Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











WhittakerW




Mathematica Notation

Traditional Notation









Hypergeometric Functions > WhittakerW[nu,mu,z] > Representations through more general functions > Through Meijer G > Generalized cases involving exp and Laguerre L





http://functions.wolfram.com/07.45.26.0045.01









  


  








Input Form





E^(z/2) LaguerreL[\[Nu] - 1/2, -z] WhittakerW[\[Nu], 0, z] == ((Cos[Pi \[Nu]] Gamma[1/2 + \[Nu]])/(Sqrt[2] Pi^(3/2))) MeijerG[{{3/4 + \[Nu]}, {3/4 - \[Nu]}}, {{1/4, 1/4, 3/4}, {1/4}}, z/2, 1/2]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["z", "/", "2"]]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["\[Nu]", "-", FractionBox["1", "2"]]], ",", RowBox[List["-", "z"]]]], "]"]], " ", RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]", ",", "0", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]], " "]], RowBox[List[SqrtBox["2"], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "4"], "+", "\[Nu]"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "4"], "-", "\[Nu]"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", FractionBox["1", "4"], ",", FractionBox["3", "4"]]], "}"]], ",", RowBox[List["{", FractionBox["1", "4"], "}"]]]], "}"]], ",", FractionBox["z", "2"], ",", FractionBox["1", "2"]]], "]"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> z </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> L </mi> <mrow> <mi> &#957; </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> W </mi> <annotation encoding='Mathematica'> TagBox[&quot;W&quot;, WhittakerW] </annotation> </semantics> <mrow> <mi> &#957; </mi> <mo> , </mo> <mn> 0 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 4 </mn> </mrow> <mrow> <mn> 3 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mi> &#957; </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mi> &#957; </mi> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[&quot;2&quot;, &quot;,&quot;, &quot;4&quot;]], RowBox[List[&quot;3&quot;, &quot;,&quot;, &quot;1&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[RowBox[List[TagBox[FractionBox[&quot;z&quot;, &quot;2&quot;], MeijerG, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;1&quot;, &quot;2&quot;], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]], MeijerG], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[RowBox[List[TagBox[RowBox[List[&quot;\[Nu]&quot;, &quot;+&quot;, FractionBox[&quot;3&quot;, &quot;4&quot;]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;3&quot;, &quot;4&quot;], &quot;-&quot;, &quot;\[Nu]&quot;]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox[FractionBox[&quot;1&quot;, &quot;4&quot;], MeijerG, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;1&quot;, &quot;4&quot;], MeijerG, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;3&quot;, &quot;4&quot;], MeijerG, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;1&quot;, &quot;4&quot;], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> LaguerreL </ci> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <ci> WhittakerW </ci> <ci> &#957; </ci> <cn type='integer'> 0 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <cos /> <apply> <times /> <pi /> <ci> &#957; </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <pi /> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> MeijerG </ci> <list> <list> <apply> <plus /> <ci> &#957; </ci> <cn type='rational'> 3 <sep /> 4 </cn> </apply> </list> <list> <apply> <plus /> <cn type='rational'> 3 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </list> </list> <list> <list> <cn type='rational'> 1 <sep /> 4 </cn> <cn type='rational'> 1 <sep /> 4 </cn> <cn type='rational'> 3 <sep /> 4 </cn> </list> <list> <cn type='rational'> 1 <sep /> 4 </cn> </list> </list> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z_", "2"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["\[Nu]_", "-", FractionBox["1", "2"]]], ",", RowBox[List["-", "z_"]]]], "]"]], " ", RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]_", ",", "0", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "4"], "+", "\[Nu]"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "4"], "-", "\[Nu]"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", FractionBox["1", "4"], ",", FractionBox["3", "4"]]], "}"]], ",", RowBox[List["{", FractionBox["1", "4"], "}"]]]], "}"]], ",", FractionBox["z", "2"], ",", FractionBox["1", "2"]]], "]"]]]], RowBox[List[SqrtBox["2"], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]]]]]]


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





2007-05-02