Assosiated Legendre function of the second kind of type 3: Representations through more general functions (formula 07.12.26.0086)

LegendreQ

$Mathematica Notation$

Hypergeometric Functions

LegendreQ[nu,mu,3,z]

Representations through more general functions

Through hypergeometric functions

Involving ₂F₁

http://functions.wolfram.com/07.12.26.0086.01

Input Form

LegendreQ[\[Nu], \[Mu], 3, z] == (2^(\[Mu] - 1) E^(I Pi \[Mu]) Sqrt[Pi] (-1 + z)^(\[Mu]/2) z^(1 - \[Mu] - \[Nu]) (1 + z)^(\[Mu]/2) ((-Sqrt[-(1/z^2)]) (Gamma[(1 + \[Mu] + \[Nu])/2]/ Gamma[(1/2) (2 - \[Mu] + \[Nu])]) Hypergeometric2F1[(\[Mu] - \[Nu])/2, (1 + \[Mu] + \[Nu])/2, 1/2, z^2] + ((2 Gamma[(2 + \[Mu] + \[Nu])/2])/Gamma[(1/2) (1 - \[Mu] + \[Nu])]) Hypergeometric2F1[(1 + \[Mu] - \[Nu])/2, 1 + (\[Mu] + \[Nu])/2, 3/2, z^2]))/(-(1/z^2))^((1/2) (\[Mu] + \[Nu])) /; !IntervalMemberQ[Interval[{-Infinity, 0}], z] && !IntervalMemberQ[Interval[{1, Infinity}], z]

Standard Form

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MathML Form

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</mo> <mrow> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>

Rule Form

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

2007-05-02

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