Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
LaguerreL






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LaguerreL[nu,lambda,z] > Complex characteristics > Imaginary part





http://functions.wolfram.com/07.03.19.0002.01









  


  










Input Form





Im[LaguerreL[n, \[Lambda], x + I y]] == Sum[((-1)^(j - 1)/(2 j + 1)!) LaguerreL[n - 2 j - 1, \[Lambda] + 2 j + 1, x] y^(2 j + 1), {j, 0, Floor[(n - 1)/2]}] /; Element[x, Reals] && Element[y, Reals] && Element[\[Lambda], Reals] && Element[n, Integers] && n >= 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Im", "[", RowBox[List["LaguerreL", "[", RowBox[List["n", ",", "\[Lambda]", ",", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]]]], "]"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "/", "2"]], "]"]]], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "j"]], "+", "1"]], ")"]], "!"]]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["n", "-", RowBox[List["2", "j"]], "-", "1"]], ",", RowBox[List["\[Lambda]", "+", RowBox[List["2", "j"]], "+", "1"]], ",", "x"]], "]"]], " ", SuperscriptBox["y", RowBox[List[RowBox[List["2", "j"]], "+", "1"]]]]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["y", "\[Element]", "Reals"]], "\[And]", RowBox[List["\[Lambda]", "\[Element]", "Reals"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "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> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msubsup> <mi> L </mi> <mi> n </mi> <mi> &#955; </mi> </msubsup> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> y </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mi> L </mi> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mi> &#955; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mtext> </mtext> <mo> /; </mo> <mrow> <mrow> <mi> x </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> y </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#955; </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <imaginary /> <apply> <ci> LaguerreL </ci> <ci> n </ci> <ci> &#955; </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> y </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> LaguerreL </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> j </ci> </apply> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <ci> &#955; </ci> <cn type='integer'> 1 </cn> </apply> <ci> x </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <in /> <ci> y </ci> <reals /> </apply> <apply> <in /> <ci> &#955; </ci> <reals /> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Im", "[", RowBox[List["LaguerreL", "[", RowBox[List["n_", ",", "\[Lambda]_", ",", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "-", "1"]]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["n", "-", RowBox[List["2", " ", "j"]], "-", "1"]], ",", RowBox[List["\[Lambda]", "+", RowBox[List["2", " ", "j"]], "+", "1"]], ",", "x"]], "]"]], " ", SuperscriptBox["y", RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["y", "\[Element]", "Reals"]], "&&", RowBox[List["\[Lambda]", "\[Element]", "Reals"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2001-10-29





© 1998-2014 Wolfram Research, Inc.