18.35: 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/18.35.E4 18.35.E4] || [[Item:Q6043|<math>\frac{\Pochhammersym{\lambda-\iunit\tau_{a,b}(\theta)}{n}}{n!}e^{\iunit n\theta}\*\genhyperF{2}{1}@@{-n,\lambda+\iunit\tau_{a,b}(\theta)}{-n-\lambda+1+\iunit\tau_{a,b}(\theta)}{e^{-2\iunit\theta}} = \sum_{\ell=0}^{n}\frac{\Pochhammersym{\lambda+\iunit\tau_{a,b}(\theta)}{\ell}}{\ell!}\frac{\Pochhammersym{\lambda-\iunit\tau_{a,b}(\theta)}{n-\ell}}{(n-\ell)!}e^{\iunit(n-2\ell)\theta}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\Pochhammersym{\lambda-\iunit\tau_{a,b}(\theta)}{n}}{n!}e^{\iunit n\theta}\*\genhyperF{2}{1}@@{-n,\lambda+\iunit\tau_{a,b}(\theta)}{-n-\lambda+1+\iunit\tau_{a,b}(\theta)}{e^{-2\iunit\theta}} = \sum_{\ell=0}^{n}\frac{\Pochhammersym{\lambda+\iunit\tau_{a,b}(\theta)}{\ell}}{\ell!}\frac{\Pochhammersym{\lambda-\iunit\tau_{a,b}(\theta)}{n-\ell}}{(n-\ell)!}e^{\iunit(n-2\ell)\theta}</syntaxhighlight> || <math>0 < \theta, \theta < \pi</math> || <syntaxhighlight lang=mathematica>(pochhammer(lambda - I*((a*cos(theta)+ b)/(sin(theta))), n))/(factorial(n))*exp(I*n*theta)* hypergeom([- n , lambda + I*((a*cos(theta)+ b)/(sin(theta)))], [- n - lambda + 1 + I*((a*cos(theta)+ b)/(sin(theta)))], exp(- 2*I*theta)) = sum((pochhammer(lambda + I*((a*cos(theta)+ b)/(sin(theta))), ell))/(factorial(ell))*(pochhammer(lambda - I*((a*cos(theta)+ b)/(sin(theta))), n - ell))/(factorial(n - ell))*exp(I*(n - 2*ell)*theta), ell = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Pochhammer[\[Lambda]- I*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]]), n],(n)!]*Exp[I*n*\[Theta]]* HypergeometricPFQ[{- n , \[Lambda]+ I*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]])}, {- n - \[Lambda]+ 1 + I*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]])}, Exp[- 2*I*\[Theta]]] == Sum[Divide[Pochhammer[\[Lambda]+ I*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]]), \[ScriptL]],(\[ScriptL])!]*Divide[Pochhammer[\[Lambda]- I*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]]), n - \[ScriptL]],(n - \[ScriptL])!]*Exp[I*(n - 2*\[ScriptL])*\[Theta]], {\[ScriptL], 0, n}, GenerateConditions->None]</syntaxhighlight> || Error || Successful || - || Successful [Tested: 300]
| [https://dlmf.nist.gov/18.35.E4 18.35.E4] || <math qid="Q6043">\frac{\Pochhammersym{\lambda-\iunit\tau_{a,b}(\theta)}{n}}{n!}e^{\iunit n\theta}\*\genhyperF{2}{1}@@{-n,\lambda+\iunit\tau_{a,b}(\theta)}{-n-\lambda+1+\iunit\tau_{a,b}(\theta)}{e^{-2\iunit\theta}} = \sum_{\ell=0}^{n}\frac{\Pochhammersym{\lambda+\iunit\tau_{a,b}(\theta)}{\ell}}{\ell!}\frac{\Pochhammersym{\lambda-\iunit\tau_{a,b}(\theta)}{n-\ell}}{(n-\ell)!}e^{\iunit(n-2\ell)\theta}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\Pochhammersym{\lambda-\iunit\tau_{a,b}(\theta)}{n}}{n!}e^{\iunit n\theta}\*\genhyperF{2}{1}@@{-n,\lambda+\iunit\tau_{a,b}(\theta)}{-n-\lambda+1+\iunit\tau_{a,b}(\theta)}{e^{-2\iunit\theta}} = \sum_{\ell=0}^{n}\frac{\Pochhammersym{\lambda+\iunit\tau_{a,b}(\theta)}{\ell}}{\ell!}\frac{\Pochhammersym{\lambda-\iunit\tau_{a,b}(\theta)}{n-\ell}}{(n-\ell)!}e^{\iunit(n-2\ell)\theta}</syntaxhighlight> || <math>0 < \theta, \theta < \pi</math> || <syntaxhighlight lang=mathematica>(pochhammer(lambda - I*((a*cos(theta)+ b)/(sin(theta))), n))/(factorial(n))*exp(I*n*theta)* hypergeom([- n , lambda + I*((a*cos(theta)+ b)/(sin(theta)))], [- n - lambda + 1 + I*((a*cos(theta)+ b)/(sin(theta)))], exp(- 2*I*theta)) = sum((pochhammer(lambda + I*((a*cos(theta)+ b)/(sin(theta))), ell))/(factorial(ell))*(pochhammer(lambda - I*((a*cos(theta)+ b)/(sin(theta))), n - ell))/(factorial(n - ell))*exp(I*(n - 2*ell)*theta), ell = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Pochhammer[\[Lambda]- I*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]]), n],(n)!]*Exp[I*n*\[Theta]]* HypergeometricPFQ[{- n , \[Lambda]+ I*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]])}, {- n - \[Lambda]+ 1 + I*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]])}, Exp[- 2*I*\[Theta]]] == Sum[Divide[Pochhammer[\[Lambda]+ I*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]]), \[ScriptL]],(\[ScriptL])!]*Divide[Pochhammer[\[Lambda]- I*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]]), n - \[ScriptL]],(n - \[ScriptL])!]*Exp[I*(n - 2*\[ScriptL])*\[Theta]], {\[ScriptL], 0, n}, GenerateConditions->None]</syntaxhighlight> || Error || Successful || - || Successful [Tested: 300]
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| [https://dlmf.nist.gov/18.35.E6 18.35.E6] || [[Item:Q6045|<math>w^{(\lambda)}(\cos@@{\theta};a,b) = \pi^{-1}\*2^{2\lambda-1}\*e^{(2\theta-\pi)\*\tau_{a,b}(\theta)}\*(\sin@@{\theta})^{2\lambda-1}\*\abs{\EulerGamma@{\lambda+\iunit\tau_{a,b}(\theta)}}^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>w^{(\lambda)}(\cos@@{\theta};a,b) = \pi^{-1}\*2^{2\lambda-1}\*e^{(2\theta-\pi)\*\tau_{a,b}(\theta)}\*(\sin@@{\theta})^{2\lambda-1}\*\abs{\EulerGamma@{\lambda+\iunit\tau_{a,b}(\theta)}}^{2}</syntaxhighlight> || <math>a \geq b, b \geq -a, \lambda > -\frac{1}{2}, 0 < \theta, \theta < \pi</math> || <syntaxhighlight lang=mathematica>(w(cos(theta); a , b))^(lambda) = (Pi)^(- 1)* (2)^(2*lambda - 1)* exp((2*theta - Pi)*((a*cos(theta)+ b)/(sin(theta))))*(sin(theta))^(2*lambda - 1)* (abs(GAMMA(lambda + I*((a*cos(theta)+ b)/(sin(theta))))))^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(w[Cos[\[Theta]]; a , b])^(\[Lambda]) == (Pi)^(- 1)* (2)^(2*\[Lambda]- 1)* Exp[(2*\[Theta]- Pi)*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]])]*(Sin[\[Theta]])^(2*\[Lambda]- 1)* (Abs[Gamma[\[Lambda]+ I*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]])]])^(2)</syntaxhighlight> || Translation Error || Translation Error || - || -
| [https://dlmf.nist.gov/18.35.E6 18.35.E6] || <math qid="Q6045">w^{(\lambda)}(\cos@@{\theta};a,b) = \pi^{-1}\*2^{2\lambda-1}\*e^{(2\theta-\pi)\*\tau_{a,b}(\theta)}\*(\sin@@{\theta})^{2\lambda-1}\*\abs{\EulerGamma@{\lambda+\iunit\tau_{a,b}(\theta)}}^{2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>w^{(\lambda)}(\cos@@{\theta};a,b) = \pi^{-1}\*2^{2\lambda-1}\*e^{(2\theta-\pi)\*\tau_{a,b}(\theta)}\*(\sin@@{\theta})^{2\lambda-1}\*\abs{\EulerGamma@{\lambda+\iunit\tau_{a,b}(\theta)}}^{2}</syntaxhighlight> || <math>a \geq b, b \geq -a, \lambda > -\frac{1}{2}, 0 < \theta, \theta < \pi</math> || <syntaxhighlight lang=mathematica>(w(cos(theta); a , b))^(lambda) = (Pi)^(- 1)* (2)^(2*lambda - 1)* exp((2*theta - Pi)*((a*cos(theta)+ b)/(sin(theta))))*(sin(theta))^(2*lambda - 1)* (abs(GAMMA(lambda + I*((a*cos(theta)+ b)/(sin(theta))))))^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(w[Cos[\[Theta]]; a , b])^(\[Lambda]) == (Pi)^(- 1)* (2)^(2*\[Lambda]- 1)* Exp[(2*\[Theta]- Pi)*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]])]*(Sin[\[Theta]])^(2*\[Lambda]- 1)* (Abs[Gamma[\[Lambda]+ I*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]])]])^(2)</syntaxhighlight> || Translation Error || Translation Error || - || -
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Latest revision as of 11:48, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
18.35.E4 ( λ - i ⁢ τ a , b ⁢ ( θ ) ) n n ! ⁢ e i ⁢ n ⁢ θ ⁢ F 1 2 ⁡ ( - n , λ + i ⁢ τ a , b ⁢ ( θ ) - n - λ + 1 + i ⁢ τ a , b ⁢ ( θ ) ; e - 2 ⁢ i ⁢ θ ) = ∑ ℓ = 0 n ( λ + i ⁢ τ a , b ⁢ ( θ ) ) ℓ ℓ ! ⁢ ( λ - i ⁢ τ a , b ⁢ ( θ ) ) n - ℓ ( n - ℓ ) ! ⁢ e i ⁢ ( n - 2 ⁢ ℓ ) ⁢ θ Pochhammer 𝜆 imaginary-unit subscript 𝜏 𝑎 𝑏 𝜃 𝑛 𝑛 superscript 𝑒 imaginary-unit 𝑛 𝜃 Gauss-hypergeometric-F-as-2F1 𝑛 𝜆 imaginary-unit subscript 𝜏 𝑎 𝑏 𝜃 𝑛 𝜆 1 imaginary-unit subscript 𝜏 𝑎 𝑏 𝜃 superscript 𝑒 2 imaginary-unit 𝜃 superscript subscript ℓ 0 𝑛 Pochhammer 𝜆 imaginary-unit subscript 𝜏 𝑎 𝑏 𝜃 ℓ ℓ Pochhammer 𝜆 imaginary-unit subscript 𝜏 𝑎 𝑏 𝜃 𝑛 ℓ 𝑛 ℓ superscript 𝑒 imaginary-unit 𝑛 2 ℓ 𝜃 {\displaystyle{\displaystyle\frac{{\left(\lambda-\mathrm{i}\tau_{a,b}(\theta)% \right)_{n}}}{n!}e^{\mathrm{i}n\theta}\*{{}_{2}F_{1}}\left({-n,\lambda+\mathrm% {i}\tau_{a,b}(\theta)\atop-n-\lambda+1+\mathrm{i}\tau_{a,b}(\theta)};e^{-2% \mathrm{i}\theta}\right)=\sum_{\ell=0}^{n}\frac{{\left(\lambda+\mathrm{i}\tau_% {a,b}(\theta)\right)_{\ell}}}{\ell!}\frac{{\left(\lambda-\mathrm{i}\tau_{a,b}(% \theta)\right)_{n-\ell}}}{(n-\ell)!}e^{\mathrm{i}(n-2\ell)\theta}}}
\frac{\Pochhammersym{\lambda-\iunit\tau_{a,b}(\theta)}{n}}{n!}e^{\iunit n\theta}\*\genhyperF{2}{1}@@{-n,\lambda+\iunit\tau_{a,b}(\theta)}{-n-\lambda+1+\iunit\tau_{a,b}(\theta)}{e^{-2\iunit\theta}} = \sum_{\ell=0}^{n}\frac{\Pochhammersym{\lambda+\iunit\tau_{a,b}(\theta)}{\ell}}{\ell!}\frac{\Pochhammersym{\lambda-\iunit\tau_{a,b}(\theta)}{n-\ell}}{(n-\ell)!}e^{\iunit(n-2\ell)\theta}		 0 < θ , θ < π formulae-sequence 0 𝜃 𝜃 𝜋 {\displaystyle{\displaystyle 0<\theta,\theta<\pi}} 
(pochhammer(lambda - I*((a*cos(theta)+ b)/(sin(theta))), n))/(factorial(n))*exp(I*n*theta)* hypergeom([- n , lambda + I*((a*cos(theta)+ b)/(sin(theta)))], [- n - lambda + 1 + I*((a*cos(theta)+ b)/(sin(theta)))], exp(- 2*I*theta)) = sum((pochhammer(lambda + I*((a*cos(theta)+ b)/(sin(theta))), ell))/(factorial(ell))*(pochhammer(lambda - I*((a*cos(theta)+ b)/(sin(theta))), n - ell))/(factorial(n - ell))*exp(I*(n - 2*ell)*theta), ell = 0..n)

Divide[Pochhammer[\[Lambda]- I*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]]), n],(n)!]*Exp[I*n*\[Theta]]* HypergeometricPFQ[{- n , \[Lambda]+ I*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]])}, {- n - \[Lambda]+ 1 + I*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]])}, Exp[- 2*I*\[Theta]]] == Sum[Divide[Pochhammer[\[Lambda]+ I*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]]), \[ScriptL]],(\[ScriptL])!]*Divide[Pochhammer[\[Lambda]- I*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]]), n - \[ScriptL]],(n - \[ScriptL])!]*Exp[I*(n - 2*\[ScriptL])*\[Theta]], {\[ScriptL], 0, n}, GenerateConditions->None]
Error Successful - Successful [Tested: 300]
18.35.E6 w ( λ ) ⁢ ( cos ⁡ θ ; a , b ) = π - 1 ⁢ 2 2 ⁢ λ - 1 ⁢ e ( 2 ⁢ θ - π ) ⁢ τ a , b ⁢ ( θ ) ⁢ ( sin ⁡ θ ) 2 ⁢ λ - 1 ⁢ | Γ ⁡ ( λ + i ⁢ τ a , b ⁢ ( θ ) ) | 2 superscript 𝑤 𝜆 𝜃 𝑎 𝑏 superscript 𝜋 1 superscript 2 2 𝜆 1 superscript 𝑒 2 𝜃 𝜋 subscript 𝜏 𝑎 𝑏 𝜃 superscript 𝜃 2 𝜆 1 Euler-Gamma 𝜆 imaginary-unit subscript 𝜏 𝑎 𝑏 𝜃 2 {\displaystyle{\displaystyle w^{(\lambda)}(\cos\theta;a,b)=\pi^{-1}\*2^{2% \lambda-1}\*e^{(2\theta-\pi)\*\tau_{a,b}(\theta)}\*(\sin\theta)^{2\lambda-1}\*% {\left|\Gamma\left(\lambda+\mathrm{i}\tau_{a,b}(\theta)\right)\right|^{2}}}}
w^{(\lambda)}(\cos@@{\theta};a,b) = \pi^{-1}\*2^{2\lambda-1}\*e^{(2\theta-\pi)\*\tau_{a,b}(\theta)}\*(\sin@@{\theta})^{2\lambda-1}\*\abs{\EulerGamma@{\lambda+\iunit\tau_{a,b}(\theta)}}^{2}		 a ≥ b , b ≥ - a , λ > - 1 2 , 0 < θ , θ < π formulae-sequence 𝑎 𝑏 formulae-sequence 𝑏 𝑎 formulae-sequence 𝜆 1 2 formulae-sequence 0 𝜃 𝜃 𝜋 {\displaystyle{\displaystyle a\geq b,b\geq-a,\lambda>-\frac{1}{2},0<\theta,% \theta<\pi}} 
(w(cos(theta); a , b))^(lambda) = (Pi)^(- 1)* (2)^(2*lambda - 1)* exp((2*theta - Pi)*((a*cos(theta)+ b)/(sin(theta))))*(sin(theta))^(2*lambda - 1)* (abs(GAMMA(lambda + I*((a*cos(theta)+ b)/(sin(theta))))))^(2)

(w[Cos[\[Theta]]; a , b])^(\[Lambda]) == (Pi)^(- 1)* (2)^(2*\[Lambda]- 1)* Exp[(2*\[Theta]- Pi)*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]])]*(Sin[\[Theta]])^(2*\[Lambda]- 1)* (Abs[Gamma[\[Lambda]+ I*(Divide[a*Cos[\[Theta]]+ b,Sin[\[Theta]]])]])^(2)
Translation Error Translation Error - -
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