Legendre function: Representations through more general functions (formula 07.07.26.0043)

LegendreP

$Mathematica Notation$

Hypergeometric Functions

LegendreP[nu,z]

Representations through more general functions

Through hypergeometric functions

Involving ₂F₁

http://functions.wolfram.com/07.07.26.0043.01

Input Form

LegendreP[\[Nu], z] == (Sin[Pi \[Nu]]/(2^\[Nu] ((1 + 2 \[Nu]) Pi^(3/2)))) (((1/z) Gamma[1 + \[Nu]] Gamma[1/2 - \[Nu]] (Cos[(Pi \[Nu])/2] + (Sqrt[-z^2] Sin[(Pi \[Nu])/2])/z) Hypergeometric2F1[(1 + \[Nu])/2, (2 + \[Nu])/2, 3/2 + \[Nu], 1/z^2])/(-z^2)^(\[Nu]/2) - 2^(1 + 2 \[Nu]) (-z^2)^(\[Nu]/2) Gamma[-\[Nu]] Gamma[3/2 + \[Nu]] (Cos[(Pi \[Nu])/2] - (Sqrt[-z^2] Sin[(Pi \[Nu])/2])/z) Hypergeometric2F1[-(\[Nu]/2), (1 - \[Nu])/2, 1/2 - \[Nu], 1/z^2]) /; !IntervalMemberQ[Interval[{-1, 0}], z]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", "\[Nu]"]]]], ")"]], SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "z"], SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "/", "2"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "+", FractionBox[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]]]], "z"]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"], ",", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["\[Nu]", "/", "2"]]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "-", FractionBox[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]]]], "z"]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]]

MathML Form

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</mo> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", FractionBox["\[Nu]", "2"]]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "+", FractionBox[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]]]], "z"]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"], ",", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]], "z"], "-", RowBox[List[SuperscriptBox["2", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["\[Nu]", "/", "2"]]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "-", FractionBox[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]]]], "z"]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]

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

2007-05-02

LegendreP[n,z]
LegendreP[n,mu,2,z]
LegendreP[nu,mu,z]
LegendreP[nu,mu,2,z]
LegendreP[nu,mu,3,z]