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{{DISPLAYTITLE:Legendre and Related Functions - 14.13 Trigonometric Expansions}}

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| [https://dlmf.nist.gov/14.13#Ex1 14.13#Ex1] || <math qid="Q4836">+\frac{1}{2}\pi i\FerrersP[\mu]{\nu}@{\cos@@{\theta}}+\FerrersQ[\mu]{\nu}@{\cos@@{\theta}} = \pi^{\frac{1}{2}}\EulerGamma@{\nu+\mu+1}(2\sin@@{\theta})^{\mu}e^{+(\nu+\mu+1)i\theta}\*\hyperOlverF@{\nu+\mu+1}{\mu+\frac{1}{2}}{\nu+\frac{3}{2}}{e^{+ 2i\theta}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>+\frac{1}{2}\pi i\FerrersP[\mu]{\nu}@{\cos@@{\theta}}+\FerrersQ[\mu]{\nu}@{\cos@@{\theta}} = \pi^{\frac{1}{2}}\EulerGamma@{\nu+\mu+1}(2\sin@@{\theta})^{\mu}e^{+(\nu+\mu+1)i\theta}\*\hyperOlverF@{\nu+\mu+1}{\mu+\frac{1}{2}}{\nu+\frac{3}{2}}{e^{+ 2i\theta}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>+(1)/(2)*Pi*I*LegendreP(nu, mu, cos(theta))+ LegendreQ(nu, mu, cos(theta)) = (Pi)^((1)/(2))* GAMMA(nu + mu + 1)*(2*sin(theta))^(mu)* exp(+(nu + mu + 1)*I*theta)* hypergeom([nu + mu + 1, mu +(1)/(2)], [nu +(3)/(2)], exp(+ 2*I*theta))/GAMMA(nu +(3)/(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>+Divide[1,2]*Pi*I*LegendreP[\[Nu], \[Mu], Cos[\[Theta]]]+ LegendreQ[\[Nu], \[Mu], Cos[\[Theta]]] == (Pi)^(Divide[1,2])* Gamma[\[Nu]+ \[Mu]+ 1]*(2*Sin[\[Theta]])^\[Mu]* Exp[+(\[Nu]+ \[Mu]+ 1)*I*\[Theta]]* Hypergeometric2F1Regularized[\[Nu]+ \[Mu]+ 1, \[Mu]+Divide[1,2], \[Nu]+Divide[3,2], Exp[+ 2*I*\[Theta]]]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [113 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5], Rule[ν, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5], Rule[ν, -0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/14.13#Ex1 14.13#Ex1] || <math qid="Q4836">-\frac{1}{2}\pi i\FerrersP[\mu]{\nu}@{\cos@@{\theta}}+\FerrersQ[\mu]{\nu}@{\cos@@{\theta}} = \pi^{\frac{1}{2}}\EulerGamma@{\nu+\mu+1}(2\sin@@{\theta})^{\mu}e^{-(\nu+\mu+1)i\theta}\*\hyperOlverF@{\nu+\mu+1}{\mu+\frac{1}{2}}{\nu+\frac{3}{2}}{e^{- 2i\theta}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>-\frac{1}{2}\pi i\FerrersP[\mu]{\nu}@{\cos@@{\theta}}+\FerrersQ[\mu]{\nu}@{\cos@@{\theta}} = \pi^{\frac{1}{2}}\EulerGamma@{\nu+\mu+1}(2\sin@@{\theta})^{\mu}e^{-(\nu+\mu+1)i\theta}\*\hyperOlverF@{\nu+\mu+1}{\mu+\frac{1}{2}}{\nu+\frac{3}{2}}{e^{- 2i\theta}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>-(1)/(2)*Pi*I*LegendreP(nu, mu, cos(theta))+ LegendreQ(nu, mu, cos(theta)) = (Pi)^((1)/(2))* GAMMA(nu + mu + 1)*(2*sin(theta))^(mu)* exp(-(nu + mu + 1)*I*theta)* hypergeom([nu + mu + 1, mu +(1)/(2)], [nu +(3)/(2)], exp(- 2*I*theta))/GAMMA(nu +(3)/(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>-Divide[1,2]*Pi*I*LegendreP[\[Nu], \[Mu], Cos[\[Theta]]]+ LegendreQ[\[Nu], \[Mu], Cos[\[Theta]]] == (Pi)^(Divide[1,2])* Gamma[\[Nu]+ \[Mu]+ 1]*(2*Sin[\[Theta]])^\[Mu]* Exp[-(\[Nu]+ \[Mu]+ 1)*I*\[Theta]]* Hypergeometric2F1Regularized[\[Nu]+ \[Mu]+ 1, \[Mu]+Divide[1,2], \[Nu]+Divide[3,2], Exp[- 2*I*\[Theta]]]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [113 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5], Rule[ν, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5], Rule[ν, -0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/14.13.E1 14.13.E1] || <math qid="Q4837">\FerrersP[\mu]{\nu}@{\cos@@{\theta}} = \frac{2^{\mu+1}(\sin@@{\theta})^{\mu}}{\pi^{1/2}}\*\sum_{k=0}^{\infty}\frac{\EulerGamma@{\nu+\mu+k+1}}{\EulerGamma@{\nu+k+\frac{3}{2}}}\frac{\Pochhammersym{\mu+\frac{1}{2}}{k}}{k!}\*\sin@{(\nu+\mu+2k+1)\theta}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersP[\mu]{\nu}@{\cos@@{\theta}} = \frac{2^{\mu+1}(\sin@@{\theta})^{\mu}}{\pi^{1/2}}\*\sum_{k=0}^{\infty}\frac{\EulerGamma@{\nu+\mu+k+1}}{\EulerGamma@{\nu+k+\frac{3}{2}}}\frac{\Pochhammersym{\mu+\frac{1}{2}}{k}}{k!}\*\sin@{(\nu+\mu+2k+1)\theta}</syntaxhighlight> || <math>\realpart@@{(\nu+\mu+k+1)} > 0, \realpart@@{(\nu+k+\frac{3}{2})} > 0</math> || <syntaxhighlight lang=mathematica>LegendreP(nu, mu, cos(theta)) = ((2)^(mu + 1)*(sin(theta))^(mu))/((Pi)^(1/2))* sum((GAMMA(nu + mu + k + 1))/(GAMMA(nu + k +(3)/(2)))*(pochhammer(mu +(1)/(2), k))/(factorial(k))* sin((nu + mu + 2*k + 1)*theta), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreP[\[Nu], \[Mu], Cos[\[Theta]]] == Divide[(2)^(\[Mu]+ 1)*(Sin[\[Theta]])^\[Mu],(Pi)^(1/2)]* Sum[Divide[Gamma[\[Nu]+ \[Mu]+ k + 1],Gamma[\[Nu]+ k +Divide[3,2]]]*Divide[Pochhammer[\[Mu]+Divide[1,2], k],(k)!]* Sin[(\[Nu]+ \[Mu]+ 2*k + 1)*\[Theta]], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Aborted || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [127 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, -1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/14.13.E2 14.13.E2] || <math qid="Q4838">\FerrersQ[\mu]{\nu}@{\cos@@{\theta}} = \pi^{1/2}2^{\mu}(\sin@@{\theta})^{\mu}\*\sum_{k=0}^{\infty}\frac{\EulerGamma@{\nu+\mu+k+1}}{\EulerGamma@{\nu+k+\frac{3}{2}}}\frac{\Pochhammersym{\mu+\frac{1}{2}}{k}}{k!}\*\cos@{(\nu+\mu+2k+1)\theta}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FerrersQ[\mu]{\nu}@{\cos@@{\theta}} = \pi^{1/2}2^{\mu}(\sin@@{\theta})^{\mu}\*\sum_{k=0}^{\infty}\frac{\EulerGamma@{\nu+\mu+k+1}}{\EulerGamma@{\nu+k+\frac{3}{2}}}\frac{\Pochhammersym{\mu+\frac{1}{2}}{k}}{k!}\*\cos@{(\nu+\mu+2k+1)\theta}</syntaxhighlight> || <math>\realpart@@{(\nu+\mu+k+1)} > 0, \realpart@@{(\nu+k+\frac{3}{2})} > 0</math> || <syntaxhighlight lang=mathematica>LegendreQ(nu, mu, cos(theta)) = (Pi)^(1/2)* (2)^(mu)*(sin(theta))^(mu)* sum((GAMMA(nu + mu + k + 1))/(GAMMA(nu + k +(3)/(2)))*(pochhammer(mu +(1)/(2), k))/(factorial(k))* cos((nu + mu + 2*k + 1)*theta), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>LegendreQ[\[Nu], \[Mu], Cos[\[Theta]]] == (Pi)^(1/2)* (2)^\[Mu]*(Sin[\[Theta]])^\[Mu]* Sum[Divide[Gamma[\[Nu]+ \[Mu]+ k + 1],Gamma[\[Nu]+ k +Divide[3,2]]]*Divide[Pochhammer[\[Mu]+Divide[1,2], k],(k)!]* Cos[(\[Nu]+ \[Mu]+ 2*k + 1)*\[Theta]], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Aborted || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [153 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.9838922770586165, -0.844402487080167]
Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.06813222813420483, 1.1810252600164224]
Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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