14.13#Ex1
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𝜋
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Ferrers-Legendre-P-first-kind
𝜇
𝜈
𝜃
Ferrers-Legendre-Q-first-kind
𝜇
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superscript
𝜋
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Euler-Gamma
𝜈
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1
superscript
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𝜃
𝜇
superscript
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scaled-hypergeometric-bold-F
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superscript
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𝑖
𝜃
{\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
Expand 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
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Ferrers-Legendre-P-first-kind
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Ferrers-Legendre-Q-first-kind
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superscript
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Euler-Gamma
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superscript
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scaled-hypergeometric-bold-F
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3
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superscript
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𝜃
{\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
Expand 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
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Ferrers-Legendre-P-first-kind
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Euler-Gamma
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{\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 }
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formulae-sequence
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{\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
Expand 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
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Ferrers-Legendre-Q-first-kind
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superscript
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Euler-Gamma
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Pochhammer
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𝜃
{\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 }
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formulae-sequence
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{\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
Expand 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