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variants of this functions
LegendreP




Mathematica Notation

Traditional Notation









Polynomials > LegendreP[n,mu,2,z] > Representations through more general functions > Through Meijer G > Classical cases involving unit step theta





http://functions.wolfram.com/05.07.26.0017.01









  


  








Input Form





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








Standard Form





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








MathML Form







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Rule Form





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


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





2001-10-29