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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/07.15.06.0031.01









  


  








Input Form





JacobiP[\[Nu], a, b, z] \[Proportional] (Pochhammer[a + b + \[Nu] + 1, \[Nu]]/(2^\[Nu] Gamma[\[Nu] + 1])) z^\[Nu] (1 + O[1/z]) + (((-1)^(a + \[Nu]) 2^(1 + a + b + \[Nu]) Sin[Pi \[Nu]] (b + \[Nu])! (a + \[Nu])!)/(Pi (1 + a + b + 2 \[Nu])!)) z^(-1 - a - b - \[Nu]) (1 + O[1/z]) /; (Abs[z] -> Infinity) && Element[a + b + 2 \[Nu] + 1, Integers] && a + b + 2 \[Nu] + 1 >= 0 && Element[a + \[Nu], Integers] && a + \[Nu] >= 0 && b + \[Nu] >= 0








Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiP", "[", RowBox[List["\[Nu]_", ",", "a_", ",", "b_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], ",", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["a", "+", "\[Nu]"]]], " ", SuperscriptBox["2", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["b", "+", "\[Nu]"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "\[Nu]"]], ")"]], "!"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "a", "-", "b", "-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], RowBox[List["\[Pi]", " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], "!"]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List[RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["a", "+", "\[Nu]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["a", "+", "\[Nu]"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["b", "+", "\[Nu]"]], "\[GreaterEqual]", "0"]]]]]]]]]]


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





2001-10-29