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variants of this functions
LegendreP




Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreP[nu,z] > Representations through more general functions > Through hypergeometric functions > Involving 2F1





http://functions.wolfram.com/07.07.26.0002.01









  


  








Input Form





LegendreP[\[Nu], z] == ((2^\[Nu] Gamma[1/2 + \[Nu]])/(Sqrt[Pi] Gamma[1 + \[Nu]])) (z - 1)^\[Nu] Hypergeometric2F1[-\[Nu], -\[Nu], -2 \[Nu], 2/(1 - z)] + ((2^(-1 - \[Nu]) Gamma[-(1/2) - \[Nu]])/(Sqrt[Pi] Gamma[-\[Nu]])) (z - 1)^(-1 - \[Nu]) Hypergeometric2F1[1 + \[Nu], 1 + \[Nu], 2 + 2 \[Nu], 2/(1 - z)] /; !IntervalMemberQ[Interval[{-1, 1}], I z] && !Element[2 \[Nu], Integers]








Standard Form





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








MathML Form







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Rule Form





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


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





2001-10-29