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
14.24.E1 P ν - μ ⁡ ( z ⁢ e s ⁢ π ⁢ i ) = e s ⁢ ν ⁢ π ⁢ i ⁢ P ν - μ ⁡ ( z ) + 2 ⁢ i ⁢ sin ⁡ ( ( ν + 1 2 ) ⁢ s ⁢ π ) ⁢ e - s ⁢ π ⁢ i / 2 cos ⁡ ( ν ⁢ π ) ⁢ Γ ⁡ ( μ - ν ) ⁢ 𝑸 ν μ ⁡ ( z ) Legendre-P-first-kind 𝜇 𝜈 𝑧 superscript 𝑒 𝑠 𝜋 𝑖 superscript 𝑒 𝑠 𝜈 𝜋 𝑖 Legendre-P-first-kind 𝜇 𝜈 𝑧 2 𝑖 𝜈 1 2 𝑠 𝜋 superscript 𝑒 𝑠 𝜋 𝑖 2 𝜈 𝜋 Euler-Gamma 𝜇 𝜈 associated-Legendre-black-Q 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(ze^{s\pi i}\right)=e^{s\nu\pi i% }P^{-\mu}_{\nu}\left(z\right)+\frac{2i\sin\left(\left(\nu+\frac{1}{2}\right)s% \pi\right)e^{-s\pi i/2}}{\cos\left(\nu\pi\right)\Gamma\left(\mu-\nu\right)}% \boldsymbol{Q}^{\mu}_{\nu}\left(z\right)}}
\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}		 ℜ ⁡ ( μ - ν ) > 0 𝜇 𝜈 0 {\displaystyle{\displaystyle\Re(\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)

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]
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 𝑸 ν μ ⁡ ( z ⁢ e s ⁢ π ⁢ i ) = ( - 1 ) s ⁢ e - s ⁢ ν ⁢ π ⁢ i ⁢ 𝑸 ν μ ⁡ ( z ) associated-Legendre-black-Q 𝜇 𝜈 𝑧 superscript 𝑒 𝑠 𝜋 𝑖 superscript 1 𝑠 superscript 𝑒 𝑠 𝜈 𝜋 𝑖 associated-Legendre-black-Q 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(ze^{s\pi i}\right)% =(-1)^{s}e^{-s\nu\pi i}\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)}}
\assLegendreOlverQ[\mu]{\nu}@{ze^{s\pi i}} = (-1)^{s}e^{-s\nu\pi i}\assLegendreOlverQ[\mu]{\nu}@{z}		 

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)

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]
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 P ν , s - μ ⁡ ( z ) = e s ⁢ μ ⁢ π ⁢ i ⁢ P ν - μ ⁡ ( z ) Legendre-P-first-kind 𝜇 𝜈 𝑠 𝑧 superscript 𝑒 𝑠 𝜇 𝜋 𝑖 Legendre-P-first-kind 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle P^{-\mu}_{\nu,s}\left(z\right)=e^{s\mu\pi i}P^{-% \mu}_{\nu}\left(z\right)}}
\assLegendreP[-\mu]{\nu,s}@{z} = e^{s\mu\pi i}\assLegendreP[-\mu]{\nu}@{z}		 

LegendreP(nu , s, - mu, z) = exp(s*mu*Pi*I)*LegendreP(nu, - mu, z)

LegendreP[\[Nu], s, - \[Mu], 3, z] == Exp[s*\[Mu]*Pi*I]*LegendreP[\[Nu], - \[Mu], 3, z]
Error Failure - Successful [Tested: 300]
14.24.E4 𝑸 ν , s μ ⁡ ( z ) = e - s ⁢ μ ⁢ π ⁢ i ⁢ 𝑸 ν μ ⁡ ( z ) - π ⁢ i ⁢ sin ⁡ ( s ⁢ μ ⁢ π ) sin ⁡ ( μ ⁢ π ) ⁢ Γ ⁡ ( ν - μ + 1 ) ⁢ P ν - μ ⁡ ( z ) associated-Legendre-black-Q 𝜇 𝜈 𝑠 𝑧 superscript 𝑒 𝑠 𝜇 𝜋 𝑖 associated-Legendre-black-Q 𝜇 𝜈 𝑧 𝜋 𝑖 𝑠 𝜇 𝜋 𝜇 𝜋 Euler-Gamma 𝜈 𝜇 1 Legendre-P-first-kind 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu,s}\left(z\right)=e^{-s% \mu\pi i}\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)-\frac{\pi i\sin\left(s\mu\pi% \right)}{\sin\left(\mu\pi\right)\Gamma\left(\nu-\mu+1\right)}P^{-\mu}_{\nu}% \left(z\right)}}
\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}		 ℜ ⁡ ( ν - μ + 1 ) > 0 𝜈 𝜇 1 0 {\displaystyle{\displaystyle\Re(\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)

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]
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
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