Whittaker hypergeometric function M: Transformations (formula 07.44.16.0008)

WhittakerM

$Mathematica Notation$

Hypergeometric Functions

WhittakerM[nu,mu,z]

Transformations

Products, sums, and powers of the direct function

Products of the direct function

http://functions.wolfram.com/07.44.16.0008.01

Input Form

WhittakerM[\[Lambda], \[Gamma], d z] WhittakerM[\[Nu], \[Mu], c z] == ((c z)^(1/2 + \[Mu]) (d z)^(1/2 + \[Gamma]) HypergeometricPFQ[ {{}, {1/2 + \[Mu] - \[Nu]}, {1/2 + \[Gamma] - \[Lambda]}}, {{}, {1 + 2 \[Mu]}, {1 + 2 \[Gamma]}}, c z, d z])/E^(((c + d)/2) z)

Standard Form

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MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <semantics> <mi> M </mi> <annotation encoding='Mathematica'> TagBox["M", WhittakerM] </annotation> </semantics> <mrow> <mi> λ </mi> <mo> , </mo> <mi> γ </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> M </mi> <annotation encoding='Mathematica'> TagBox["M", WhittakerM] </annotation> </semantics> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo>  </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> μ </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> γ </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <msubsup> <mi> F </mi> <mrow> <mrow> <mn> 0 </mn> <mo> : </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mn> 1 </mn> </mrow> <mrow> <mrow> <mn> 0 </mn> <mo> : </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mtable> <mtr> <mtd> <mrow> <mrow> <mo> : </mo> <mrow> <mi> μ </mi> <mo> - </mo> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mi> γ </mi> <mo> - </mo> <mi> λ </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo> : </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> γ </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> </mrow> </mtd> </mtr> </mtable> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> , </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <msub> <semantics> <mi> M </mi> <annotation encoding='Mathematica'> TagBox["M", WhittakerM] </annotation> </semantics> <mrow> <mi> λ </mi> <mo> , </mo> <mi> γ </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> M </mi> <annotation encoding='Mathematica'> TagBox["M", WhittakerM] </annotation> </semantics> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo>  </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> μ </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> γ </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <msubsup> <mi> F </mi> <mrow> <mrow> <mn> 0 </mn> <mo> : </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mn> 1 </mn> </mrow> <mrow> <mrow> <mn> 0 </mn> <mo> : </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mtable> <mtr> <mtd> <mrow> <mrow> <mo> : </mo> <mrow> <mi> μ </mi> <mo> - </mo> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mi> γ </mi> <mo> - </mo> <mi> λ </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo> : </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> γ </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> </mrow> </mtd> </mtr> </mtable> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> , </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>

Rule Form

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

2007-05-02