Legendre and Related Functions - 14.12 Integral Representations

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14.12.E1 𝖯 Ξ½ ΞΌ ⁑ ( cos ⁑ ΞΈ ) = 2 1 / 2 ⁒ ( sin ⁑ ΞΈ ) ΞΌ Ο€ 1 / 2 ⁒ Ξ“ ⁑ ( 1 2 - ΞΌ ) ⁒ ∫ 0 ΞΈ cos ⁑ ( ( Ξ½ + 1 2 ) ⁒ t ) ( cos ⁑ t - cos ⁑ ΞΈ ) ΞΌ + ( 1 / 2 ) ⁒ d t Ferrers-Legendre-P-first-kind πœ‡ 𝜈 πœƒ superscript 2 1 2 superscript πœƒ πœ‡ superscript πœ‹ 1 2 Euler-Gamma 1 2 πœ‡ superscript subscript 0 πœƒ 𝜈 1 2 𝑑 superscript 𝑑 πœƒ πœ‡ 1 2 𝑑 {\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu}\left(\cos\theta\right)=% \frac{2^{1/2}(\sin\theta)^{\mu}}{\pi^{1/2}\Gamma\left(\frac{1}{2}-\mu\right)}% \int_{0}^{\theta}\frac{\cos\left(\left(\nu+\frac{1}{2}\right)t\right)}{(\cos t% -\cos\theta)^{\mu+(1/2)}}\mathrm{d}t}}
\FerrersP[\mu]{\nu}@{\cos@@{\theta}} = \frac{2^{1/2}(\sin@@{\theta})^{\mu}}{\pi^{1/2}\EulerGamma@{\frac{1}{2}-\mu}}\int_{0}^{\theta}\frac{\cos@{\left(\nu+\frac{1}{2}\right)t}}{(\cos@@{t}-\cos@@{\theta})^{\mu+(1/2)}}\diff{t}		 0 < ΞΈ , ΞΈ < Ο€ , β„œ ⁑ ( 1 2 - ΞΌ ) > 0 formulae-sequence 0 πœƒ formulae-sequence πœƒ πœ‹ 1 2 πœ‡ 0 {\displaystyle{\displaystyle 0<\theta,\theta<\pi,\Re(\frac{1}{2}-\mu)>0}} 
LegendreP(nu, mu, cos(theta)) = ((2)^(1/2)*(sin(theta))^(mu))/((Pi)^(1/2)* GAMMA((1)/(2)- mu))*int((cos((nu +(1)/(2))*t))/((cos(t)- cos(theta))^(mu +(1/2))), t = 0..theta)

LegendreP[\[Nu], \[Mu], Cos[\[Theta]]] == Divide[(2)^(1/2)*(Sin[\[Theta]])^\[Mu],(Pi)^(1/2)* Gamma[Divide[1,2]- \[Mu]]]*Integrate[Divide[Cos[(\[Nu]+Divide[1,2])*t],(Cos[t]- Cos[\[Theta]])^(\[Mu]+(1/2))], {t, 0, \[Theta]}, GenerateConditions->None]
Error Aborted - Skipped - Because timed out
14.12.E2 𝖯 Ξ½ - ΞΌ ⁑ ( x ) = ( 1 - x 2 ) - ΞΌ / 2 Ξ“ ⁑ ( ΞΌ ) ⁒ ∫ x 1 𝖯 Ξ½ ⁑ ( t ) ⁒ ( t - x ) ΞΌ - 1 ⁒ d t Ferrers-Legendre-P-first-kind πœ‡ 𝜈 π‘₯ superscript 1 superscript π‘₯ 2 πœ‡ 2 Euler-Gamma πœ‡ superscript subscript π‘₯ 1 shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑑 superscript 𝑑 π‘₯ πœ‡ 1 𝑑 {\displaystyle{\displaystyle\mathsf{P}^{-\mu}_{\nu}\left(x\right)=\frac{\left(% 1-x^{2}\right)^{-\mu/2}}{\Gamma\left(\mu\right)}\int_{x}^{1}\mathsf{P}_{\nu}% \left(t\right)(t-x)^{\mu-1}\mathrm{d}t}}
\FerrersP[-\mu]{\nu}@{x} = \frac{\left(1-x^{2}\right)^{-\mu/2}}{\EulerGamma@{\mu}}\int_{x}^{1}\FerrersP[]{\nu}@{t}(t-x)^{\mu-1}\diff{t}		 β„œ ⁑ ΞΌ > 0 , β„œ ⁑ ( ΞΌ ) > 0 , | ( 1 2 - 1 2 ⁒ x ) | < 1 formulae-sequence πœ‡ 0 formulae-sequence πœ‡ 0 1 2 1 2 π‘₯ 1 {\displaystyle{\displaystyle\Re\mu>0,\Re(\mu)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|% <1}} 
LegendreP(nu, - mu, x) = ((1 - (x)^(2))^(- mu/2))/(GAMMA(mu))*int(LegendreP(nu, t)*(t - x)^(mu - 1), t = x..1)

LegendreP[\[Nu], - \[Mu], x] == Divide[(1 - (x)^(2))^(- \[Mu]/2),Gamma[\[Mu]]]*Integrate[LegendreP[\[Nu], t]*(t - x)^(\[Mu]- 1), {t, x, 1}, GenerateConditions->None]
Failure Failure Skipped - Because timed out Skipped - Because timed out
14.12.E3 𝖰 Ξ½ ΞΌ ⁑ ( cos ⁑ ΞΈ ) = Ο€ 1 / 2 ⁒ Ξ“ ⁑ ( Ξ½ + ΞΌ + 1 ) ⁒ ( sin ⁑ ΞΈ ) ΞΌ 2 ΞΌ + 1 ⁒ Ξ“ ⁑ ( ΞΌ + 1 2 ) ⁒ Ξ“ ⁑ ( Ξ½ - ΞΌ + 1 ) ⁒ ( ∫ 0 ∞ ( sinh ⁑ t ) 2 ⁒ ΞΌ ( cos ⁑ ΞΈ + i ⁒ sin ⁑ ΞΈ ⁒ cosh ⁑ t ) Ξ½ + ΞΌ + 1 ⁒ d t + ∫ 0 ∞ ( sinh ⁑ t ) 2 ⁒ ΞΌ ( cos ⁑ ΞΈ - i ⁒ sin ⁑ ΞΈ ⁒ cosh ⁑ t ) Ξ½ + ΞΌ + 1 ⁒ d t ) Ferrers-Legendre-Q-first-kind πœ‡ 𝜈 πœƒ superscript πœ‹ 1 2 Euler-Gamma 𝜈 πœ‡ 1 superscript πœƒ πœ‡ superscript 2 πœ‡ 1 Euler-Gamma πœ‡ 1 2 Euler-Gamma 𝜈 πœ‡ 1 superscript subscript 0 superscript 𝑑 2 πœ‡ superscript πœƒ 𝑖 πœƒ 𝑑 𝜈 πœ‡ 1 𝑑 superscript subscript 0 superscript 𝑑 2 πœ‡ superscript πœƒ 𝑖 πœƒ 𝑑 𝜈 πœ‡ 1 𝑑 {\displaystyle{\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(\cos\theta\right)=% \frac{\pi^{1/2}\Gamma\left(\nu+\mu+1\right)(\sin\theta)^{\mu}}{2^{\mu+1}\Gamma% \left(\mu+\frac{1}{2}\right)\Gamma\left(\nu-\mu+1\right)}\*\left(\int_{0}^{% \infty}\frac{(\sinh t)^{2\mu}}{(\cos\theta+i\sin\theta\cosh t)^{\nu+\mu+1}}% \mathrm{d}t+\int_{0}^{\infty}\frac{(\sinh t)^{2\mu}}{(\cos\theta-i\sin\theta% \cosh t)^{\nu+\mu+1}}\mathrm{d}t\right)}}
\FerrersQ[\mu]{\nu}@{\cos@@{\theta}} = \frac{\pi^{1/2}\EulerGamma@{\nu+\mu+1}(\sin@@{\theta})^{\mu}}{2^{\mu+1}\EulerGamma@{\mu+\frac{1}{2}}\EulerGamma@{\nu-\mu+1}}\*\left(\int_{0}^{\infty}\frac{(\sinh@@{t})^{2\mu}}{(\cos@@{\theta}+i\sin@@{\theta}\cosh@@{t})^{\nu+\mu+1}}\diff{t}+\int_{0}^{\infty}\frac{(\sinh@@{t})^{2\mu}}{(\cos@@{\theta}-i\sin@@{\theta}\cosh@@{t})^{\nu+\mu+1}}\diff{t}\right)		 0 < ΞΈ , ΞΈ < Ο€ , β„œ ⁑ ΞΌ > - 1 2 , β„œ ⁑ Ξ½ + ΞΌ > - 1 , β„œ ⁑ Ξ½ - ΞΌ > - 1 , β„œ ⁑ ( Ξ½ + ΞΌ + 1 ) > 0 , β„œ ⁑ ( ΞΌ + 1 2 ) > 0 , β„œ ⁑ ( Ξ½ - ΞΌ + 1 ) > 0 formulae-sequence 0 πœƒ formulae-sequence πœƒ πœ‹ formulae-sequence πœ‡ 1 2 formulae-sequence 𝜈 πœ‡ 1 formulae-sequence 𝜈 πœ‡ 1 formulae-sequence 𝜈 πœ‡ 1 0 formulae-sequence πœ‡ 1 2 0 𝜈 πœ‡ 1 0 {\displaystyle{\displaystyle 0<\theta,\theta<\pi,\Re\mu>-\tfrac{1}{2},\Re\nu+% \mu>-1,\Re\nu-\mu>-1,\Re(\nu+\mu+1)>0,\Re(\mu+\frac{1}{2})>0,\Re(\nu-\mu+1)>0}} 
LegendreQ(nu, mu, cos(theta)) = ((Pi)^(1/2)* GAMMA(nu + mu + 1)*(sin(theta))^(mu))/((2)^(mu + 1)* GAMMA(mu +(1)/(2))*GAMMA(nu - mu + 1))*(int(((sinh(t))^(2*mu))/((cos(theta)+ I*sin(theta)*cosh(t))^(nu + mu + 1)), t = 0..infinity)+ int(((sinh(t))^(2*mu))/((cos(theta)- I*sin(theta)*cosh(t))^(nu + mu + 1)), t = 0..infinity))

LegendreQ[\[Nu], \[Mu], Cos[\[Theta]]] == Divide[(Pi)^(1/2)* Gamma[\[Nu]+ \[Mu]+ 1]*(Sin[\[Theta]])^\[Mu],(2)^(\[Mu]+ 1)* Gamma[\[Mu]+Divide[1,2]]*Gamma[\[Nu]- \[Mu]+ 1]]*(Integrate[Divide[(Sinh[t])^(2*\[Mu]),(Cos[\[Theta]]+ I*Sin[\[Theta]]*Cosh[t])^(\[Nu]+ \[Mu]+ 1)], {t, 0, Infinity}, GenerateConditions->None]+ Integrate[Divide[(Sinh[t])^(2*\[Mu]),(Cos[\[Theta]]- I*Sin[\[Theta]]*Cosh[t])^(\[Nu]+ \[Mu]+ 1)], {t, 0, Infinity}, GenerateConditions->None])
Error Aborted - Skipped - Because timed out
14.12.E4 P Ξ½ - ΞΌ ⁑ ( x ) = 2 1 / 2 ⁒ Ξ“ ⁑ ( ΞΌ + 1 2 ) ⁒ ( x 2 - 1 ) ΞΌ / 2 Ο€ 1 / 2 ⁒ Ξ“ ⁑ ( Ξ½ + ΞΌ + 1 ) ⁒ Ξ“ ⁑ ( ΞΌ - Ξ½ ) ⁒ ∫ 0 ∞ cosh ⁑ ( ( Ξ½ + 1 2 ) ⁒ t ) ( x + cosh ⁑ t ) ΞΌ + ( 1 / 2 ) ⁒ d t Legendre-P-first-kind πœ‡ 𝜈 π‘₯ superscript 2 1 2 Euler-Gamma πœ‡ 1 2 superscript superscript π‘₯ 2 1 πœ‡ 2 superscript πœ‹ 1 2 Euler-Gamma 𝜈 πœ‡ 1 Euler-Gamma πœ‡ 𝜈 superscript subscript 0 𝜈 1 2 𝑑 superscript π‘₯ 𝑑 πœ‡ 1 2 𝑑 {\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=\frac{2^{1/2}\Gamma% \left(\mu+\frac{1}{2}\right)\left(x^{2}-1\right)^{\mu/2}}{\pi^{1/2}\Gamma\left% (\nu+\mu+1\right)\Gamma\left(\mu-\nu\right)}\*\int_{0}^{\infty}\frac{\cosh% \left(\left(\nu+\frac{1}{2}\right)t\right)}{(x+\cosh t)^{\mu+(1/2)}}\mathrm{d}% t}}
\assLegendreP[-\mu]{\nu}@{x} = \frac{2^{1/2}\EulerGamma@{\mu+\frac{1}{2}}\left(x^{2}-1\right)^{\mu/2}}{\pi^{1/2}\EulerGamma@{\nu+\mu+1}\EulerGamma@{\mu-\nu}}\*\int_{0}^{\infty}\frac{\cosh@{\left(\nu+\frac{1}{2}\right)t}}{(x+\cosh@@{t})^{\mu+(1/2)}}\diff{t}		 β„œ ⁑ ( ΞΌ - Ξ½ ) > 0 , β„œ ⁑ ( ΞΌ + 1 2 ) > 0 , β„œ ⁑ ( Ξ½ + ΞΌ + 1 ) > 0 , β„œ ⁑ ( ΞΌ - Ξ½ ) > 0 formulae-sequence πœ‡ 𝜈 0 formulae-sequence πœ‡ 1 2 0 formulae-sequence 𝜈 πœ‡ 1 0 πœ‡ 𝜈 0 {\displaystyle{\displaystyle\Re\left(\mu-\nu\right)>0,\Re(\mu+\frac{1}{2})>0,% \Re(\nu+\mu+1)>0,\Re(\mu-\nu)>0}} 
LegendreP(nu, - mu, x) = ((2)^(1/2)* GAMMA(mu +(1)/(2))*((x)^(2)- 1)^(mu/2))/((Pi)^(1/2)* GAMMA(nu + mu + 1)*GAMMA(mu - nu))* int((cosh((nu +(1)/(2))*t))/((x + cosh(t))^(mu +(1/2))), t = 0..infinity)

LegendreP[\[Nu], - \[Mu], 3, x] == Divide[(2)^(1/2)* Gamma[\[Mu]+Divide[1,2]]*((x)^(2)- 1)^(\[Mu]/2),(Pi)^(1/2)* Gamma[\[Nu]+ \[Mu]+ 1]*Gamma[\[Mu]- \[Nu]]]* Integrate[Divide[Cosh[(\[Nu]+Divide[1,2])*t],(x + Cosh[t])^(\[Mu]+(1/2))], {t, 0, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
14.12.E5 P Ξ½ - ΞΌ ⁑ ( x ) = ( x 2 - 1 ) - ΞΌ / 2 Ξ“ ⁑ ( ΞΌ ) ⁒ ∫ 1 x P Ξ½ ⁑ ( t ) ⁒ ( x - t ) ΞΌ - 1 ⁒ d t Legendre-P-first-kind πœ‡ 𝜈 π‘₯ superscript superscript π‘₯ 2 1 πœ‡ 2 Euler-Gamma πœ‡ superscript subscript 1 π‘₯ Legendre-spherical-polynomial 𝜈 𝑑 superscript π‘₯ 𝑑 πœ‡ 1 𝑑 {\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=\frac{\left(x^{2}-1% \right)^{-\mu/2}}{\Gamma\left(\mu\right)}\int_{1}^{x}P_{\nu}\left(t\right)(x-t% )^{\mu-1}\mathrm{d}t}}
\assLegendreP[-\mu]{\nu}@{x} = \frac{\left(x^{2}-1\right)^{-\mu/2}}{\EulerGamma@{\mu}}\int_{1}^{x}\LegendrepolyP{\nu}@{t}(x-t)^{\mu-1}\diff{t}		 β„œ ⁑ ΞΌ > 0 , β„œ ⁑ ( ΞΌ ) > 0 formulae-sequence πœ‡ 0 πœ‡ 0 {\displaystyle{\displaystyle\Re\mu>0,\Re(\mu)>0}} 
LegendreP(nu, - mu, x) = (((x)^(2)- 1)^(- mu/2))/(GAMMA(mu))*int(LegendreP(nu, t)*(x - t)^(mu - 1), t = 1..x)

LegendreP[\[Nu], - \[Mu], 3, x] == Divide[((x)^(2)- 1)^(- \[Mu]/2),Gamma[\[Mu]]]*Integrate[LegendreP[\[Nu], t]*(x - t)^(\[Mu]- 1), {t, 1, x}, GenerateConditions->None]
Failure Failure Skipped - Because timed out Skipped - Because timed out
14.12.E6 𝑸 Ξ½ ΞΌ ⁑ ( x ) = Ο€ 1 / 2 ⁒ ( x 2 - 1 ) ΞΌ / 2 2 ΞΌ ⁒ Ξ“ ⁑ ( ΞΌ + 1 2 ) ⁒ Ξ“ ⁑ ( Ξ½ - ΞΌ + 1 ) ⁒ ∫ 0 ∞ ( sinh ⁑ t ) 2 ⁒ ΞΌ ( x + ( x 2 - 1 ) 1 / 2 ⁒ cosh ⁑ t ) Ξ½ + ΞΌ + 1 ⁒ d t associated-Legendre-black-Q πœ‡ 𝜈 π‘₯ superscript πœ‹ 1 2 superscript superscript π‘₯ 2 1 πœ‡ 2 superscript 2 πœ‡ Euler-Gamma πœ‡ 1 2 Euler-Gamma 𝜈 πœ‡ 1 superscript subscript 0 superscript 𝑑 2 πœ‡ superscript π‘₯ superscript superscript π‘₯ 2 1 1 2 𝑑 𝜈 πœ‡ 1 𝑑 {\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=\frac{\pi% ^{1/2}\left(x^{2}-1\right)^{\mu/2}}{2^{\mu}\Gamma\left(\mu+\frac{1}{2}\right)% \Gamma\left(\nu-\mu+1\right)}\*\int_{0}^{\infty}\frac{(\sinh t)^{2\mu}}{\left(% x+(x^{2}-1)^{1/2}\cosh t\right)^{\nu+\mu+1}}\mathrm{d}t}}
\assLegendreOlverQ[\mu]{\nu}@{x} = \frac{\pi^{1/2}\left(x^{2}-1\right)^{\mu/2}}{2^{\mu}\EulerGamma@{\mu+\frac{1}{2}}\EulerGamma@{\nu-\mu+1}}\*\int_{0}^{\infty}\frac{(\sinh@@{t})^{2\mu}}{\left(x+(x^{2}-1)^{1/2}\cosh@@{t}\right)^{\nu+\mu+1}}\diff{t}		 β„œ ⁑ ( Ξ½ + 1 ) > β„œ ⁑ ΞΌ , β„œ ⁑ ΞΌ > - 1 2 , β„œ ⁑ ( ΞΌ + 1 2 ) > 0 , β„œ ⁑ ( Ξ½ - ΞΌ + 1 ) > 0 formulae-sequence 𝜈 1 πœ‡ formulae-sequence πœ‡ 1 2 formulae-sequence πœ‡ 1 2 0 𝜈 πœ‡ 1 0 {\displaystyle{\displaystyle\Re\left(\nu+1\right)>\Re\mu,\Re\mu>-\tfrac{1}{2},% \Re(\mu+\frac{1}{2})>0,\Re(\nu-\mu+1)>0}} 
exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x)/GAMMA(nu+mu+1) = ((Pi)^(1/2)*((x)^(2)- 1)^(mu/2))/((2)^(mu)* GAMMA(mu +(1)/(2))*GAMMA(nu - mu + 1))* int(((sinh(t))^(2*mu))/((x +((x)^(2)- 1)^(1/2)* cosh(t))^(nu + mu + 1)), t = 0..infinity)

Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x]/Gamma[\[Nu] + \[Mu] + 1] == Divide[(Pi)^(1/2)*((x)^(2)- 1)^(\[Mu]/2),(2)^\[Mu]* Gamma[\[Mu]+Divide[1,2]]*Gamma[\[Nu]- \[Mu]+ 1]]* Integrate[Divide[(Sinh[t])^(2*\[Mu]),(x +((x)^(2)- 1)^(1/2)* Cosh[t])^(\[Nu]+ \[Mu]+ 1)], {t, 0, Infinity}, GenerateConditions->None]
Error Aborted - Skipped - Because timed out
14.12.E7 P Ξ½ m ⁑ ( x ) = ( Ξ½ + 1 ) m Ο€ ⁒ ∫ 0 Ο€ ( x + ( x 2 - 1 ) 1 / 2 ⁒ cos ⁑ Ο• ) Ξ½ ⁒ cos ⁑ ( m ⁒ Ο• ) ⁒ d Ο• Legendre-P-first-kind π‘š 𝜈 π‘₯ Pochhammer 𝜈 1 π‘š πœ‹ superscript subscript 0 πœ‹ superscript π‘₯ superscript superscript π‘₯ 2 1 1 2 italic-Ο• 𝜈 π‘š italic-Ο• italic-Ο• {\displaystyle{\displaystyle P^{m}_{\nu}\left(x\right)=\frac{{\left(\nu+1% \right)_{m}}}{\pi}\*\int_{0}^{\pi}\left(x+\left(x^{2}-1\right)^{1/2}\cos\phi% \right)^{\nu}\cos\left(m\phi\right)\mathrm{d}\phi}}
\assLegendreP[m]{\nu}@{x} = \frac{\Pochhammersym{\nu+1}{m}}{\pi}\*\int_{0}^{\pi}\left(x+\left(x^{2}-1\right)^{1/2}\cos@@{\phi}\right)^{\nu}\cos@{m\phi}\diff{\phi}		 

LegendreP(nu, m, x) = (pochhammer(nu + 1, m))/(Pi)* int((x +((x)^(2)- 1)^(1/2)* cos(phi))^(nu)* cos(m*phi), phi = 0..Pi)

LegendreP[\[Nu], m, 3, x] == Divide[Pochhammer[\[Nu]+ 1, m],Pi]* Integrate[(x +((x)^(2)- 1)^(1/2)* Cos[\[Phi]])^\[Nu]* Cos[m*\[Phi]], {\[Phi], 0, Pi}, GenerateConditions->None]
Failure Failure Skipped - Because timed out Successful [Tested: 90]
14.12.E8 P n m ⁑ ( x ) = 2 m ⁒ m ! ⁒ ( n + m ) ! ⁒ ( x 2 - 1 ) m / 2 ( 2 ⁒ m ) ! ⁒ ( n - m ) ! ⁒ Ο€ ⁒ ∫ 0 Ο€ ( x + ( x 2 - 1 ) 1 / 2 ⁒ cos ⁑ Ο• ) n - m ⁒ ( sin ⁑ Ο• ) 2 ⁒ m ⁒ d Ο• Legendre-P-first-kind π‘š 𝑛 π‘₯ superscript 2 π‘š π‘š 𝑛 π‘š superscript superscript π‘₯ 2 1 π‘š 2 2 π‘š 𝑛 π‘š πœ‹ superscript subscript 0 πœ‹ superscript π‘₯ superscript superscript π‘₯ 2 1 1 2 italic-Ο• 𝑛 π‘š superscript italic-Ο• 2 π‘š italic-Ο• {\displaystyle{\displaystyle P^{m}_{n}\left(x\right)=\frac{2^{m}m!(n+m)!\left(% x^{2}-1\right)^{m/2}}{(2m)!(n-m)!\pi}\int_{0}^{\pi}\left(x+\left(x^{2}-1\right% )^{1/2}\cos\phi\right)^{n-m}(\sin\phi)^{2m}\mathrm{d}\phi}}
\assLegendreP[m]{n}@{x} = \frac{2^{m}m!(n+m)!\left(x^{2}-1\right)^{m/2}}{(2m)!(n-m)!\pi}\int_{0}^{\pi}\left(x+\left(x^{2}-1\right)^{1/2}\cos@@{\phi}\right)^{n-m}(\sin@@{\phi})^{2m}\diff{\phi}		 n β‰₯ m 𝑛 π‘š {\displaystyle{\displaystyle n\geq m}} 
LegendreP(n, m, x) = ((2)^(m)* factorial(m)*factorial(n + m)*((x)^(2)- 1)^(m/2))/(factorial(2*m)*factorial(n - m)*Pi)*int((x +((x)^(2)- 1)^(1/2)* cos(phi))^(n - m)*(sin(phi))^(2*m), phi = 0..Pi)

LegendreP[n, m, 3, x] == Divide[(2)^(m)* (m)!*(n + m)!*((x)^(2)- 1)^(m/2),(2*m)!*(n - m)!*Pi]*Integrate[(x +((x)^(2)- 1)^(1/2)* Cos[\[Phi]])^(n - m)*(Sin[\[Phi]])^(2*m), {\[Phi], 0, Pi}, GenerateConditions->None]
Error Aborted - Successful [Tested: 18]
14.12.E9 𝑸 n m ⁑ ( x ) = 1 n ! ⁒ ∫ 0 u ( x - ( x 2 - 1 ) 1 / 2 ⁒ cosh ⁑ t ) n ⁒ cosh ⁑ ( m ⁒ t ) ⁒ d t associated-Legendre-black-Q π‘š 𝑛 π‘₯ 1 𝑛 superscript subscript 0 𝑒 superscript π‘₯ superscript superscript π‘₯ 2 1 1 2 𝑑 𝑛 π‘š 𝑑 𝑑 {\displaystyle{\displaystyle\boldsymbol{Q}^{m}_{n}\left(x\right)=\frac{1}{n!}% \int_{0}^{u}\left(x-\left(x^{2}-1\right)^{1/2}\cosh t\right)^{n}\cosh\left(mt% \right)\mathrm{d}t}}
\assLegendreOlverQ[m]{n}@{x} = \frac{1}{n!}\int_{0}^{u}\left(x-\left(x^{2}-1\right)^{1/2}\cosh@@{t}\right)^{n}\cosh@{mt}\diff{t}		 

exp(-(m)*Pi*I)*LegendreQ(n,m,x)/GAMMA(n+m+1) = (1)/(factorial(n))*int((x -((x)^(2)- 1)^(1/2)* cosh(t))^(n)* cosh(m*t), t = 0..u)

Exp[-(m) Pi I] LegendreQ[n, m, 3, x]/Gamma[n + m + 1] == Divide[1,(n)!]*Integrate[(x -((x)^(2)- 1)^(1/2)* Cosh[t])^(n)* Cosh[m*t], {t, 0, u}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
14.12.E10 u = 1 2 ⁒ ln ⁑ ( x + 1 x - 1 ) 𝑒 1 2 π‘₯ 1 π‘₯ 1 {\displaystyle{\displaystyle u=\frac{1}{2}\ln\left(\frac{x+1}{x-1}\right)}}
u = \frac{1}{2}\ln@{\frac{x+1}{x-1}}		 

u = (1)/(2)*ln((x + 1)/(x - 1))

u == Divide[1,2]*Log[Divide[x + 1,x - 1]]
Failure Failure
Failed [30 / 30]
Result: .613064480e-1+.5000000000*I
Test Values: {u = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: .3167192595-1.070796327*I
Test Values: {u = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[0.06130644756738857, 0.49999999999999994]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5]}


Result: Complex[0.3167192594503838, -1.0707963267948966]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 0.5]}

... skip entries to safe data
14.12.E11 𝑸 n m ⁑ ( x ) = ( x 2 - 1 ) m / 2 2 n + 1 ⁒ n ! ⁒ ∫ - 1 1 ( 1 - t 2 ) n ( x - t ) n + m + 1 ⁒ d t associated-Legendre-black-Q π‘š 𝑛 π‘₯ superscript superscript π‘₯ 2 1 π‘š 2 superscript 2 𝑛 1 𝑛 superscript subscript 1 1 superscript 1 superscript 𝑑 2 𝑛 superscript π‘₯ 𝑑 𝑛 π‘š 1 𝑑 {\displaystyle{\displaystyle\boldsymbol{Q}^{m}_{n}\left(x\right)=\frac{\left(x% ^{2}-1\right)^{m/2}}{2^{n+1}n!}\int_{-1}^{1}\frac{\left(1-t^{2}\right)^{n}}{(x% -t)^{n+m+1}}\mathrm{d}t}}
\assLegendreOlverQ[m]{n}@{x} = \frac{\left(x^{2}-1\right)^{m/2}}{2^{n+1}n!}\int_{-1}^{1}\frac{\left(1-t^{2}\right)^{n}}{(x-t)^{n+m+1}}\diff{t}		 

exp(-(m)*Pi*I)*LegendreQ(n,m,x)/GAMMA(n+m+1) = (((x)^(2)- 1)^(m/2))/((2)^(n + 1)* factorial(n))*int(((1 - (t)^(2))^(n))/((x - t)^(n + m + 1)), t = - 1..1)

Exp[-(m) Pi I] LegendreQ[n, m, 3, x]/Gamma[n + m + 1] == Divide[((x)^(2)- 1)^(m/2),(2)^(n + 1)* (n)!]*Integrate[Divide[(1 - (t)^(2))^(n),(x - t)^(n + m + 1)], {t, - 1, 1}, GenerateConditions->None]
Failure Failure
Failed [9 / 27]
Result: -.6801747617+Float(undefined)*I
Test Values: {x = 1/2, m = 1, n = 1}


Result: -.3400873809-Float(infinity)*I
Test Values: {x = 1/2, m = 1, n = 2}

... skip entries to safe data
 Successful [Tested: 27]
14.12.E12 𝑸 n m ⁑ ( x ) = 1 ( n - m ) ! ⁒ P n m ⁑ ( x ) ⁒ ∫ x ∞ d t ( t 2 - 1 ) ⁒ ( P n m ⁑ ( t ) ) 2 associated-Legendre-black-Q π‘š 𝑛 π‘₯ 1 𝑛 π‘š Legendre-P-first-kind π‘š 𝑛 π‘₯ superscript subscript π‘₯ 𝑑 superscript 𝑑 2 1 superscript Legendre-P-first-kind π‘š 𝑛 𝑑 2 {\displaystyle{\displaystyle\boldsymbol{Q}^{m}_{n}\left(x\right)=\frac{1}{(n-m% )!}P^{m}_{n}\left(x\right)\int_{x}^{\infty}\frac{\mathrm{d}t}{\left(t^{2}-1% \right)\left(\displaystyle P^{m}_{n}\left(t\right)\right)^{2}}}}
\assLegendreOlverQ[m]{n}@{x} = \frac{1}{(n-m)!}\assLegendreP[m]{n}@{x}\int_{x}^{\infty}\frac{\diff{t}}{\left(t^{2}-1\right)\left(\displaystyle\assLegendreP[m]{n}@{t}\right)^{2}}		 n β‰₯ m 𝑛 π‘š {\displaystyle{\displaystyle n\geq m}} 
exp(-(m)*Pi*I)*LegendreQ(n,m,x)/GAMMA(n+m+1) = (1)/(factorial(n - m))*LegendreP(n, m, x)*int((1)/(((t)^(2)- 1)*(LegendreP(n, m, t))^(2)), t = x..infinity)

Exp[-(m) Pi I] LegendreQ[n, m, 3, x]/Gamma[n + m + 1] == Divide[1,(n - m)!]*LegendreP[n, m, 3, x]*Integrate[Divide[1,((t)^(2)- 1)*(LegendreP[n, m, 3, t])^(2)], {t, x, Infinity}, GenerateConditions->None]
Failure Aborted
Failed [6 / 18]
Result: -.6801747617-Float(infinity)*I
Test Values: {x = 1/2, m = 1, n = 1}


Result: -.3400873809-Float(infinity)*I
Test Values: {x = 1/2, m = 1, n = 2}

... skip entries to safe data
 Skipped - Because timed out
14.12.E13 𝑸 n ⁑ ( x ) = 1 2 ⁒ ( n ! ) ⁒ ∫ - 1 1 P n ⁑ ( t ) x - t ⁒ d t shorthand-associated-Legendre-black-Q 𝑛 π‘₯ 1 2 𝑛 superscript subscript 1 1 Legendre-spherical-polynomial 𝑛 𝑑 π‘₯ 𝑑 𝑑 {\displaystyle{\displaystyle\boldsymbol{Q}_{n}\left(x\right)=\frac{1}{2(n!)}% \int_{-1}^{1}\frac{P_{n}\left(t\right)}{x-t}\mathrm{d}t}}
\assLegendreOlverQ[]{n}@{x} = \frac{1}{2(n!)}\int_{-1}^{1}\frac{\LegendrepolyP{n}@{t}}{x-t}\diff{t}		 

LegendreQ(n,x)/GAMMA(n+1) = (1)/(2*(factorial(n)))*int((LegendreP(n, t))/(x - t), t = - 1..1)

Exp[-(n) Pi I] LegendreQ[n, 2, 3, x]/Gamma[n + 3] == Divide[1,2*((n)!)]*Integrate[Divide[LegendreP[n, t],x - t], {t, - 1, 1}, GenerateConditions->None]
Failure Aborted
Failed [3 / 9]
Result: Float(undefined)-.7853981634*I
Test Values: {x = 1/2, n = 1}


Result: Float(undefined)+.9817477045e-1*I
Test Values: {x = 1/2, n = 2}

... skip entries to safe data
 Skipped - Because timed out
14.12.E14 𝑸 n ⁑ ( x ) = 1 n ! ⁒ ∫ 0 ∞ d t ( x + ( x 2 - 1 ) 1 / 2 ⁒ cosh ⁑ t ) n + 1 shorthand-associated-Legendre-black-Q 𝑛 π‘₯ 1 𝑛 superscript subscript 0 𝑑 superscript π‘₯ superscript superscript π‘₯ 2 1 1 2 𝑑 𝑛 1 {\displaystyle{\displaystyle\boldsymbol{Q}_{n}\left(x\right)=\frac{1}{n!}\int_% {0}^{\infty}\frac{\mathrm{d}t}{\left(x+(x^{2}-1)^{1/2}\cosh t\right)^{n+1}}}}
\assLegendreOlverQ[]{n}@{x} = \frac{1}{n!}\int_{0}^{\infty}\frac{\diff{t}}{\left(x+(x^{2}-1)^{1/2}\cosh@@{t}\right)^{n+1}}		 

LegendreQ(n,x)/GAMMA(n+1) = (1)/(factorial(n))*int((1)/((x +((x)^(2)- 1)^(1/2)* cosh(t))^(n + 1)), t = 0..infinity)

Exp[-(n) Pi I] LegendreQ[n, 2, 3, x]/Gamma[n + 3] == Divide[1,(n)!]*Integrate[Divide[1,(x +((x)^(2)- 1)^(1/2)* Cosh[t])^(n + 1)], {t, 0, Infinity}, GenerateConditions->None]
Aborted Aborted Successful [Tested: 9] Skipped - Because timed out
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