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variants of this functions
LaguerreL




Mathematica Notation

Traditional Notation









Hypergeometric Functions > LaguerreL[nu,z] > Series representations > Asymptotic series expansions





http://functions.wolfram.com/07.02.06.0019.01









  


  








Input Form





LaguerreL[\[Nu], z] \[Proportional] ((-z)^\[Nu]/\[Nu]!) (1 - \[Nu]^2/z + ((1 - \[Nu])^2 \[Nu]^2)/(2 z^2) - \[Ellipsis]) + (1/Gamma[-\[Nu]]) z^(-\[Nu] - 1) E^z (1 + (1 + \[Nu])^2/z + ((1 + \[Nu])^2 (2 + \[Nu])^2)/(2 z^2) + \[Ellipsis]) /; (Abs[z] -> Infinity)








Standard Form





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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> <mi> L </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mrow> <mfrac> <mrow> <msup> <mi> &#8519; </mi> <mi> z </mi> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mi> z </mi> </mfrac> <mo> + </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> &#957; </mi> </msup> <mrow> <mi> &#957; </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> &#957; </mi> <mn> 2 </mn> </msup> <mi> z </mi> </mfrac> <mo> + </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> &#957; </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> - </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> LaguerreL </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <exponentiale /> <ci> z </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <ci> &#957; </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#8230; </ci> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LaguerreL", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "\[Nu]"], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[SuperscriptBox["\[Nu]", "2"], "z"], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], "2"], " ", SuperscriptBox["\[Nu]", "2"]]], RowBox[List["2", " ", SuperscriptBox["z", "2"]]]], "-", "\[Ellipsis]"]], ")"]]]], RowBox[List["\[Nu]", "!"]]], "+", FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", "z"], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], "2"], "z"], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]], "2"]]], RowBox[List["2", " ", SuperscriptBox["z", "2"]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]


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





2007-05-02