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variants of this functions
LegendreP






Mathematica Notation

Traditional Notation









Polynomials > LegendreP[n,z] > Differential equations > Ordinary linear differential equations and wronskians > For the direct function itself





http://functions.wolfram.com/05.03.13.0007.01









  


  










Input Form





z^2 Derivative[2][w][z] - z (-1 + 2 s + (r (1 + a^2 z^(2 r)))/ (1 - a^2 z^(2 r))) Derivative[1][w][z] + (s^2 - (a^2 n (1 + n) r^2 z^(2 r) (-1 + a^2 z^(2 r)))/ (1 - a^2 z^(2 r))^2 + (r s (1 + a^2 z^(2 r)))/(1 - a^2 z^(2 r))) w[z] == 0 /; w[z] == Subscript[c, 1] z^s LegendreP[n, a z^r] + Subscript[c, 2] z^s LegendreQ[n, a z^r]










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[SuperscriptBox["z_", "2"], " ", RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "-", RowBox[List["z_", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "s_"]], "+", FractionBox[RowBox[List["r_", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["z_", RowBox[List["2", " ", "r_"]]]]]]], ")"]]]], RowBox[List["1", "-", RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["z_", RowBox[List["2", " ", "r_"]]]]]]]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["s_", "2"], "-", FractionBox[RowBox[List[SuperscriptBox["a_", "2"], " ", "n_", " ", RowBox[List["(", RowBox[List["1", "+", "n_"]], ")"]], " ", SuperscriptBox["r_", "2"], " ", SuperscriptBox["z_", RowBox[List["2", " ", "r_"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["z_", RowBox[List["2", " ", "r_"]]]]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["z_", RowBox[List["2", " ", "r_"]]]]]]], ")"]], "2"]], "+", FractionBox[RowBox[List["r_", " ", "s_", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["z_", RowBox[List["2", " ", "r_"]]]]]]], ")"]]]], RowBox[List["1", "-", RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["z_", RowBox[List["2", " ", "r_"]]]]]]]]]], ")"]], " ", RowBox[List["w", "[", "z_", "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], " ", SuperscriptBox["z", "s"], " ", RowBox[List["LegendreP", "[", RowBox[List["n", ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", SuperscriptBox["z", "s"], " ", RowBox[List["LegendreQ", "[", RowBox[List["n", ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]]]]]]]]]]]]]]










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





2007-05-02





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