14.25: Difference between revisions

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! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/14.25.E1 14.25.E1] || [[Item:Q4958|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\realpart@@{\mu} > \realpart@@{\nu}, \realpart@@{\nu} > -1, \realpart@@{(\mu-\nu)} > 0, \realpart@@{(\nu+1)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Error || Aborted || - || Skipped - Because timed out
| [https://dlmf.nist.gov/14.25.E1 14.25.E1] || <math qid="Q4958">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\realpart@@{\mu} > \realpart@@{\nu}, \realpart@@{\nu} > -1, \realpart@@{(\mu-\nu)} > 0, \realpart@@{(\nu+1)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Error || Aborted || - || Skipped - Because timed out
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| [https://dlmf.nist.gov/14.25.E2 14.25.E2] || [[Item:Q4959|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\realpart@{\nu+1} > \realpart@@{\mu}, \realpart@@{\mu} > -\tfrac{1}{2}, \realpart@@{(\mu+\frac{1}{2})} > 0, \realpart@@{(\nu-\mu+1)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Error || Aborted || - || Skipped - Because timed out
| [https://dlmf.nist.gov/14.25.E2 14.25.E2] || <math qid="Q4959">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>\realpart@{\nu+1} > \realpart@@{\mu}, \realpart@@{\mu} > -\tfrac{1}{2}, \realpart@@{(\mu+\frac{1}{2})} > 0, \realpart@@{(\nu-\mu+1)} > 0</math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Error || Aborted || - || Skipped - Because timed out
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Latest revision as of 11:38, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
14.25.E1 P ν - μ ⁡ ( z ) = ( z 2 - 1 ) μ / 2 2 ν ⁢ Γ ⁡ ( μ - ν ) ⁢ Γ ⁡ ( ν + 1 ) ⁢ ∫ 0 ∞ ( sinh ⁡ t ) 2 ⁢ ν + 1 ( z + cosh ⁡ t ) ν + μ + 1 ⁢ d t Legendre-P-first-kind 𝜇 𝜈 𝑧 superscript superscript 𝑧 2 1 𝜇 2 superscript 2 𝜈 Euler-Gamma 𝜇 𝜈 Euler-Gamma 𝜈 1 superscript subscript 0 superscript 𝑡 2 𝜈 1 superscript 𝑧 𝑡 𝜈 𝜇 1 𝑡 {\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}		 ℜ ⁡ μ > ℜ ⁡ ν , ℜ ⁡ ν > - 1 , ℜ ⁡ ( μ - ν ) > 0 , ℜ ⁡ ( ν + 1 ) > 0 formulae-sequence 𝜇 𝜈 formulae-sequence 𝜈 1 formulae-sequence 𝜇 𝜈 0 𝜈 1 0 {\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 𝑸 ν μ ⁡ ( z ) = π 1 / 2 ⁢ ( z 2 - 1 ) μ / 2 2 μ ⁢ Γ ⁡ ( μ + 1 2 ) ⁢ Γ ⁡ ( ν - μ + 1 ) ⁢ ∫ 0 ∞ ( sinh ⁡ t ) 2 ⁢ μ ( z + ( z 2 - 1 ) 1 / 2 ⁢ cosh ⁡ t ) ν + μ + 1 ⁢ d t associated-Legendre-black-Q 𝜇 𝜈 𝑧 superscript 𝜋 1 2 superscript superscript 𝑧 2 1 𝜇 2 superscript 2 𝜇 Euler-Gamma 𝜇 1 2 Euler-Gamma 𝜈 𝜇 1 superscript subscript 0 superscript 𝑡 2 𝜇 superscript 𝑧 superscript superscript 𝑧 2 1 1 2 𝑡 𝜈 𝜇 1 𝑡 {\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}		 ℜ ⁡ ( ν + 1 ) > ℜ ⁡ μ , ℜ ⁡ μ > - 1 2 , ℜ ⁡ ( μ + 1 2 ) > 0 , ℜ ⁡ ( ν - μ + 1 ) > 0 formulae-sequence 𝜈 1 𝜇 formulae-sequence 𝜇 1 2 formulae-sequence 𝜇 1 2 0 𝜈 𝜇 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,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
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