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WhittakerM




Mathematica Notation

Traditional Notation









Hypergeometric Functions > WhittakerM[nu,mu,z] > Identities > Functional identities > Relations between contiguous functions





http://functions.wolfram.com/07.44.17.0013.01









  


  








Input Form





(1 + 2 \[Mu] - 2 \[Nu]) (-1 + 2 z + 2 \[Mu] - 2 \[Nu]) WhittakerM[-(1/2) + \[Nu], 1/2 + \[Mu], z] - 4 Sqrt[z] (1 + 2 \[Mu]) (-1 + z - 2 \[Nu]) WhittakerM[\[Nu], \[Mu], z] - (3 + 4 \[Mu]^2 + 8 \[Nu] + 4 \[Nu]^2 + 8 \[Mu] (1 + \[Nu])) WhittakerM[3/2 + \[Nu], 1/2 + \[Mu], z] == 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "z"]], "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["WhittakerM", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]], ",", "z"]], "]"]]]], "-", RowBox[List["4", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["3", "+", RowBox[List["4", " ", SuperscriptBox["\[Mu]", "2"]]], "+", RowBox[List["8", " ", "\[Nu]"]], "+", RowBox[List["4", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["8", " ", "\[Mu]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]]]]]], ")"]], " ", RowBox[List["WhittakerM", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]], ",", "z"]], "]"]]]]]], "\[Equal]", "0"]]]]








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> <mrow> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <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> <mi> &#956; </mi> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> &#957; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> M </mi> <annotation encoding='Mathematica'> TagBox[&quot;M&quot;, WhittakerM] </annotation> </semantics> <mrow> <mrow> <mi> &#957; </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> M </mi> <annotation encoding='Mathematica'> TagBox[&quot;M&quot;, WhittakerM] </annotation> </semantics> <mrow> <mrow> <mi> &#957; </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#63449; </mo> <mn> 0 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <times /> <cn type='integer'> -4 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <ci> WhittakerM </ci> <ci> &#957; </ci> <ci> &#956; </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <ci> &#956; </ci> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <ci> WhittakerM </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> &#956; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <ci> WhittakerM </ci> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <ci> &#956; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]_"]], "-", RowBox[List["2", " ", "\[Nu]_"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "z_"]], "+", RowBox[List["2", " ", "\[Mu]_"]], "-", RowBox[List["2", " ", "\[Nu]_"]]]], ")"]], " ", RowBox[List["WhittakerM", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]_"]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]_"]], ",", "z_"]], "]"]]]], "-", RowBox[List["4", " ", SqrtBox["z_"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]_"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z_", "-", RowBox[List["2", " ", "\[Nu]_"]]]], ")"]], " ", RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "z_"]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["3", "+", RowBox[List["4", " ", SuperscriptBox["\[Mu]_", "2"]]], "+", RowBox[List["8", " ", "\[Nu]_"]], "+", RowBox[List["4", " ", SuperscriptBox["\[Nu]_", "2"]]], "+", RowBox[List["8", " ", "\[Mu]_", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]_"]], ")"]]]]]], ")"]], " ", RowBox[List["WhittakerM", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "\[Nu]_"]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]_"]], ",", "z_"]], "]"]]]]]], "]"]], "\[RuleDelayed]", "0"]]]]


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





2007-05-02