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variants of this functions
LegendreP






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreP[nu,mu,2,z] > Representations through more general functions > Through Meijer G > Classical cases involving sgn





http://functions.wolfram.com/07.08.26.0023.01









  


  










Input Form





(1/(((1 - z) Sign[1 - Abs[z]])^\[Mu] Sqrt[1 + z])) LegendreP[-(1/4), \[Mu], 2, (1 - 6 z + z^2)/(1 + z)^2] == (Gamma[1/2 - \[Mu]]/(2^\[Mu] Sqrt[Pi])) MeijerG[{{(1 - \[Mu])/2}, {(1 - 3 \[Mu])/2}}, {{-(\[Mu]/2)}, {\[Mu]/2}}, z] /; !IntervalMemberQ[Interval[{-Infinity, 0}], z]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], " ", RowBox[List["Sign", "[", RowBox[List["1", "-", RowBox[List["Abs", "[", "z", "]"]]]], "]"]]]], ")"]], RowBox[List["-", "\[Mu]"]]], SqrtBox[RowBox[List["1", "+", "z"]]]], RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], ",", "\[Mu]", ",", "2", ",", FractionBox[RowBox[List["1", "-", RowBox[List["6", " ", "z"]], "+", SuperscriptBox["z", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], "2"]]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]"]], "]"]], RowBox[List[SuperscriptBox["2", "\[Mu]"], " ", SqrtBox["\[Pi]"]]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox[RowBox[List["1", "-", "\[Mu]"]], "2"], "}"]], ",", RowBox[List["{", FractionBox[RowBox[List["1", "-", RowBox[List["3", "\[Mu]"]]]], "2"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["-", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", FractionBox["\[Mu]", "2"], "}"]]]], "}"]], ",", "z"]], "]"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]]










MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "z_"]], ")"]], " ", RowBox[List["Sign", "[", RowBox[List["1", "-", RowBox[List["Abs", "[", "z_", "]"]]]], "]"]]]], ")"]], RowBox[List["-", "\[Mu]_"]]], " ", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], ",", "\[Mu]_", ",", "2", ",", FractionBox[RowBox[List["1", "-", RowBox[List["6", " ", "z_"]], "+", SuperscriptBox["z_", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z_"]], ")"]], "2"]]]], "]"]]]], SqrtBox[RowBox[List["1", "+", "z_"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]"]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox[RowBox[List["1", "-", "\[Mu]"]], "2"], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", "\[Mu]"]]]], ")"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["-", FractionBox["\[Mu]", "2"]]], "}"]], ",", RowBox[List["{", FractionBox["\[Mu]", "2"], "}"]]]], "}"]], ",", "z"]], "]"]]]], RowBox[List[SuperscriptBox["2", "\[Mu]"], " ", SqrtBox["\[Pi]"]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]










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





2001-10-29





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