Legendre and Related Functions - 14.25 Integral Representations
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
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| 14.25.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}
\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} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{\mu} > \realpart@@{\nu}, \realpart@@{\nu} > -1, \realpart@@{(\mu-\nu)} > 0, \realpart@@{(\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)
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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]
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Error | Aborted | - | Skipped - Because timed out |
| 14.25.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}
\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} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@{\nu+1} > \realpart@@{\mu}, \realpart@@{\mu} > -\tfrac{1}{2}, \realpart@@{(\mu+\frac{1}{2})} > 0, \realpart@@{(\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)
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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]
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Error | Aborted | - | Skipped - Because timed out |