*r]], {p, 2*k, 3*k}, GenerateConditions->None]
Translation Error Translation Error - -
33.20.E5 π–₯ k ⁒ ( β„“ ; r ) = βˆ‘ p = 2 ⁒ k 3 ⁒ k ( - 1 ) β„“ + 1 + p ⁒ ( 2 ⁒ | r | ) ( p + 1 ) / 2 ⁒ C k , p ⁒ I 2 ⁒ β„“ + 1 + p ⁑ ( 8 ⁒ | r | ) subscript π–₯ π‘˜ β„“ π‘Ÿ superscript subscript 𝑝 2 π‘˜ 3 π‘˜ superscript 1 β„“ 1 𝑝 superscript 2 π‘Ÿ 𝑝 1 2 subscript 𝐢 π‘˜ 𝑝 modified-Bessel-first-kind 2 β„“ 1 𝑝 8 π‘Ÿ {\displaystyle{\displaystyle{\sf F}_{k}(\ell;r)=\sum_{p=2k}^{3k}(-1)^{\ell+1+p% }(2|r|)^{(p+1)/2}C_{k,p}I_{2\ell+1+p}\left(\sqrt{8|r|}\right)}}
{\sf F}_{k}(\ell;r) = \sum_{p=2k}^{3k}(-1)^{\ell+1+p}(2|r|)^{(p+1)/2}C_{k,p}\modBesselI{2\ell+1+p}@{\sqrt{8|r|}}		 r < 0 , β„œ ⁑ ( ( 2 ⁒ β„“ + 1 + p ) + k + 1 ) > 0 formulae-sequence π‘Ÿ 0 2 β„“ 1 𝑝 π‘˜ 1 0 {\displaystyle{\displaystyle r<0,\Re((2\ell+1+p)+k+1)>0}} 
F[k](ell ; r) = sum((- 1)^(ell + 1 + p)*(2*abs(r))^((p + 1)/2)* C[k , p]*BesselI(2*ell + 1 + p, sqrt(8*abs(r))), p = 2*k..3*k)

Subscript[F, k][\[ScriptL]; r] == Sum[(- 1)^(\[ScriptL]+ 1 + p)*(2*Abs[r])^((p + 1)/2)* Subscript[C, k , p]*BesselI[2*\[ScriptL]+ 1 + p, Sqrt[8*Abs[r]]], {p, 2*k, 3*k}, GenerateConditions->None]
Translation Error Translation Error - -
33.20#Ex5 C k , p = 0 subscript 𝐢 π‘˜ 𝑝 0 {\displaystyle{\displaystyle C_{k,p}=0}}
C_{k,p} = 0		 p < 2 ⁒ k , p > 3 ⁒ k formulae-sequence 𝑝 2 π‘˜ 𝑝 3 π‘˜ {\displaystyle{\displaystyle p<2k,p>3k}} 
C[k , p] = 0
 
Subscript[C, k , p] == 0
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33.20#Ex6 C k , p = ( - ( 2 ⁒ β„“ + p ) ⁒ C k - 1 , p - 2 + C k - 1 , p - 3 ) / ( 4 ⁒ p ) subscript 𝐢 π‘˜ 𝑝 2 β„“ 𝑝 subscript 𝐢 π‘˜ 1 𝑝 2 subscript 𝐢 π‘˜ 1 𝑝 3 4 𝑝 {\displaystyle{\displaystyle C_{k,p}=\left(-(2\ell+p)C_{k-1,p-2}+C_{k-1,p-3}% \right)/(4p)}}
C_{k,p} = \left(-(2\ell+p)C_{k-1,p-2}+C_{k-1,p-3}\right)/(4p)		 k > 0 , 2 ⁒ k ≀ p , p ≀ 3 ⁒ k formulae-sequence π‘˜ 0 formulae-sequence 2 π‘˜ 𝑝 𝑝 3 π‘˜ {\displaystyle{\displaystyle k>0,2k\leq p,p\leq 3k}} 
C[k , p] = (-(2*ell + p)*C[k - 1 , p - 2]+ C[k - 1 , p - 3])/(4*p)
 
Subscript[C, k , p] == (-(2*\[ScriptL]+ p)*Subscript[C, k - 1 , p - 2]+ Subscript[C, k - 1 , p - 3])/(4*p)
 Skipped - no semantic math Skipped - no semantic math - -
33.20.E8 𝖧 k ⁒ ( β„“ ; r ) = βˆ‘ p = 2 ⁒ k 3 ⁒ k ( 2 ⁒ r ) ( p + 1 ) / 2 ⁒ C k , p ⁒ Y 2 ⁒ β„“ + 1 + p ⁑ ( 8 ⁒ r ) subscript 𝖧 π‘˜ β„“ π‘Ÿ superscript subscript 𝑝 2 π‘˜ 3 π‘˜ superscript 2 π‘Ÿ 𝑝 1 2 subscript 𝐢 π‘˜ 𝑝 Bessel-Y-Weber 2 β„“ 1 𝑝 8 π‘Ÿ {\displaystyle{\displaystyle{\sf H}_{k}(\ell;r)=\sum_{p=2k}^{3k}(2r)^{(p+1)/2}% C_{k,p}Y_{2\ell+1+p}\left(\sqrt{8r}\right)}}
{\sf H}_{k}(\ell;r) = \sum_{p=2k}^{3k}(2r)^{(p+1)/2}C_{k,p}\BesselY{2\ell+1+p}@{\sqrt{8r}}		 r > 0 , β„œ ⁑ ( ( 2 ⁒ β„“ + 1 + p ) + k + 1 ) > 0 , β„œ ⁑ ( ( - ( 2 ⁒ β„“ + 1 + p ) ) + k + 1 ) > 0 formulae-sequence π‘Ÿ 0 formulae-sequence 2 β„“ 1 𝑝 π‘˜ 1 0 2 β„“ 1 𝑝 π‘˜ 1 0 {\displaystyle{\displaystyle r>0,\Re((2\ell+1+p)+k+1)>0,\Re((-(2\ell+1+p))+k+1% )>0}} 
H[k](ell ; r) = sum((2*r)^((p + 1)/2)* C[k , p]*BesselY(2*ell + 1 + p, sqrt(8*r)), p = 2*k..3*k)

Subscript[H, k][\[ScriptL]; r] == Sum[(2*r)^((p + 1)/2)* Subscript[C, k , p]*BesselY[2*\[ScriptL]+ 1 + p, Sqrt[8*r]], {p, 2*k, 3*k}, GenerateConditions->None]
Translation Error Translation Error - -
33.20.E9 𝖧 k ⁒ ( β„“ ; r ) = ( - 1 ) β„“ + 1 ⁒ 2 Ο€ ⁒ βˆ‘ p = 2 ⁒ k 3 ⁒ k ( 2 ⁒ | r | ) ( p + 1 ) / 2 ⁒ C k , p ⁒ K 2 ⁒ β„“ + 1 + p ⁑ ( 8 ⁒ | r | ) subscript 𝖧 π‘˜ β„“ π‘Ÿ superscript 1 β„“ 1 2 πœ‹ superscript subscript 𝑝 2 π‘˜ 3 π‘˜ superscript 2 π‘Ÿ 𝑝 1 2 subscript 𝐢 π‘˜ 𝑝 modified-Bessel-second-kind 2 β„“ 1 𝑝 8 π‘Ÿ {\displaystyle{\displaystyle{\sf H}_{k}(\ell;r)=(-1)^{\ell+1}\frac{2}{\pi}\sum% _{p=2k}^{3k}(2|r|)^{(p+1)/2}C_{k,p}K_{2\ell+1+p}\left(\sqrt{8|r|}\right)}}
{\sf H}_{k}(\ell;r) = (-1)^{\ell+1}\frac{2}{\pi}\sum_{p=2k}^{3k}(2|r|)^{(p+1)/2}C_{k,p}\modBesselK{2\ell+1+p}@{\sqrt{8|r|}}		 r < 0 π‘Ÿ 0 {\displaystyle{\displaystyle r<0}} 
H[k](ell ; r) = (- 1)^(ell + 1)*(2)/(Pi)*sum((2*abs(r))^((p + 1)/2)* C[k , p]*BesselK(2*ell + 1 + p, sqrt(8*abs(r))), p = 2*k..3*k)

Subscript[H, k][\[ScriptL]; r] == (- 1)^(\[ScriptL]+ 1)*Divide[2,Pi]*Sum[(2*Abs[r])^((p + 1)/2)* Subscript[C, k , p]*BesselK[2*\[ScriptL]+ 1 + p, Sqrt[8*Abs[r]]], {p, 2*k, 3*k}, GenerateConditions->None]
Translation Error Translation Error - -
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