Assosiated Legendre function of the second kind of type 3: Differentiation (formula 07.12.20.0003)

LegendreQ

$Mathematica Notation$

Hypergeometric Functions

LegendreQ[nu,mu,3,z]

Differentiation

Low-order differentiation

With respect to nu

http://functions.wolfram.com/07.12.20.0003.01

Input Form

D[LegendreQ[\[Nu], \[Mu], 3, z], \[Nu]] == (Pi/2) E^(I \[Mu] Pi) Csc[\[Mu] Pi] Pochhammer[1 - \[Mu] + \[Nu], 2 \[Mu]] (PolyGamma[1 - \[Mu] + \[Nu]] - PolyGamma[1 + \[Mu] + \[Nu]]) LegendreP[\[Nu], -\[Mu], 3, z] - ((E^(I \[Mu] Pi) (1 + 2 \[Nu]) (z - 1))/ (4 Gamma[1 - \[Mu] + \[Nu]])) ((z - 1)^(\[Mu]/2)/(z + 1)^(\[Mu]/2)) (((z + 1)^\[Mu]/(z - 1)^\[Mu]) Gamma[\[Mu] - 1] Gamma[1 - \[Mu] + \[Nu]] HypergeometricPFQ[{{1 - \[Nu], 2 + \[Nu]}, {1}, {1, -\[Nu], 1 + \[Nu]}}, {{2, 2 - \[Mu]}, {}, {2 + \[Nu], 1 - \[Nu]}}, (1 - z)/2, (1 - z)/2] + Gamma[-1 - \[Mu]] Gamma[1 + \[Mu] + \[Nu]] HypergeometricPFQ[ {{1 - \[Nu], 2 + \[Nu]}, {1}, {1, -\[Nu], 1 + \[Nu]}}, {{2, 2 + \[Mu]}, {}, {2 + \[Nu], 1 - \[Nu]}}, (1 - z)/2, (1 - z)/2]) /; !Element[\[Mu], Integers]

Standard Form

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MathML Form

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</ms> <ms> μ </ms> </list> </apply> <ms> ; </ms> </list> </apply> <ms> ; </ms> <apply> <ci> RowBox </ci> <list> <ms> ν </ms> <ms> + </ms> <ms> 2 </ms> </list> </apply> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> ν </ms> </list> </apply> <ms> ; </ms> </list> </apply> </list> </apply> </list> </list> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> z </ms> </list> </apply> <ms> 2 </ms> </apply> </list> </apply> <ms> , </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> z </ms> </list> </apply> <ms> 2 </ms> </apply> <ms> ] </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> Γ </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> μ </ms> </list> </apply> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> Γ </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> μ </ms> <ms> + </ms> <ms> ν </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> F </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> 0 </ms> <ms> 2 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> 1 </ms> <ms> 3 </ms> </list> </apply> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> ν </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ν </ms> <ms> + </ms> <ms> 2 </ms> </list> </apply> <ms> ; </ms> <ms> 1 </ms> <ms> ; </ms> <ms> 1 </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> ν </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ν </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ; </ms> </list> </apply> </list> </apply> </list> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> + </ms> <ms> μ </ms> </list> </apply> <ms> ; </ms> </list> </apply> <ms> ; </ms> <apply> <ci> RowBox </ci> <list> <ms> ν </ms> <ms> + </ms> <ms> 2 </ms> </list> </apply> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> ν </ms> </list> </apply> <ms> ; </ms> </list> </apply> </list> </apply> </list> </list> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> z </ms> </list> </apply> <ms> 2 </ms> </apply> </list> </apply> <ms> , </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> z </ms> </list> </apply> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <ms> μ </ms> <ms> ∉ </ms> <ms> ℤ </ms> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>

Rule Form

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Date Added to functions.wolfram.com (modification date)

2001-10-29

LegendreQ[nu,z]
LegendreQ[nu,mu,z]
LegendreQ[nu,mu,2,z]