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 JacobiP

 http://functions.wolfram.com/07.15.06.0033.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiP", "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]]], " ", SuperscriptBox["2", RowBox[List["1", "+", "a"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a"]], "]"]]]], 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["1", "+", "a", "+", "b", "+", "\[Nu]"]], "]"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "a"]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], "]"]]]]]], RowBox[List[" ", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], "]"]]]]]]], " ", SuperscriptBox["z", "\[Nu]"], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]]], " ", SuperscriptBox["2", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["a", "+", "\[Nu]", "+", "1"]], "]"]]]]]], RowBox[List["\[Pi]", " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", "\[Nu]", "-", "1"]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], "!"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "a", "-", "b", "-", "\[Nu]"]]], 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[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", 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["1", "+", "a", "+", "\[Nu]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["1", "+", "a", "+", "\[Nu]"]], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[RowBox[List["b", "+", "\[Nu]"]], "<", "0"]], "&&", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]]]]

 MathML Form

 P ν ( a , b ) ( z ) ( - 1 ) a + b + 2 ν 2 a + 1 sin ( π ν ) Γ ( a + 1 ) z - a - 1 π ( - b - ν ) ! ( a + ν + 1 ) Γ ( a + b + ν + 1 ) ( 1 + O ( 1 z ) ) + ( - 1 ) a + b + 2 ν 2 a + b + ν + 1 sin ( π ν ) Γ ( a + ν + 1 ) π ( - b - ν - 1 ) ! ( a + b + 2 ν + 1 ) ! z - a - b - ν - 1 ( 1 + O ( 1 z ) ) ( 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]] ) + 2 - ν Γ ( a + b + 2 ν + 1 ) z ν Γ ( ν + 1 ) Γ ( a + b + ν + 1 ) ( 1 + O ( 1 z ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) a + b + 2 ν a + ν + 1 b + ν < 0 z { - 1 , 1 } Condition Proportional JacobiP ν a b z -1 a b 2 ν 2 a 1 ν Gamma a 1 z -1 a -1 -1 b -1 ν a ν 1 Gamma a b ν 1 -1 1 O 1 z -1 -1 a b 2 ν 2 a b ν 1 ν Gamma a ν 1 -1 b -1 ν -1 a b 2 ν 1 -1 z -1 a -1 b -1 ν -1 1 O 1 z -1 z -1 2 -1 -1 PolyGamma -1 b -1 ν -1 PolyGamma a b ν 1 PolyGamma a b 2 ν 2 -1 2 -1 ν Gamma a b 2 ν 1 z ν Gamma ν 1 Gamma a b ν 1 -1 1 O 1 z -1 Rule z a b 2 ν a ν 1 b ν 0 z -1 1 [/itex]

 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[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]]], " ", 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["1", "+", "a", "+", "b", "+", "\[Nu]"]], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], "]"]]]], ")"]], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]]], " ", SuperscriptBox["2", RowBox[List["1", "+", "a", "+", "b", "+", "\[Nu]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "\[Nu]", "+", "1"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "a", "-", "b", "-", "\[Nu]"]]], " ", 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[RowBox[List["-", "b"]], "-", "\[Nu]", "-", "1"]], ")"]], "!"]], " ", 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]"]]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], "\[GreaterEqual]", "0"]], "&&", 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