Whittaker hypergeometric function W: Representations through equivalent functions (formula 07.45.27.0005)

WhittakerW

$Mathematica Notation$

Hypergeometric Functions

WhittakerW[nu,mu,z]

Representations through equivalent functions

With related functions

http://functions.wolfram.com/07.45.27.0005.01

Input Form

WhittakerW[\[Nu], \[Mu], z] == (Gamma[-2 \[Mu]]/Gamma[1/2 - \[Mu] - \[Nu]]) WhittakerM[\[Nu], \[Mu], z] + (Gamma[2 \[Mu]]/Gamma[1/2 + \[Mu] - \[Nu]]) WhittakerM[\[Nu], -\[Mu], z] /; !Element[2 \[Mu], Integers]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "]"]]], RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["Gamma", "[", RowBox[List["2", " ", "\[Mu]"]], "]"]], " "]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], "]"]]], RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]", ",", RowBox[List["-", "\[Mu]"]], ",", "z"]], "]"]]]]]]]], "/;", " ", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List["2", "\[Mu]"]], ",", "Integers"]], "]"]], "]"]]]]]]

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> W </mi> <annotation encoding='Mathematica'> TagBox["W", WhittakerW] </annotation> </semantics> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mrow> <mfrac> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> - </mo> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> M </mi> <annotation encoding='Mathematica'> TagBox["M", WhittakerM] </annotation> </semantics> <mrow> <mi> ν </mi> <mo> , </mo> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> <mo> - </mo> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mfrac> <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> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <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[List[], Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> WhittakerW </ci> <ci> ν </ci> <ci> μ </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> μ </ci> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> WhittakerM </ci> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -2 </cn> <ci> μ </ci> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> WhittakerM </ci> <ci> ν </ci> <ci> μ </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <not /> <apply> <in /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> μ </ci> </apply> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]], "]"]], " ", RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "]"]]], "+", FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["2", " ", "\[Mu]"]], "]"]], " ", RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]", ",", RowBox[List["-", "\[Mu]"]], ",", "z"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], "]"]]]]], "/;", RowBox[List["!", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "\[Element]", "Integers"]]]]]]]]]]

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

2007-05-02