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WhittakerM






Mathematica Notation

Traditional Notation









Hypergeometric Functions > WhittakerM[nu,mu,z] > Representations through more general functions > Through Meijer G > Generalized cases for products of WhittakerM M





http://functions.wolfram.com/07.44.26.0063.01









  


  










Input Form





WhittakerM[0, \[Gamma], -z] WhittakerM[0, \[Mu], z] == 2^(1/2 + \[Gamma] + 2 \[Mu]) Sqrt[Pi] (-z)^(1/2 + \[Gamma]) Gamma[1 + \[Gamma]] Gamma[1 + \[Mu]] MeijerG[{{3/4 - \[Gamma]/2, 1/4 - \[Gamma]/2}, {3/4 + \[Mu]/2}}, {{1/4 + \[Mu]/2}, {1/4 - \[Gamma] - \[Mu]/2, 1/4 - \[Gamma] + \[Mu]/2, 1/4 - \[Mu]/2, 3/4 + \[Mu]/2}}, z/2, 1/2]










Standard Form





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MathML Form







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</mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#947; </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#947; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 3 </mn> <mo> , </mo> <mn> 5 </mn> </mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 2 </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> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mfrac> <mi> &#947; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mfrac> <mi> &#947; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mi> &#956; </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mfrac> <mi> &#956; </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mi> &#947; </mi> </mrow> <mo> - </mo> <mfrac> <mi> &#956; </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mi> &#947; </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#956; </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mfrac> <mi> &#956; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mi> &#956; </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[&quot;3&quot;, &quot;,&quot;, &quot;5&quot;]], RowBox[List[&quot;1&quot;, &quot;,&quot;, &quot;2&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[FractionBox[&quot;3&quot;, &quot;4&quot;], &quot;-&quot;, FractionBox[&quot;\[Gamma]&quot;, &quot;2&quot;]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;1&quot;, &quot;4&quot;], &quot;-&quot;, FractionBox[&quot;\[Gamma]&quot;, &quot;2&quot;]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;\[Mu]&quot;, &quot;2&quot;], &quot;+&quot;, FractionBox[&quot;3&quot;, &quot;4&quot;]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox[RowBox[List[FractionBox[&quot;\[Mu]&quot;, &quot;2&quot;], &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;4&quot;]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[RowBox[List[&quot;-&quot;, &quot;\[Gamma]&quot;]], &quot;-&quot;, FractionBox[&quot;\[Mu]&quot;, &quot;2&quot;], &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;4&quot;]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[RowBox[List[&quot;-&quot;, &quot;\[Gamma]&quot;]], &quot;+&quot;, FractionBox[&quot;\[Mu]&quot;, &quot;2&quot;], &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;4&quot;]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;1&quot;, &quot;4&quot;], &quot;-&quot;, FractionBox[&quot;\[Mu]&quot;, &quot;2&quot;]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;\[Mu]&quot;, &quot;2&quot;], &quot;+&quot;, FractionBox[&quot;3&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> <ci> WhittakerM </ci> <cn type='integer'> 0 </cn> <ci> &#947; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <ci> WhittakerM </ci> <cn type='integer'> 0 </cn> <ci> &#956; </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> &#947; </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <plus /> <ci> &#947; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#947; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> MeijerG </ci> <list> <list> <apply> <plus /> <cn type='rational'> 3 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#947; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#947; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </list> <list> <apply> <plus /> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 3 <sep /> 4 </cn> </apply> </list> </list> <list> <list> <apply> <plus /> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </list> <list> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#947; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#947; </ci> </apply> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <plus /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 3 <sep /> 4 </cn> </apply> </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[RowBox[List["WhittakerM", "[", RowBox[List["0", ",", "\[Gamma]_", ",", RowBox[List["-", "z_"]]]], "]"]], " ", RowBox[List["WhittakerM", "[", RowBox[List["0", ",", "\[Mu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "+", "\[Gamma]", "+", RowBox[List["2", " ", "\[Mu]"]]]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "\[Gamma]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Gamma]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]"]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Gamma]", "2"]]], ",", RowBox[List[FractionBox["1", "4"], "-", FractionBox["\[Gamma]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Mu]", "2"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], "+", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], "-", "\[Gamma]", "-", FractionBox["\[Mu]", "2"]]], ",", RowBox[List[FractionBox["1", "4"], "-", "\[Gamma]", "+", FractionBox["\[Mu]", "2"]]], ",", RowBox[List[FractionBox["1", "4"], "-", FractionBox["\[Mu]", "2"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Mu]", "2"]]]]], "}"]]]], "}"]], ",", FractionBox["z", "2"], ",", FractionBox["1", "2"]]], "]"]]]]]]]]










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





2007-05-02





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