Legendre and Related Functions - 14.24 Analytic Continuation
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
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| 14.24.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \assLegendreP[-\mu]{\nu}@{ze^{s\pi i}} = e^{s\nu\pi i}\assLegendreP[-\mu]{\nu}@{z}+\frac{2i\sin@{\left(\nu+\frac{1}{2}\right)s\pi}e^{-s\pi i/2}}{\cos@{\nu\pi}\EulerGamma@{\mu-\nu}}\assLegendreOlverQ[\mu]{\nu}@{z}}
\assLegendreP[-\mu]{\nu}@{ze^{s\pi i}} = e^{s\nu\pi i}\assLegendreP[-\mu]{\nu}@{z}+\frac{2i\sin@{\left(\nu+\frac{1}{2}\right)s\pi}e^{-s\pi i/2}}{\cos@{\nu\pi}\EulerGamma@{\mu-\nu}}\assLegendreOlverQ[\mu]{\nu}@{z} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\mu-\nu)} > 0} | LegendreP(nu, - mu, z*exp(s*Pi*I)) = exp(s*nu*Pi*I)*LegendreP(nu, - mu, z)+(2*I*sin((nu +(1)/(2))*s*Pi)*exp(- s*Pi*I/2))/(cos(nu*Pi)*GAMMA(mu - nu))*exp(-(mu)*Pi*I)*LegendreQ(nu,mu,z)/GAMMA(nu+mu+1)
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LegendreP[\[Nu], - \[Mu], 3, z*Exp[s*Pi*I]] == Exp[s*\[Nu]*Pi*I]*LegendreP[\[Nu], - \[Mu], 3, z]+Divide[2*I*Sin[(\[Nu]+Divide[1,2])*s*Pi]*Exp[- s*Pi*I/2],Cos[\[Nu]*Pi]*Gamma[\[Mu]- \[Nu]]]*Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, z]/Gamma[\[Nu] + \[Mu] + 1]
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Failure | Failure | Manual Skip! | Failed [299 / 300]
Result: Complex[-21.32728052513349, -8.911336897051166]
Test Values: {Rule[s, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
Result: Complex[13.892460412350314, 1.7999110613880858]
Test Values: {Rule[s, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], 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 |
| 14.24.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \assLegendreOlverQ[\mu]{\nu}@{ze^{s\pi i}} = (-1)^{s}e^{-s\nu\pi i}\assLegendreOlverQ[\mu]{\nu}@{z}}
\assLegendreOlverQ[\mu]{\nu}@{ze^{s\pi i}} = (-1)^{s}e^{-s\nu\pi i}\assLegendreOlverQ[\mu]{\nu}@{z} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | exp(-(mu)*Pi*I)*LegendreQ(nu,mu,z*exp(s*Pi*I))/GAMMA(nu+mu+1) = (- 1)^(s)* exp(- s*nu*Pi*I)*exp(-(mu)*Pi*I)*LegendreQ(nu,mu,z)/GAMMA(nu+mu+1)
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Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, z*Exp[s*Pi*I]]/Gamma[\[Nu] + \[Mu] + 1] == (- 1)^(s)* Exp[- s*\[Nu]*Pi*I]*Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, z]/Gamma[\[Nu] + \[Mu] + 1]
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Failure | Failure | Failed [300 / 300] Result: -.2140796977+.7286338337*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, s = -3/2, z = 1/2*3^(1/2)+1/2*I}
Result: -.1549543426-.1299026639*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, s = -3/2, z = -1/2+1/2*I*3^(1/2)}
... skip entries to safe data |
Failed [300 / 300]
Result: Complex[-0.2140796979538467, 0.7286338343398007]
Test Values: {Rule[s, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], 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[2.2472082058834166, -8.359397493451592]
Test Values: {Rule[s, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 14.24.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \assLegendreP[-\mu]{\nu,s}@{z} = e^{s\mu\pi i}\assLegendreP[-\mu]{\nu}@{z}}
\assLegendreP[-\mu]{\nu,s}@{z} = e^{s\mu\pi i}\assLegendreP[-\mu]{\nu}@{z} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | LegendreP(nu , s, - mu, z) = exp(s*mu*Pi*I)*LegendreP(nu, - mu, z)
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LegendreP[\[Nu], s, - \[Mu], 3, z] == Exp[s*\[Mu]*Pi*I]*LegendreP[\[Nu], - \[Mu], 3, z]
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Error | Failure | - | Successful [Tested: 300] |
| 14.24.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \assLegendreOlverQ[\mu]{\nu,s}@{z} = e^{-s\mu\pi i}\assLegendreOlverQ[\mu]{\nu}@{z}-\frac{\pi i\sin@{s\mu\pi}}{\sin@{\mu\pi}\EulerGamma@{\nu-\mu+1}}\assLegendreP[-\mu]{\nu}@{z}}
\assLegendreOlverQ[\mu]{\nu,s}@{z} = e^{-s\mu\pi i}\assLegendreOlverQ[\mu]{\nu}@{z}-\frac{\pi i\sin@{s\mu\pi}}{\sin@{\mu\pi}\EulerGamma@{\nu-\mu+1}}\assLegendreP[-\mu]{\nu}@{z} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\nu-\mu+1)} > 0} | exp(-(mu)*Pi*I)*LegendreQ(nu , s,mu,z)/GAMMA(nu , s+mu+1) = exp(- s*mu*Pi*I)*exp(-(mu)*Pi*I)*LegendreQ(nu,mu,z)/GAMMA(nu+mu+1)-(Pi*I*sin(s*mu*Pi))/(sin(mu*Pi)*GAMMA(nu - mu + 1))*LegendreP(nu, - mu, z)
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Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], s, \[Mu], 3, z]/Gamma[\[Nu], s + \[Mu] + 1] == Exp[- s*\[Mu]*Pi*I]*Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, z]/Gamma[\[Nu] + \[Mu] + 1]-Divide[Pi*I*Sin[s*\[Mu]*Pi],Sin[\[Mu]*Pi]*Gamma[\[Nu]- \[Mu]+ 1]]*LegendreP[\[Nu], - \[Mu], 3, z]
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Error | Failure | - | Failed [69 / 300]
Result: Indeterminate
Test Values: {Rule[s, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5], Rule[ν, -1.5]}
Result: Indeterminate
Test Values: {Rule[s, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, -1.5], Rule[ν, -0.5]}
... skip entries to safe data |