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WhittakerM






Mathematica Notation

Traditional Notation









Hypergeometric Functions > WhittakerM[nu,mu,z] > Continued fraction representations





http://functions.wolfram.com/07.44.10.0001.01









  


  










Input Form





WhittakerM[\[Nu], \[Mu], z] == (z^(1/2 + \[Mu]) (1 + ((1/2 + \[Mu] - \[Nu]) z)/ ((2 \[Mu] + 1) (1 - ((3/2 + \[Mu] - \[Nu]) z)/(2 (2 \[Mu] + 2))/ (1 + ((3/2 + \[Mu] - \[Nu]) z)/(2 (2 \[Mu] + 2)) - ((5/2 + \[Mu] - \[Nu]) z)/(3 (2 \[Mu] + 3))/ (1 + ((5/2 + \[Mu] - \[Nu]) z)/(3 (2 \[Mu] + 3)) - ((7/2 + \[Mu] - \[Nu]) z)/(4 (2 \[Mu] + 4))/ (1 + ((7/2 + \[Mu] - \[Nu]) z)/(4 (2 \[Mu] + 4)) - ((9/2 + \[Mu] - \[Nu]) z)/(5 (2 \[Mu] + 5))/ (1 + ((9/2 + \[Mu] - \[Nu]) z)/(5 (2 \[Mu] + 5)) + \[Ellipsis]))))))))/E^(z/2)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "z"]], "/", "2"]]], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ")"]], "z"]], RowBox[List[RowBox[List["2", "\[Mu]"]], "+", "1"]]], RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["3", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ")"]], "z"]], RowBox[List["2", RowBox[List["(", RowBox[List[RowBox[List["2", "\[Mu]"]], "+", "2"]], ")"]]]]]]], RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["3", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ")"]], "z"]], RowBox[List["2", RowBox[List["(", RowBox[List[RowBox[List["2", "\[Mu]"]], "+", "2"]], ")"]]]]], "+", FractionBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["5", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ")"]], "z"]], RowBox[List["3", RowBox[List["(", RowBox[List[RowBox[List["2", "\[Mu]"]], "+", "3"]], ")"]]]]]]], RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["5", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ")"]], "z"]], RowBox[List["3", RowBox[List["(", RowBox[List[RowBox[List["2", "\[Mu]"]], "+", "3"]], ")"]]]]], "+", FractionBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["7", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ")"]], "z"]], RowBox[List["4", RowBox[List["(", RowBox[List[RowBox[List["2", "\[Mu]"]], "+", "4"]], ")"]]]]]]], RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["7", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ")"]], "z"]], RowBox[List["4", RowBox[List["(", RowBox[List[RowBox[List["2", "\[Mu]"]], "+", "4"]], ")"]]]]], "+", FractionBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["9", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ")"]], "z"]], RowBox[List["5", RowBox[List["(", RowBox[List[RowBox[List["2", "\[Mu]"]], "+", "5"]], ")"]]]]]]], RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["9", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ")"]], "z"]], RowBox[List["5", RowBox[List["(", RowBox[List[RowBox[List["2", "\[Mu]"]], "+", "5"]], ")"]]]]], "+", "\[Ellipsis]"]]]]]]]]]]], ")"]]]]], ")"]]]]], ")"]]]]]]]]










MathML Form







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</mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 7 </mn> <mn> 2 </mn> </mfrac> <mo> + </mo> <mi> &#956; </mi> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 9 </mn> <mn> 2 </mn> </mfrac> <mo> + </mo> <mi> &#956; </mi> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 9 </mn> <mn> 2 </mn> </mfrac> <mo> + </mo> <mi> &#956; </mi> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mo> &#8230; </mo> </mrow> </mfrac> </mrow> </mfrac> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> WhittakerM </ci> <ci> &#957; </ci> <ci> &#956; </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> &#956; 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</ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <cn type='rational'> 5 <sep /> 2 </cn> <ci> &#956; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <cn type='rational'> 5 <sep /> 2 </cn> <ci> &#956; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <cn type='rational'> 7 <sep /> 2 </cn> <ci> &#956; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <cn type='rational'> 7 <sep /> 2 </cn> <ci> &#956; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <cn type='rational'> 9 <sep /> 2 </cn> <ci> &#956; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> <cn type='integer'> 5 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <cn type='rational'> 9 <sep /> 2 </cn> <ci> &#956; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> <cn type='integer'> 5 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", "2"]]]], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ")"]], " ", "z"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["3", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ")"]], " ", "z"]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "2"]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["3", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ")"]], " ", "z"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "2"]], ")"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["5", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ")"]], " ", "z"]], RowBox[List[RowBox[List["(", RowBox[List["3", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "3"]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["5", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ")"]], " ", "z"]], RowBox[List["3", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "3"]], ")"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["7", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ")"]], " ", "z"]], RowBox[List[RowBox[List["(", RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "4"]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["7", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ")"]], " ", "z"]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "4"]], ")"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["9", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ")"]], " ", "z"]], RowBox[List[RowBox[List["(", RowBox[List["5", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "5"]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["9", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ")"]], " ", "z"]], RowBox[List["5", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "5"]], ")"]]]]], "+", "\[Ellipsis]"]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]]]










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





2007-05-02





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