Legendre and Related Functions - 14.13 Trigonometric Expansions

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14.13#Ex1 + 1 2 ⁒ Ο€ ⁒ i ⁒ 𝖯 Ξ½ ΞΌ ⁑ ( cos ⁑ ΞΈ ) + 𝖰 Ξ½ ΞΌ ⁑ ( cos ⁑ ΞΈ ) = Ο€ 1 2 ⁒ Ξ“ ⁑ ( Ξ½ + ΞΌ + 1 ) ⁒ ( 2 ⁒ sin ⁑ ΞΈ ) ΞΌ ⁒ e + ( Ξ½ + ΞΌ + 1 ) ⁒ i ⁒ ΞΈ ⁒ 𝐅 ⁑ ( Ξ½ + ΞΌ + 1 , ΞΌ + 1 2 ; Ξ½ + 3 2 ; e + 2 ⁒ i ⁒ ΞΈ ) 1 2 πœ‹ 𝑖 Ferrers-Legendre-P-first-kind πœ‡ 𝜈 πœƒ Ferrers-Legendre-Q-first-kind πœ‡ 𝜈 πœƒ superscript πœ‹ 1 2 Euler-Gamma 𝜈 πœ‡ 1 superscript 2 πœƒ πœ‡ superscript 𝑒 𝜈 πœ‡ 1 𝑖 πœƒ scaled-hypergeometric-bold-F 𝜈 πœ‡ 1 πœ‡ 1 2 𝜈 3 2 superscript 𝑒 2 𝑖 πœƒ {\displaystyle{\displaystyle+\frac{1}{2}\pi i\mathsf{P}^{\mu}_{\nu}\left(\cos% \theta\right)+\mathsf{Q}^{\mu}_{\nu}\left(\cos\theta\right)=\pi^{\frac{1}{2}}% \Gamma\left(\nu+\mu+1\right)(2\sin\theta)^{\mu}e^{+(\nu+\mu+1)i\theta}\*% \mathbf{F}\left(\nu+\mu+1,\mu+\frac{1}{2};\nu+\frac{3}{2};e^{+2i\theta}\right)}}
+\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}}		 

+(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[ΞΈ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΌ, -1.5], Rule[Ξ½, -1.5]}


Result: Indeterminate
Test Values: {Rule[ΞΈ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΌ, -1.5], Rule[Ξ½, -0.5]}

... skip entries to safe data
14.13#Ex1 - 1 2 ⁒ Ο€ ⁒ i ⁒ 𝖯 Ξ½ ΞΌ ⁑ ( cos ⁑ ΞΈ ) + 𝖰 Ξ½ ΞΌ ⁑ ( cos ⁑ ΞΈ ) = Ο€ 1 2 ⁒ Ξ“ ⁑ ( Ξ½ + ΞΌ + 1 ) ⁒ ( 2 ⁒ sin ⁑ ΞΈ ) ΞΌ ⁒ e - ( Ξ½ + ΞΌ + 1 ) ⁒ i ⁒ ΞΈ ⁒ 𝐅 ⁑ ( Ξ½ + ΞΌ + 1 , ΞΌ + 1 2 ; Ξ½ + 3 2 ; e - 2 ⁒ i ⁒ ΞΈ ) 1 2 πœ‹ 𝑖 Ferrers-Legendre-P-first-kind πœ‡ 𝜈 πœƒ Ferrers-Legendre-Q-first-kind πœ‡ 𝜈 πœƒ superscript πœ‹ 1 2 Euler-Gamma 𝜈 πœ‡ 1 superscript 2 πœƒ πœ‡ superscript 𝑒 𝜈 πœ‡ 1 𝑖 πœƒ scaled-hypergeometric-bold-F 𝜈 πœ‡ 1 πœ‡ 1 2 𝜈 3 2 superscript 𝑒 2 𝑖 πœƒ {\displaystyle{\displaystyle-\frac{1}{2}\pi i\mathsf{P}^{\mu}_{\nu}\left(\cos% \theta\right)+\mathsf{Q}^{\mu}_{\nu}\left(\cos\theta\right)=\pi^{\frac{1}{2}}% \Gamma\left(\nu+\mu+1\right)(2\sin\theta)^{\mu}e^{-(\nu+\mu+1)i\theta}\*% \mathbf{F}\left(\nu+\mu+1,\mu+\frac{1}{2};\nu+\frac{3}{2};e^{-2i\theta}\right)}}
-\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}}		 

-(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[ΞΈ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΌ, -1.5], Rule[Ξ½, -1.5]}


Result: Indeterminate
Test Values: {Rule[ΞΈ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΌ, -1.5], Rule[Ξ½, -0.5]}

... skip entries to safe data
14.13.E1 𝖯 Ξ½ ΞΌ ⁑ ( cos ⁑ ΞΈ ) = 2 ΞΌ + 1 ⁒ ( sin ⁑ ΞΈ ) ΞΌ Ο€ 1 / 2 ⁒ βˆ‘ k = 0 ∞ Ξ“ ⁑ ( Ξ½ + ΞΌ + k + 1 ) Ξ“ ⁑ ( Ξ½ + k + 3 2 ) ⁒ ( ΞΌ + 1 2 ) k k ! ⁒ sin ⁑ ( ( Ξ½ + ΞΌ + 2 ⁒ k + 1 ) ⁒ ΞΈ ) Ferrers-Legendre-P-first-kind πœ‡ 𝜈 πœƒ superscript 2 πœ‡ 1 superscript πœƒ πœ‡ superscript πœ‹ 1 2 superscript subscript π‘˜ 0 Euler-Gamma 𝜈 πœ‡ π‘˜ 1 Euler-Gamma 𝜈 π‘˜ 3 2 Pochhammer πœ‡ 1 2 π‘˜ π‘˜ 𝜈 πœ‡ 2 π‘˜ 1 πœƒ {\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu}\left(\cos\theta\right)=% \frac{2^{\mu+1}(\sin\theta)^{\mu}}{\pi^{1/2}}\*\sum_{k=0}^{\infty}\frac{\Gamma% \left(\nu+\mu+k+1\right)}{\Gamma\left(\nu+k+\frac{3}{2}\right)}\frac{{\left(% \mu+\frac{1}{2}\right)_{k}}}{k!}\*\sin\left((\nu+\mu+2k+1)\theta\right)}}
\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}		 β„œ ⁑ ( Ξ½ + ΞΌ + k + 1 ) > 0 , β„œ ⁑ ( Ξ½ + k + 3 2 ) > 0 formulae-sequence 𝜈 πœ‡ π‘˜ 1 0 𝜈 π‘˜ 3 2 0 {\displaystyle{\displaystyle\Re(\nu+\mu+k+1)>0,\Re(\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[ΞΈ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΌ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, -1.5]}


Result: Indeterminate
Test Values: {Rule[ΞΈ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΌ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[Ξ½, -1.5]}

... skip entries to safe data
14.13.E2 𝖰 Ξ½ ΞΌ ⁑ ( cos ⁑ ΞΈ ) = Ο€ 1 / 2 ⁒ 2 ΞΌ ⁒ ( sin ⁑ ΞΈ ) ΞΌ ⁒ βˆ‘ k = 0 ∞ Ξ“ ⁑ ( Ξ½ + ΞΌ + k + 1 ) Ξ“ ⁑ ( Ξ½ + k + 3 2 ) ⁒ ( ΞΌ + 1 2 ) k k ! ⁒ cos ⁑ ( ( Ξ½ + ΞΌ + 2 ⁒ k + 1 ) ⁒ ΞΈ ) Ferrers-Legendre-Q-first-kind πœ‡ 𝜈 πœƒ superscript πœ‹ 1 2 superscript 2 πœ‡ superscript πœƒ πœ‡ superscript subscript π‘˜ 0 Euler-Gamma 𝜈 πœ‡ π‘˜ 1 Euler-Gamma 𝜈 π‘˜ 3 2 Pochhammer πœ‡ 1 2 π‘˜ π‘˜ 𝜈 πœ‡ 2 π‘˜ 1 πœƒ {\displaystyle{\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(\cos\theta\right)=\pi^% {1/2}2^{\mu}(\sin\theta)^{\mu}\*\sum_{k=0}^{\infty}\frac{\Gamma\left(\nu+\mu+k% +1\right)}{\Gamma\left(\nu+k+\frac{3}{2}\right)}\frac{{\left(\mu+\frac{1}{2}% \right)_{k}}}{k!}\*\cos\left((\nu+\mu+2k+1)\theta\right)}}
\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}		 β„œ ⁑ ( Ξ½ + ΞΌ + k + 1 ) > 0 , β„œ ⁑ ( Ξ½ + k + 3 2 ) > 0 formulae-sequence 𝜈 πœ‡ π‘˜ 1 0 𝜈 π‘˜ 3 2 0 {\displaystyle{\displaystyle\Re(\nu+\mu+k+1)>0,\Re(\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[ΞΈ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ΞΌ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.06813222813420483, 1.1810252600164224]
Test Values: {Rule[ΞΈ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ΞΌ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
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