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variants of this functions
JacobiP






Mathematica Notation

Traditional Notation









Hypergeometric Functions > JacobiP[nu,a,b,z] > Operations > Orthogonality, completeness, and Fourier expansions





http://functions.wolfram.com/07.15.25.0006.01









  


  










Input Form





Integrate[(Sqrt[(m! (a + b + 2 m + 1) Gamma[a + b + m + 1])/ (2^(a + b + 1) Gamma[a + m + 1] Gamma[b + m + 1])] (1 - t)^(a/2) (1 + t)^(b/2) JacobiP[m, a, b, t]) (Sqrt[(n! (a + b + 2 n + 1) Gamma[a + b + n + 1])/ (2^(a + b + 1) Gamma[a + n + 1] Gamma[b + n + 1])] (1 - t)^(a/2) (1 + t)^(b/2) JacobiP[n, a, b, t]), {t, -1, 1}] == KroneckerDelta[m, n] /; Re[a] > -1 && Re[b] > -1










Standard Form





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MathML Form







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</mo> <mrow> <msubsup> <mi> P </mi> <mi> n </mi> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </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> a </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </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> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> a </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> b </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </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> <ci> t </ci> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> t </ci> </apply> <apply> <times /> <ci> b </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> JacobiP </ci> <ci> m </ci> <ci> a </ci> <ci> b </ci> <ci> t </ci> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> n </ci> </apply> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> a </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> b </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </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> <ci> t </ci> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> t </ci> </apply> <apply> <times /> <ci> b </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> JacobiP </ci> <ci> n </ci> <ci> a </ci> <ci> b </ci> <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> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <gt /> <apply> <real /> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29





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