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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==infinity





http://functions.wolfram.com/07.15.06.0034.01









  


  










Input Form





JacobiP[\[Nu], a, b, z] \[Proportional] (-((2^(1 + a) Sin[Pi \[Nu]] Gamma[1 + a])/(Pi (-b - \[Nu])! (1 + a + \[Nu]) Gamma[-\[Nu]]))) z^(-1 - a) (1 + O[1/z]) - (Sin[Pi \[Nu]]/(2^\[Nu] Pi)) z^\[Nu] (Log[(z - 1)/2] - 2 EulerGamma - PolyGamma[-b - \[Nu]] - PolyGamma[-\[Nu]]) (1 + O[1/z]) /; (Abs[z] -> Infinity) && a + b + 2 \[Nu] == -1 && Element[1 + a + \[Nu], Integers] && 1 + a + \[Nu] >= 0 && b + \[Nu] < 0 && !IntervalMemberQ[Interval[{-1, 1}], z]










Standard Form





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MathML Form







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</mo> <msup> <mi> z </mi> <mi> &#957; </mi> </msup> </mrow> <mi> &#960; </mi> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[EulerGamma]] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mtext> </mtext> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> b </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> &lt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> JacobiP </ci> <ci> &#957; </ci> <ci> a </ci> <ci> b </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <sin /> <apply> <times /> <pi /> <ci> &#957; </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <pi /> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <plus /> <ci> a </ci> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <sin /> <apply> <times /> <pi /> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> &#957; </ci> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ln /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; 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</ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <notin /> <ci> z </ci> <list> <cn type='integer'> -1 </cn> <cn type='integer'> 1 </cn> </list> </apply> </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[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "+", "a"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "a"]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], RowBox[List["\[Pi]", " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", "\[Nu]"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "\[Nu]"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", FractionBox[RowBox[List["z", "-", "1"]], "2"], "]"]], "-", RowBox[List["2", " ", "EulerGamma"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "b"]], "-", "\[Nu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], "\[Pi]"]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List[RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], "\[Equal]", RowBox[List["-", "1"]]]], "&&", RowBox[List[RowBox[List["1", "+", "a", "+", "\[Nu]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["1", "+", "a", "+", "\[Nu]"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["b", "+", "\[Nu]"]], "<", "0"]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]]]










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





2001-10-29





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