Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

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
JacobiP




Mathematica Notation

Traditional Notation









Polynomials > JacobiP[n,a,b,z] > Integration > Definite integration > Involving the direct function





http://functions.wolfram.com/05.06.21.0006.01









  


  








Input Form





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








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "1"]], "1"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "t"]], ")"]], "a"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "t"]], ")"]], "c"], " ", RowBox[List["JacobiP", "[", RowBox[List["n", ",", "a", ",", "b", ",", "t"]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["a", "+", "c", "+", "1"]]], " ", RowBox[List["Gamma", "[", RowBox[List["c", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "n", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["c", "-", "b", "+", "1"]], "]"]]]], RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["c", "-", "b", "-", "n", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "c", "+", "n", "+", "2"]], "]"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "a", "]"]], ">", RowBox[List["-", "1"]]]], "\[And]", RowBox[List[RowBox[List["Re", "[", "b", "]"]], ">", RowBox[List["-", "1"]]]], "\[And]", RowBox[List[RowBox[List["Re", "[", "c", "]"]], ">", RowBox[List["-", "1"]]]]]]]]]]








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> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> t </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> c </mi> </msup> <mo> &#8290; </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> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> a </mi> <mo> + </mo> <mi> c </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> + </mo> <mi> c </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> + </mo> <mi> c </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> c </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </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> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> c </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> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <ci> a </ci> </apply> <apply> <power /> <apply> <plus /> <ci> t </ci> <cn type='integer'> 1 </cn> </apply> <ci> c </ci> </apply> <apply> <ci> JacobiP </ci> <ci> n </ci> <ci> a </ci> <ci> b </ci> <ci> t </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> a </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> c </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 /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> n </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <ci> c </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </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> <gt /> <apply> <real /> <ci> c </ci> </apply> <cn type='integer'> -1 </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[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "t_"]], ")"]], "a_"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "t_"]], ")"]], "c_"], " ", RowBox[List["JacobiP", "[", RowBox[List["n_", ",", "a_", ",", "b_", ",", "t_"]], "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["a", "+", "c", "+", "1"]]], " ", RowBox[List["Gamma", "[", RowBox[List["c", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "n", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["c", "-", "b", "+", "1"]], "]"]]]], RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["c", "-", "b", "-", "n", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "c", "+", "n", "+", "2"]], "]"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "a", "]"]], ">", RowBox[List["-", "1"]]]], "&&", RowBox[List[RowBox[List["Re", "[", "b", "]"]], ">", RowBox[List["-", "1"]]]], "&&", RowBox[List[RowBox[List["Re", "[", "c", "]"]], ">", RowBox[List["-", "1"]]]]]]]]]]]]


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





2002-03-07