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
GegenbauerC






Mathematica Notation

Traditional Notation









Polynomials > GegenbauerC[n,lambda,z] > Operations > Orthogonality, completeness, and Fourier expansions





http://functions.wolfram.com/05.09.25.0006.01









  


  










Input Form





Integrate[(Sqrt[(m! (\[Lambda] + m) Gamma[\[Lambda]]^2)/ (Pi 2^(1 - 2 \[Lambda]) Gamma[2 \[Lambda] + m])] (1 - t^2)^((2 \[Lambda] - 1)/4) GegenbauerC[m, \[Lambda], t]) (Sqrt[(n! (\[Lambda] + n) Gamma[\[Lambda]]^2)/(Pi 2^(1 - 2 \[Lambda]) Gamma[2 \[Lambda] + n])] (1 - t^2)^((2 \[Lambda] - 1)/4) GegenbauerC[n, \[Lambda], t]), {t, -1, 1}] == KroneckerDelta[m, n] /; Re[\[Lambda]] > -(1/2) && \[Lambda] != 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "1"]], "1"], RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["m", "!"]], " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "m"]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", "\[Lambda]", "]"]], "2"]]], RowBox[List["\[Pi]", " ", SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["2", " ", "\[Lambda]"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "m"]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["t", "2"]]], ")"]], FractionBox[RowBox[List[RowBox[List["2", "\[Lambda]"]], "-", "1"]], "4"]], RowBox[List["GegenbauerC", "[", RowBox[List["m", ",", "\[Lambda]", ",", "t"]], "]"]]]], ")"]], RowBox[List["(", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "n"]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", "\[Lambda]", "]"]], "2"]]], RowBox[List["\[Pi]", " ", SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["2", " ", "\[Lambda]"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "n"]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["t", "2"]]], ")"]], FractionBox[RowBox[List[RowBox[List["2", "\[Lambda]"]], "-", "1"]], "4"]], RowBox[List["GegenbauerC", "[", RowBox[List["n", ",", "\[Lambda]", ",", "t"]], "]"]]]], ")"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List["KroneckerDelta", "[", RowBox[List["m", ",", "n"]], "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Lambda]", "]"]], ">", RowBox[List["-", FractionBox["1", "2"]]]]], "\[And]", RowBox[List["\[Lambda]", "\[NotEqual]", "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> <mo> &#8747; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 1 </mn> </msubsup> <mrow> <mrow> <mo> ( </mo> <mrow> <msqrt> <mfrac> <mrow> <mrow> <mi> m </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> &#955; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#955; </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> t </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <msubsup> <mi> C </mi> <mi> m </mi> <mi> &#955; </mi> </msubsup> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mfrac> <mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> &#955; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#955; </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> t </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <msubsup> <mi> C </mi> <mi> n </mi> <mi> &#955; </mi> </msubsup> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> </msub> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#955; </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#955; </mi> <mo> &#8800; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> -1 </cn> </lowlimit> <uplimit> <cn type='integer'> 1 </cn> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> m </ci> </apply> <apply> <plus /> <ci> m </ci> <ci> &#955; </ci> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <ci> &#955; </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> t </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <ci> m </ci> </apply> <ci> &#955; </ci> </apply> <ci> t </ci> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> n </ci> </apply> <apply> <plus /> <ci> n </ci> <ci> &#955; </ci> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <ci> &#955; </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> t </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <ci> n </ci> </apply> <ci> &#955; </ci> </apply> <ci> t </ci> </apply> </apply> </apply> </apply> <apply> <ci> KroneckerDelta </ci> <ci> m </ci> <ci> n </ci> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <real /> <ci> &#955; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <neq /> <ci> &#955; </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "1"]], "1"], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["m_", "!"]], " ", RowBox[List["(", RowBox[List["\[Lambda]_", "+", "m_"]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", "\[Lambda]_", "]"]], "2"]]], RowBox[List["\[Pi]", " ", SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["2", " ", "\[Lambda]_"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "\[Lambda]_"]], "+", "m_"]], "]"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["t_", "2"]]], ")"]], RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Lambda]_"]], "-", "1"]], ")"]]]]], " ", RowBox[List["GegenbauerC", "[", RowBox[List["m_", ",", "\[Lambda]_", ",", "t_"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["n_", "!"]], " ", RowBox[List["(", RowBox[List["\[Lambda]_", "+", "n_"]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", "\[Lambda]_", "]"]], "2"]]], RowBox[List["\[Pi]", " ", SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["2", " ", "\[Lambda]_"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "\[Lambda]_"]], "+", "n_"]], "]"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["t_", "2"]]], ")"]], RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Lambda]_"]], "-", "1"]], ")"]]]]], " ", RowBox[List["GegenbauerC", "[", RowBox[List["n_", ",", "\[Lambda]_", ",", "t_"]], "]"]]]], ")"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List["m", ",", "n"]], "]"]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Lambda]", "]"]], ">", RowBox[List["-", FractionBox["1", "2"]]]]], "&&", RowBox[List["\[Lambda]", "\[NotEqual]", "0"]]]]]]]]]]










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





2001-10-29





© 1998- Wolfram Research, Inc.