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variants of this functions
LaguerreL






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/07.03.06.0027.01









  


  










Input Form





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










Standard Form





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MathML Form







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Rule Form





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










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





2007-05-02





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