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 ∂ ∂ ⁡ ν ⁡ 𝖯 ν μ ⁡ ( x ) = π ⁢ cot ⁡ ( ν ⁢ π ) ⁢ 𝖯 ν μ ⁡ ( x ) - 1 π ⁢ 𝖠 ν μ ⁢ ( x ) partial-derivative 𝜈 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 𝜋 𝜈 𝜋 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 1 𝜋 superscript subscript 𝖠 𝜈 𝜇 𝑥 {\displaystyle{\displaystyle\frac{\partial}{\partial\nu}\mathsf{P}^{\mu}_{\nu}% \left(x\right)=\pi\cot\left(\nu\pi\right)\mathsf{P}^{\mu}_{\nu}\left(x\right)-% \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)		 | ( 1 2 - 1 2 ⁢ x ) | < 1 1 2 1 2 𝑥 1 {\displaystyle{\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 ∂ ∂ ⁡ ν ⁡ 𝖰 ν μ ⁡ ( x ) = - 1 2 ⁢ π 2 ⁢ 𝖯 ν μ ⁡ ( x ) + π ⁢ sin ⁡ ( μ ⁢ π ) sin ⁡ ( ν ⁢ π ) ⁢ sin ⁡ ( ( ν + μ ) ⁢ π ) ⁢ 𝖰 ν μ ⁡ ( x ) - 1 2 ⁢ cot ⁡ ( ( ν + μ ) ⁢ π ) ⁢ 𝖠 ν μ ⁢ ( x ) + 1 2 ⁢ csc ⁡ ( ( ν + μ ) ⁢ π ) ⁢ 𝖠 ν μ ⁢ ( - x ) partial-derivative 𝜈 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 1 2 superscript 𝜋 2 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 𝜋 𝜇 𝜋 𝜈 𝜋 𝜈 𝜇 𝜋 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 1 2 𝜈 𝜇 𝜋 superscript subscript 𝖠 𝜈 𝜇 𝑥 1 2 𝜈 𝜇 𝜋 superscript subscript 𝖠 𝜈 𝜇 𝑥 {\displaystyle{\displaystyle\frac{\partial}{\partial\nu}\mathsf{Q}^{\mu}_{\nu}% \left(x\right)=-\tfrac{1}{2}\pi^{2}\mathsf{P}^{\mu}_{\nu}\left(x\right)+\frac{% \pi\sin\left(\mu\pi\right)}{\sin\left(\nu\pi\right)\sin\left((\nu+\mu)\pi% \right)}\mathsf{Q}^{\mu}_{\nu}\left(x\right)-\tfrac{1}{2}\cot\left((\nu+\mu)% \pi\right)\mathsf{A}_{\nu}^{\mu}(x)+\tfrac{1}{2}\csc\left((\nu+\mu)\pi\right)% \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)		 | ( 1 2 - 1 2 ⁢ x ) | < 1 , ℜ ⁡ ( ν + μ + 1 ) > 0 , ℜ ⁡ ( ν - μ + 1 ) > 0 formulae-sequence 1 2 1 2 𝑥 1 formulae-sequence 𝜈 𝜇 1 0 𝜈 𝜇 1 0 {\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1,\Re(\nu+\mu+1)>0,% \Re(\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 𝖠 ν μ ⁢ ( x ) = sin ⁡ ( ν ⁢ π ) ⁢ ( 1 + x 1 - x ) μ / 2 ⁢ ∑ k = 0 ∞ ( 1 2 - 1 2 ⁢ x ) k ⁢ Γ ⁡ ( k - ν ) ⁢ Γ ⁡ ( k + ν + 1 ) k ! ⁢ Γ ⁡ ( k - μ + 1 ) ⁢ ( ψ ⁡ ( k + ν + 1 ) - ψ ⁡ ( k - ν ) ) superscript subscript 𝖠 𝜈 𝜇 𝑥 𝜈 𝜋 superscript 1 𝑥 1 𝑥 𝜇 2 superscript subscript 𝑘 0 superscript 1 2 1 2 𝑥 𝑘 Euler-Gamma 𝑘 𝜈 Euler-Gamma 𝑘 𝜈 1 𝑘 Euler-Gamma 𝑘 𝜇 1 digamma 𝑘 𝜈 1 digamma 𝑘 𝜈 {\displaystyle{\displaystyle\mathsf{A}_{\nu}^{\mu}(x)=\sin\left(\nu\pi\right)% \left(\frac{1+x}{1-x}\right)^{\mu/2}\*\sum_{k=0}^{\infty}\frac{\left(\frac{1}{% 2}-\frac{1}{2}x\right)^{k}\Gamma\left(k-\nu\right)\Gamma\left(k+\nu+1\right)}{% k!\Gamma\left(k-\mu+1\right)}\*\left(\psi\left(k+\nu+1\right)-\psi\left(k-\nu% \right)\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)		 ℜ ⁡ ( k - ν ) > 0 , ℜ ⁡ ( k + ν + 1 ) > 0 , ℜ ⁡ ( k - μ + 1 ) > 0 formulae-sequence 𝑘 𝜈 0 formulae-sequence 𝑘 𝜈 1 0 𝑘 𝜇 1 0 {\displaystyle{\displaystyle\Re(k-\nu)>0,\Re(k+\nu+1)>0,\Re(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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