Legendre and Related Functions - 14.11 Derivatives with Respect to Degree or Order

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
14.11.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \pderiv{}{\nu}\FerrersP[\mu]{\nu}@{x} = \pi\cot@{\nu\pi}\FerrersP[\mu]{\nu}@{x}-\frac{1}{\pi}\mathsf{A}_{\nu}^{\mu}(x)}
\pderiv{}{\nu}\FerrersP[\mu]{\nu}@{x} = \pi\cot@{\nu\pi}\FerrersP[\mu]{\nu}@{x}-\frac{1}{\pi}\mathsf{A}_{\nu}^{\mu}(x)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1} 
diff(LegendreP(nu, mu, x), nu) = Pi*cot(nu*Pi)*LegendreP(nu, mu, x)-(1)/(Pi)*(A[nu])^(mu)(x)

D[LegendreP[\[Nu], \[Mu], x], \[Nu]] == Pi*Cot[\[Nu]*Pi]*LegendreP[\[Nu], \[Mu], x]-Divide[1,Pi]*(Subscript[A, \[Nu]])^\[Mu][x]
Failure Failure
Failed [300 / 300]
Result: 11.90824559-1.654502830*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2, A[nu] = 1/2*3^(1/2)+1/2*I}


Result: 11.53757926-1.652858974*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2, A[nu] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[11.90824558684297, -1.654502826549051]
Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[A, ν], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[11.53757925943594, -1.6528589711511499]
Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[A, ν], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
14.11.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \pderiv{}{\nu}\FerrersQ[\mu]{\nu}@{x} = -\tfrac{1}{2}\pi^{2}\FerrersP[\mu]{\nu}@{x}+\frac{\pi\sin@{\mu\pi}}{\sin@{\nu\pi}\sin@{(\nu+\mu)\pi}}\FerrersQ[\mu]{\nu}@{x}-\tfrac{1}{2}\cot@{(\nu+\mu)\pi}\mathsf{A}_{\nu}^{\mu}(x)+\tfrac{1}{2}\csc@{(\nu+\mu)\pi}\mathsf{A}_{\nu}^{\mu}(-x)}
\pderiv{}{\nu}\FerrersQ[\mu]{\nu}@{x} = -\tfrac{1}{2}\pi^{2}\FerrersP[\mu]{\nu}@{x}+\frac{\pi\sin@{\mu\pi}}{\sin@{\nu\pi}\sin@{(\nu+\mu)\pi}}\FerrersQ[\mu]{\nu}@{x}-\tfrac{1}{2}\cot@{(\nu+\mu)\pi}\mathsf{A}_{\nu}^{\mu}(x)+\tfrac{1}{2}\csc@{(\nu+\mu)\pi}\mathsf{A}_{\nu}^{\mu}(-x)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1, \realpart@@{(\nu+\mu+1)} > 0, \realpart@@{(\nu-\mu+1)} > 0} 
diff(LegendreQ(nu, mu, x), nu) = -(1)/(2)*(Pi)^(2)* LegendreP(nu, mu, x)+(Pi*sin(mu*Pi))/(sin(nu*Pi)*sin((nu + mu)*Pi))*LegendreQ(nu, mu, x)-(1)/(2)*cot((nu + mu)*Pi)*(A[nu])^(mu)(x)+(1)/(2)*csc((nu + mu)*Pi)*(A[nu])^(mu)(- x)

D[LegendreQ[\[Nu], \[Mu], x], \[Nu]] == -Divide[1,2]*(Pi)^(2)* LegendreP[\[Nu], \[Mu], x]+Divide[Pi*Sin[\[Mu]*Pi],Sin[\[Nu]*Pi]*Sin[(\[Nu]+ \[Mu])*Pi]]*LegendreQ[\[Nu], \[Mu], x]-Divide[1,2]*Cot[(\[Nu]+ \[Mu])*Pi]*(Subscript[A, \[Nu]])^\[Mu][x]+Divide[1,2]*Csc[(\[Nu]+ \[Mu])*Pi]*(Subscript[A, \[Nu]])^\[Mu][- x]
Failure Failure
Failed [300 / 300]
Result: -2.639260453-18.83790600*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2, A[nu] = 1/2*3^(1/2)+1/2*I}


Result: -2.596785248-18.22264548*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2, A[nu] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-2.639260449912798, -18.837906001053177]
Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[A, ν], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.5967852433828247, -18.222645474383306]
Test Values: {Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[A, ν], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
14.11.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mathsf{A}_{\nu}^{\mu}(x) = \sin@{\nu\pi}\left(\frac{1+x}{1-x}\right)^{\mu/2}\*\sum_{k=0}^{\infty}\frac{\left(\frac{1}{2}-\frac{1}{2}x\right)^{k}\EulerGamma@{k-\nu}\EulerGamma@{k+\nu+1}}{k!\EulerGamma@{k-\mu+1}}\*\left(\digamma@{k+\nu+1}-\digamma@{k-\nu}\right)}
\mathsf{A}_{\nu}^{\mu}(x) = \sin@{\nu\pi}\left(\frac{1+x}{1-x}\right)^{\mu/2}\*\sum_{k=0}^{\infty}\frac{\left(\frac{1}{2}-\frac{1}{2}x\right)^{k}\EulerGamma@{k-\nu}\EulerGamma@{k+\nu+1}}{k!\EulerGamma@{k-\mu+1}}\*\left(\digamma@{k+\nu+1}-\digamma@{k-\nu}\right)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(k-\nu)} > 0, \realpart@@{(k+\nu+1)} > 0, \realpart@@{(k-\mu+1)} > 0} 
(A[nu])^(mu)(x) = sin(nu*Pi)*((1 + x)/(1 - x))^(mu/2)* sum((((1)/(2)-(1)/(2)*x)^(k)* GAMMA(k - nu)*GAMMA(k + nu + 1))/(factorial(k)*GAMMA(k - mu + 1))*(Psi(k + nu + 1)- Psi(k - nu)), k = 0..infinity)

(Subscript[A, \[Nu]])^\[Mu][x] == Sin[\[Nu]*Pi]*(Divide[1 + x,1 - x])^(\[Mu]/2)* Sum[Divide[(Divide[1,2]-Divide[1,2]*x)^(k)* Gamma[k - \[Nu]]*Gamma[k + \[Nu]+ 1],(k)!*Gamma[k - \[Mu]+ 1]]*(PolyGamma[k + \[Nu]+ 1]- PolyGamma[k - \[Nu]]), {k, 0, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
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