Assosiated Legendre function of the second kind of type 2: Representations through more general functions (formula 07.11.26.0031)

LegendreQ

$Mathematica Notation$

Hypergeometric Functions

LegendreQ[nu,mu,2,z]

Representations through more general functions

Through hypergeometric functions

Involving ₂F^~₁

http://functions.wolfram.com/07.11.26.0031.01

Input Form

LegendreQ[\[Nu], \[Mu], 2, z] == (1/(z Gamma[-\[Mu] - \[Nu]])) ((2^(-3 - \[Nu]) Sqrt[Pi] (-z^2)^((1/2) (-1 + \[Mu] - \[Nu])) Sec[Pi \[Nu]] (1 - 1/z^2)^\[Mu] Sin[Pi (\[Mu] - \[Nu])] (4^(1 + \[Nu]) z (-z^2)^\[Nu] Gamma[-\[Mu] - \[Nu]] Gamma[\[Mu] - \[Nu]] Hypergeometric2F1Regularized[(\[Mu] - \[Nu])/2, (1/2) (1 + \[Mu] - \[Nu]), 1/2 - \[Nu], 1/z^2] (z Cos[(1/2) Pi (\[Mu] + \[Nu])] - Sqrt[-z^2] Sin[(1/2) Pi (\[Mu] + \[Nu])]) + Pi Hypergeometric2F1Regularized[ (1/2) (1 + \[Mu] + \[Nu]), (1/2) (2 + \[Mu] + \[Nu]), 3/2 + \[Nu], 1/z^2] ((-Sqrt[-z^2]) Cot[(1/2) Pi (\[Mu] + \[Nu])] Csc[(1/2) Pi (\[Mu] - \[Nu])] + z Sec[(1/2) Pi (\[Mu] - \[Nu])] Tan[(1/2) Pi (\[Mu] + \[Nu])])))/(1 - z^2)^(\[Mu]/2)) /; !IntervalMemberQ[Interval[{-1, 1}], z]

Standard Form

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MathML Form

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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,3,z]