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variants of this functions
LaguerreL




Mathematica Notation

Traditional Notation









Hypergeometric Functions > LaguerreL[nu,lambda,z] > Series representations > Generalized power series > Expansions at z==0 > For the function itself > General case





http://functions.wolfram.com/07.03.06.0001.02









  


  








Input Form





LaguerreL[\[Nu], \[Lambda], z] \[Proportional] (Gamma[1 + \[Nu] + \[Lambda]]/Gamma[\[Nu] + 1]) (1/Gamma[1 + \[Lambda]] - (\[Nu] z)/Gamma[2 + \[Lambda]] - ((1 - \[Nu]) \[Nu] z^2)/(2 Gamma[3 + \[Lambda]]) - \[Ellipsis]) /; (z -> 0)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LaguerreL", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]", "+", "\[Lambda]"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]], RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Lambda]"]], "]"]]], "-", FractionBox[RowBox[List["\[Nu]", " ", "z"]], RowBox[List["Gamma", "[", RowBox[List["2", "+", "\[Lambda]"]], "]"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], " ", "\[Nu]", " ", SuperscriptBox["z", "2"]]], RowBox[List["2", " ", RowBox[List["Gamma", "[", RowBox[List["3", "+", "\[Lambda]"]], "]"]]]]], "-", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "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> <msubsup> <mi> L </mi> <mi> &#957; </mi> <mi> &#955; </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mfrac> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mi> &#955; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#957; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> - </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 0 </mn> </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> &#955; </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#957; </ci> <ci> &#955; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#955; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#957; </ci> <ci> z </ci> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#955; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <ci> &#957; </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#955; </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#8230; </ci> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LaguerreL", "[", RowBox[List["\[Nu]_", ",", "\[Lambda]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]", "+", "\[Lambda]"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Lambda]"]], "]"]]], "-", FractionBox[RowBox[List["\[Nu]", " ", "z"]], RowBox[List["Gamma", "[", RowBox[List["2", "+", "\[Lambda]"]], "]"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], " ", "\[Nu]", " ", SuperscriptBox["z", "2"]]], RowBox[List["2", " ", RowBox[List["Gamma", "[", RowBox[List["3", "+", "\[Lambda]"]], "]"]]]]], "-", "\[Ellipsis]"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]]]]]]]]


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





2001-10-29