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WhittakerM




Mathematica Notation

Traditional Notation









Hypergeometric Functions > WhittakerM[nu,mu,z] > Specific values > Specialized values > For fixed nu, z





http://functions.wolfram.com/07.44.03.0005.01









  


  








Input Form





WhittakerM[\[Nu], -(1/4), z] - ((2 Sqrt[-z] Gamma[3/4 + \[Nu]])/ (Sqrt[z] Gamma[1/4 + \[Nu]])) WhittakerM[\[Nu], 1/4, z] == ((2^(1/2 + 2 \[Nu]) E^(z/2) z^(1/4) Gamma[3/4 + \[Nu]])/Sqrt[Pi]) HermiteH[-(1/2) - 2 \[Nu], Sqrt[-z]]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]", ",", RowBox[List["-", FractionBox["1", "4"]]], ",", "z"]], "]"]], "-", RowBox[List[FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "4"], "+", "\[Nu]"]], "]"]], " "]], RowBox[List[SqrtBox["z"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], "+", "\[Nu]"]], "]"]]]]], RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]", ",", FractionBox["1", "4"], ",", "z"]], "]"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["2", " ", "\[Nu]"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["z", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "4"], "+", "\[Nu]"]], "]"]], " "]], SqrtBox["\[Pi]"]], RowBox[List["HermiteH", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ",", SqrtBox[RowBox[List["-", "z"]]]]], "]"]]]]]]]]








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[&quot;M&quot;, WhittakerM] </annotation> </semantics> <mrow> <mi> &#957; </mi> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> M </mi> <annotation encoding='Mathematica'> TagBox[&quot;M&quot;, WhittakerM] </annotation> </semantics> <mrow> <mi> &#957; </mi> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> z </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <mroot> <mi> z </mi> <mn> 4 </mn> </mroot> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <msqrt> <mi> &#960; </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mi> H </mi> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <ci> WhittakerM </ci> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='rational'> 3 <sep /> 4 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> WhittakerM </ci> <ci> &#957; </ci> <cn type='rational'> 1 <sep /> 4 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='rational'> 3 <sep /> 4 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HermiteH </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]_", ",", RowBox[List["-", FractionBox["1", "4"]]], ",", "z_"]], "]"]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z_"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "4"], "+", "\[Nu]_"]], "]"]]]], ")"]], " ", RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]_", ",", FractionBox["1", "4"], ",", "z_"]], "]"]]]], RowBox[List[SqrtBox["z_"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], "+", "\[Nu]_"]], "]"]]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["2", " ", "\[Nu]"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["z", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "4"], "+", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["HermiteH", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ",", SqrtBox[RowBox[List["-", "z"]]]]], "]"]]]], SqrtBox["\[Pi]"]]]]]]


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





2007-05-02