Legendre and Related Functions - 14.13 Trigonometric Expansions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
14.13#Ex1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle +\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}}}
+\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}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
+(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))

+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]]]
Failure Failure Skipped - Because timed out
Failed [113 / 300]
Result: Indeterminate
Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5], Rule[ν, -1.5]}


Result: Indeterminate
Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5], Rule[ν, -0.5]}

... skip entries to safe data
14.13#Ex1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -\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}}}
-\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}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
-(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))

-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]]]
Failure Failure Skipped - Because timed out
Failed [113 / 300]
Result: Indeterminate
Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5], Rule[ν, -1.5]}


Result: Indeterminate
Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5], Rule[ν, -0.5]}

... skip entries to safe data
14.13.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}
\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\nu+\mu+k+1)} > 0, \realpart@@{(\nu+k+\frac{3}{2})} > 0} 
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)

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]
Aborted Failure Skipped - Because timed out
Failed [127 / 300]
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]}


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]}

... skip entries to safe data
14.13.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}
\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\nu+\mu+k+1)} > 0, \realpart@@{(\nu+k+\frac{3}{2})} > 0} 
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)

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]
Aborted Failure Skipped - Because timed out
Failed [153 / 300]
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]]]}


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]]]}

... skip entries to safe data
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