33.20: Difference between revisions

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! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/33.20.E4 33.20.E4] || [[Item:Q9672|<math>{\sf F}_{k}(\ell;r) = \sum_{p=2k}^{3k}(2r)^{(p+1)/2}C_{k,p}\BesselJ{2\ell+1+p}@{\sqrt{8r}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>{\sf F}_{k}(\ell;r) = \sum_{p=2k}^{3k}(2r)^{(p+1)/2}C_{k,p}\BesselJ{2\ell+1+p}@{\sqrt{8r}}</syntaxhighlight> || <math>r > 0, \realpart@@{((2\ell+1+p)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Translation Error || Translation Error || - || -
| [https://dlmf.nist.gov/33.20.E4 33.20.E4] || <math qid="Q9672">{\sf F}_{k}(\ell;r) = \sum_{p=2k}^{3k}(2r)^{(p+1)/2}C_{k,p}\BesselJ{2\ell+1+p}@{\sqrt{8r}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>{\sf F}_{k}(\ell;r) = \sum_{p=2k}^{3k}(2r)^{(p+1)/2}C_{k,p}\BesselJ{2\ell+1+p}@{\sqrt{8r}}</syntaxhighlight> || <math>r > 0, \realpart@@{((2\ell+1+p)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Translation Error || Translation Error || - || -
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| [https://dlmf.nist.gov/33.20.E5 33.20.E5] || [[Item:Q9673|<math>{\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|}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>{\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|}}</syntaxhighlight> || <math>r < 0, \realpart@@{((2\ell+1+p)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Translation Error || Translation Error || - || -
| [https://dlmf.nist.gov/33.20.E5 33.20.E5] || <math qid="Q9673">{\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|}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>{\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|}}</syntaxhighlight> || <math>r < 0, \realpart@@{((2\ell+1+p)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Translation Error || Translation Error || - || -
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| [https://dlmf.nist.gov/33.20#Ex5 33.20#Ex5] || [[Item:Q9674|<math>C_{k,p} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>C_{k,p} = 0</syntaxhighlight> || <math>p < 2k, p > 3k</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">C[k , p] = 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[C, k , p] == 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/33.20#Ex5 33.20#Ex5] || <math qid="Q9674">C_{k,p} = 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>C_{k,p} = 0</syntaxhighlight> || <math>p < 2k, p > 3k</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">C[k , p] = 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[C, k , p] == 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/33.20#Ex6 33.20#Ex6] || [[Item:Q9675|<math>C_{k,p} = \left(-(2\ell+p)C_{k-1,p-2}+C_{k-1,p-3}\right)/(4p)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>C_{k,p} = \left(-(2\ell+p)C_{k-1,p-2}+C_{k-1,p-3}\right)/(4p)</syntaxhighlight> || <math>k > 0, 2k \leq p, p \leq 3k</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">C[k , p] = (-(2*ell + p)*C[k - 1 , p - 2]+ C[k - 1 , p - 3])/(4*p)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[C, k , p] == (-(2*\[ScriptL]+ p)*Subscript[C, k - 1 , p - 2]+ Subscript[C, k - 1 , p - 3])/(4*p)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/33.20#Ex6 33.20#Ex6] || <math qid="Q9675">C_{k,p} = \left(-(2\ell+p)C_{k-1,p-2}+C_{k-1,p-3}\right)/(4p)</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>C_{k,p} = \left(-(2\ell+p)C_{k-1,p-2}+C_{k-1,p-3}\right)/(4p)</syntaxhighlight> || <math>k > 0, 2k \leq p, p \leq 3k</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">C[k , p] = (-(2*ell + p)*C[k - 1 , p - 2]+ C[k - 1 , p - 3])/(4*p)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[C, k , p] == (-(2*\[ScriptL]+ p)*Subscript[C, k - 1 , p - 2]+ Subscript[C, k - 1 , p - 3])/(4*p)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/33.20.E8 33.20.E8] || [[Item:Q9677|<math>{\sf H}_{k}(\ell;r) = \sum_{p=2k}^{3k}(2r)^{(p+1)/2}C_{k,p}\BesselY{2\ell+1+p}@{\sqrt{8r}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>{\sf H}_{k}(\ell;r) = \sum_{p=2k}^{3k}(2r)^{(p+1)/2}C_{k,p}\BesselY{2\ell+1+p}@{\sqrt{8r}}</syntaxhighlight> || <math>r > 0, \realpart@@{((2\ell+1+p)+k+1)} > 0, \realpart@@{((-(2\ell+1+p))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Translation Error || Translation Error || - || -
| [https://dlmf.nist.gov/33.20.E8 33.20.E8] || <math qid="Q9677">{\sf H}_{k}(\ell;r) = \sum_{p=2k}^{3k}(2r)^{(p+1)/2}C_{k,p}\BesselY{2\ell+1+p}@{\sqrt{8r}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>{\sf H}_{k}(\ell;r) = \sum_{p=2k}^{3k}(2r)^{(p+1)/2}C_{k,p}\BesselY{2\ell+1+p}@{\sqrt{8r}}</syntaxhighlight> || <math>r > 0, \realpart@@{((2\ell+1+p)+k+1)} > 0, \realpart@@{((-(2\ell+1+p))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Translation Error || Translation Error || - || -
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| [https://dlmf.nist.gov/33.20.E9 33.20.E9] || [[Item:Q9678|<math>{\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|}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>{\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|}}</syntaxhighlight> || <math>r < 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Translation Error || Translation Error || - || -
| [https://dlmf.nist.gov/33.20.E9 33.20.E9] || <math qid="Q9678">{\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|}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>{\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|}}</syntaxhighlight> || <math>r < 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Translation Error || Translation Error || - || -
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Latest revision as of 12:14, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
33.20.E4 𝖥 k ⁢ ( ℓ ; r ) = ∑ p = 2 ⁢ k 3 ⁢ k ( 2 ⁢ r ) ( p + 1 ) / 2 ⁢ C k , p ⁢ J 2 ⁢ ℓ + 1 + p ⁡ ( 8 ⁢ r ) subscript 𝖥 𝑘 ℓ 𝑟 superscript subscript 𝑝 2 𝑘 3 𝑘 superscript 2 𝑟 𝑝 1 2 subscript 𝐶 𝑘 𝑝 Bessel-J 2 ℓ 1 𝑝 8 𝑟 {\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 , ℜ ⁡ ( ( 2 ⁢ ℓ + 1 + p ) + k + 1 ) > 0 formulae-sequence 𝑟 0 2 ℓ 1 𝑝 𝑘 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 𝖥 k ⁢ ( ℓ ; r ) = ∑ p = 2 ⁢ k 3 ⁢ k ( - 1 ) ℓ + 1 + p ⁢ ( 2 ⁢ | r | ) ( p + 1 ) / 2 ⁢ C k , p ⁢ I 2 ⁢ ℓ + 1 + p ⁡ ( 8 ⁢ | r | ) subscript 𝖥 𝑘 ℓ 𝑟 superscript subscript 𝑝 2 𝑘 3 𝑘 superscript 1 ℓ 1 𝑝 superscript 2 𝑟 𝑝 1 2 subscript 𝐶 𝑘 𝑝 modified-Bessel-first-kind 2 ℓ 1 𝑝 8 𝑟 {\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 , ℜ ⁡ ( ( 2 ⁢ ℓ + 1 + p ) + k + 1 ) > 0 formulae-sequence 𝑟 0 2 ℓ 1 𝑝 𝑘 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 𝐶 𝑘 𝑝 0 {\displaystyle{\displaystyle C_{k,p}=0}}
C_{k,p} = 0		 p < 2 ⁢ k , p > 3 ⁢ k formulae-sequence 𝑝 2 𝑘 𝑝 3 𝑘 {\displaystyle{\displaystyle p<2k,p>3k}} 
C[k , p] = 0
 
Subscript[C, k , p] == 0
 Skipped - no semantic math Skipped - no semantic math - -
33.20#Ex6 C k , p = ( - ( 2 ⁢ ℓ + p ) ⁢ C k - 1 , p - 2 + C k - 1 , p - 3 ) / ( 4 ⁢ p ) subscript 𝐶 𝑘 𝑝 2 ℓ 𝑝 subscript 𝐶 𝑘 1 𝑝 2 subscript 𝐶 𝑘 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 𝑘 0 formulae-sequence 2 𝑘 𝑝 𝑝 3 𝑘 {\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 ⁢ ( ℓ ; r ) = ∑ p = 2 ⁢ k 3 ⁢ k ( 2 ⁢ r ) ( p + 1 ) / 2 ⁢ C k , p ⁢ Y 2 ⁢ ℓ + 1 + p ⁡ ( 8 ⁢ r ) subscript 𝖧 𝑘 ℓ 𝑟 superscript subscript 𝑝 2 𝑘 3 𝑘 superscript 2 𝑟 𝑝 1 2 subscript 𝐶 𝑘 𝑝 Bessel-Y-Weber 2 ℓ 1 𝑝 8 𝑟 {\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 , ℜ ⁡ ( ( 2 ⁢ ℓ + 1 + p ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ( 2 ⁢ ℓ + 1 + p ) ) + k + 1 ) > 0 formulae-sequence 𝑟 0 formulae-sequence 2 ℓ 1 𝑝 𝑘 1 0 2 ℓ 1 𝑝 𝑘 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 ⁢ ( ℓ ; r ) = ( - 1 ) ℓ + 1 ⁢ 2 π ⁢ ∑ p = 2 ⁢ k 3 ⁢ k ( 2 ⁢ | r | ) ( p + 1 ) / 2 ⁢ C k , p ⁢ K 2 ⁢ ℓ + 1 + p ⁡ ( 8 ⁢ | r | ) subscript 𝖧 𝑘 ℓ 𝑟 superscript 1 ℓ 1 2 𝜋 superscript subscript 𝑝 2 𝑘 3 𝑘 superscript 2 𝑟 𝑝 1 2 subscript 𝐶 𝑘 𝑝 modified-Bessel-second-kind 2 ℓ 1 𝑝 8 𝑟 {\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 𝑟 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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