Legendre and Related Functions - 14.25 Integral Representations

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
14.25.E1	${\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(z\right)=\frac{\left(z^{2}-1\right)^{\mu/2}}{2^{\nu}\Gamma\left(\mu-\nu\right)\Gamma\left(\nu+1\right)}\int_{0}^{\infty}\frac{(\sinh t)^{2\nu+1}}{(z+\cosh t)^{\nu+\mu+1}}\mathrm{d}t}}$ \assLegendreP[-\mu]{\nu}@{z} = \frac{\left(z^{2}-1\right)^{\mu/2}}{2^{\nu}\EulerGamma@{\mu-\nu}\EulerGamma@{\nu+1}}\int_{0}^{\infty}\frac{(\sinh@@{t})^{2\nu+1}}{(z+\cosh@@{t})^{\nu+\mu+1}}\diff{t}	${\displaystyle{\displaystyle\Re\mu>\Re\nu,\Re\nu>-1,\Re(\mu-\nu)>0,\Re(\nu+1)>0}}$	LegendreP(nu, - mu, z) = (((z)^(2)- 1)^(mu/2))/((2)^(nu)* GAMMA(mu - nu)*GAMMA(nu + 1))*int(((sinh(t))^(2*nu + 1))/((z + cosh(t))^(nu + mu + 1)), t = 0..infinity)	LegendreP[\[Nu], - \[Mu], 3, z] == Divide[((z)^(2)- 1)^(\[Mu]/2),(2)^\[Nu]* Gamma[\[Mu]- \[Nu]]*Gamma[\[Nu]+ 1]]*Integrate[Divide[(Sinh[t])^(2*\[Nu]+ 1),(z + Cosh[t])^(\[Nu]+ \[Mu]+ 1)], {t, 0, Infinity}, GenerateConditions->None]	Error	Aborted	-	Skipped - Because timed out
14.25.E2	${\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)=\frac{\pi^{1/2}\left(z^{2}-1\right)^{\mu/2}}{2^{\mu}\Gamma\left(\mu+\frac{1}{2}\right)\Gamma\left(\nu-\mu+1\right)}\*\int_{0}^{\infty}\frac{(\sinh t)^{2\mu}}{\left(z+(z^{2}-1)^{1/2}\cosh t\right)^{\nu+\mu+1}}\mathrm{d}t}}$ \assLegendreOlverQ[\mu]{\nu}@{z} = \frac{\pi^{1/2}\left(z^{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(z+(z^{2}-1)^{1/2}\cosh@@{t}\right)^{\nu+\mu+1}}\diff{t}	${\displaystyle{\displaystyle\Re\left(\nu+1\right)>\Re\mu,\Re\mu>-\tfrac{1}{2},\Re(\mu+\frac{1}{2})>0,\Re(\nu-\mu+1)>0}}$	exp(-(mu)*Pi*I)*LegendreQ(nu,mu,z)/GAMMA(nu+mu+1) = ((Pi)^(1/2)*((z)^(2)- 1)^(mu/2))/((2)^(mu)* GAMMA(mu +(1)/(2))*GAMMA(nu - mu + 1))* int(((sinh(t))^(2*mu))/((z +((z)^(2)- 1)^(1/2)* cosh(t))^(nu + mu + 1)), t = 0..infinity)	Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, z]/Gamma[\[Nu] + \[Mu] + 1] == Divide[(Pi)^(1/2)*((z)^(2)- 1)^(\[Mu]/2),(2)^\[Mu]* Gamma[\[Mu]+Divide[1,2]]*Gamma[\[Nu]- \[Mu]+ 1]]* Integrate[Divide[(Sinh[t])^(2*\[Mu]),(z +((z)^(2)- 1)^(1/2)* Cosh[t])^(\[Nu]+ \[Mu]+ 1)], {t, 0, Infinity}, GenerateConditions->None]	Error	Aborted	-	Skipped - Because timed out