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 JacobiP

 http://functions.wolfram.com/07.15.06.0028.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiP", "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["b", "+", "\[Nu]", "+", "1"]], "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "a"]], "-", "\[Nu]"]], "]"]]]]]]], SuperscriptBox["z", RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", "\[Nu]", "-", "1"]]], RowBox[List["(", RowBox[List[RowBox[List["Log", "[", FractionBox[RowBox[List["z", "-", "1"]], "2"], "]"]], "-", "EulerGamma", "-", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "b"]], "-", "\[Nu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["2", "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], "]"]]]], ")"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], ",", "\[Nu]"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " "]]], SuperscriptBox["z", "\[Nu]"], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List[RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List["a", "+", "\[Nu]"]], ",", "Integers"]], "]"]], "]"]]]]]]]]

 MathML Form

 P ν ( a , b ) ( z ) 2 - ν ( a + b + ν + 1 ) ν TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", "b", "+", "\[Nu]", "+", "1"]], ")"]], "\[Nu]"], Pochhammer] z ν Γ ( ν + 1 ) ( 1 + O ( 1 z ) ) - 2 a + b + ν + 1 sin ( ν π ) Γ ( b + ν + 1 ) π ( a + b + 2 ν + 1 ) ! Γ ( - a - ν ) z - a - b - ν - 1 ( log ( z - 1 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( - b - ν ) - ψ TagBox["\[Psi]", PolyGamma] ( a + b + ν + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( a + b + 2 ν + 2 ) - TagBox["\[DoubledGamma]", Function[EulerGamma]] ) ( 1 + O ( 1 z ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) a + b + 2 ν a + ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition Proportional JacobiP ν a b z 2 -1 ν Pochhammer a b ν 1 ν z ν Gamma ν 1 -1 1 O 1 z -1 -1 2 a b ν 1 ν Gamma b ν 1 a b 2 ν 1 Gamma -1 a -1 ν -1 z -1 a -1 b -1 ν -1 z -1 2 -1 -1 PolyGamma -1 b -1 ν -1 PolyGamma a b ν 1 PolyGamma a b 2 ν 2 -1 1 O 1 z -1 Rule z a b 2 ν a ν [/itex]

 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["a", "+", "b", "+", "\[Nu]", "+", "1"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["b", "+", "\[Nu]", "+", "1"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", "\[Nu]", "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", FractionBox[RowBox[List["z", "-", "1"]], "2"], "]"]], "-", "EulerGamma", "-", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "b"]], "-", "\[Nu]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["2", "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], RowBox[List["\[Pi]", " ", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "a"]], "-", "\[Nu]"]], "]"]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], ",", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List[RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], "\[GreaterEqual]", "0"]], "&&", RowBox[List["!", RowBox[List[RowBox[List["a", "+", "\[Nu]"]], "\[Element]", "Integers"]]]]]]]]]]]]

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

 2001-10-29