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variants of this functions
JacobiP




Mathematica Notation

Traditional Notation









Hypergeometric Functions > JacobiP[nu,a,b,z] > Series representations > Generalized power series > Expansions at z==1





http://functions.wolfram.com/07.15.06.0008.01









  


  








Input Form





JacobiP[\[Nu], a, b, z] == (Gamma[a + \[Nu] + 1]/Gamma[\[Nu] + 1]) (1/Gamma[1 + a] + ((\[Nu] (1 + a + b + \[Nu]))/(2 Gamma[2 + a])) (z - 1) - (((1 - \[Nu]) \[Nu] (1 + a + b + \[Nu]) (2 + a + b + \[Nu]))/ (8 Gamma[3 + a])) (z - 1)^2 + \[Ellipsis]) /; Abs[(1 - z)/2] < 1








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiP", "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "\[Nu]", "+", "1"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " "]]], RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a"]], "]"]]], "+", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], ")"]]]]]], RowBox[List["2", " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "a"]], "]"]]]]], RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], " ", "\[Nu]", " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "a", "+", "b", "+", "\[Nu]"]], ")"]]]], RowBox[List["8", " ", RowBox[List["Gamma", "[", RowBox[List["3", "+", "a"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "2"]]], " ", "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["1", "-", "z"]], "2"], "]"]], "<", "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> <mi> P </mi> <mi> &#957; </mi> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </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> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </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> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> &#957; </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </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> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> JacobiP </ci> <ci> &#957; </ci> <ci> a </ci> <ci> b </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <ci> &#957; </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> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> &#957; </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </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> <plus /> <ci> a </ci> <ci> b </ci> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> a </ci> <ci> b </ci> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> <apply> <lt /> <apply> <abs /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiP", "[", RowBox[List["\[Nu]_", ",", "a_", ",", "b_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", "+", "\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a"]], "]"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], RowBox[List["2", " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "a"]], "]"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], " ", "\[Nu]", " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "a", "+", "b", "+", "\[Nu]"]], ")"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "2"]]], RowBox[List["8", " ", RowBox[List["Gamma", "[", RowBox[List["3", "+", "a"]], "]"]]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]], "/;", RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["1", "-", "z"]], "2"], "]"]], "<", "1"]]]]]]]]


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





2001-10-29