Whittaker hypergeometric function W: Representations through more general functions (formula 07.45.26.0003)

WhittakerW

$Mathematica Notation$

Hypergeometric Functions

WhittakerW[nu,mu,z]

Representations through more general functions

Through hypergeometric functions

Involving _pF_q

http://functions.wolfram.com/07.45.26.0003.01

Input Form

WhittakerW[\[Nu], \[Mu], -z] WhittakerW[\[Nu], \[Mu], z] == (1/((-z^2)^\[Mu] (2 \[Mu] (4 \[Mu]^2 - 1) Gamma[1/2 - \[Mu] - \[Nu]]^2 Gamma[1/2 + \[Mu] - \[Nu]]^2))) ((-Pi) (-z^2)^(1/2 + \[Mu]) (-1 + 4 \[Mu]^2) Csc[Pi \[Mu]] Gamma[1/2 - \[Mu] - \[Nu]] Gamma[1/2 + \[Mu] - \[Nu]] HypergeometricPFQ[{1/2 - \[Nu], 1/2 + \[Nu]}, {1/2, 1 - \[Mu], 1 + \[Mu]}, z^2/4] + 2 Sqrt[-z^2] \[Mu] (-1 + 4 \[Mu]^2) Gamma[2 \[Mu]]^2 Gamma[1/2 - \[Mu] - \[Nu]]^2 HypergeometricPFQ[{1/2 - \[Mu] - \[Nu], 1/2 - \[Mu] + \[Nu]}, {1 - 2 \[Mu], 1/2 - \[Mu], 1 - \[Mu]}, z^2/4] + 2 (-z^2)^(1/2 + 2 \[Mu]) \[Mu] (-1 + 4 \[Mu]^2) Gamma[-2 \[Mu]]^2 Gamma[1/2 + \[Mu] - \[Nu]]^2 HypergeometricPFQ[{1/2 + \[Mu] - \[Nu], 1/2 + \[Mu] + \[Nu]}, {1/2 + \[Mu], 1 + \[Mu], 1 + 2 \[Mu]}, z^2/4] + 4 Pi (-z)^\[Mu] z^(2 + \[Mu]) \[Mu] \[Nu] Sec[Pi \[Mu]] Gamma[1/2 - \[Mu] - \[Nu]] Gamma[1/2 + \[Mu] - \[Nu]] HypergeometricPFQ[{1 - \[Nu], 1 + \[Nu]}, {3/2, 3/2 - \[Mu], 3/2 + \[Mu]}, z^2/4]) /; !Element[2 \[Mu], Integers]

Standard Form

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MathML Form

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HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <ci> WhittakerW </ci> <ci> ν </ci> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <ci> WhittakerW </ci> <ci> ν </ci> <ci> μ </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> μ 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Rule Form

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

2007-05-02