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 LegendreP

 http://functions.wolfram.com/05.07.13.0007.01

 Input Form

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

 Standard Form

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

 MathML Form

 z 2 w ′′ ( z ) - z ( 2 s + r ( a 2 z 2 r + 1 ) 1 - a 2 z 2 r - 1 ) w ( z ) + ( - a 2 r 2 ( μ 2 + ( a 2 z 2 r - 1 ) n ( n + 1 ) ) z 2 r ( 1 - a 2 z 2 r ) 2 + s 2 + r s ( a 2 z 2 r + 1 ) 1 - a 2 z 2 r ) w ( z ) 0 /; w ( z ) c 1 z s P TagBox["P", LegendreP] n μ ( a z r TagBox[RowBox[List["a", " ", SuperscriptBox["z", "r"]]], HoldComplete[LegendreP, 2]] ) + c 2 z s Q TagBox["Q", LegendreQ] n μ ( a z r TagBox[RowBox[List["a", " ", SuperscriptBox["z", "r"]]], HoldComplete[LegendreQ, 2]] ) Condition z 2 z 2 w z -1 z 2 s r a 2 z 2 r 1 1 -1 a 2 z 2 r -1 -1 z w z -1 a 2 r 2 μ 2 a 2 z 2 r -1 n n 1 z 2 r 1 -1 a 2 z 2 r 2 -1 s 2 r s a 2 z 2 r 1 1 -1 a 2 z 2 r -1 w z 0 w z Subscript c 1 z s LegendreP n μ 2 a z r Subscript c 2 z s LegendreQ n μ 2 a z r [/itex]

 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[RowBox[List["(", RowBox[List[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_"]]]]]]]], "-", "1", "+", RowBox[List["2", " ", "s_"]]]], ")"]], " ", "z_", " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["s_", "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_"]]]]]]]], "-", FractionBox[RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["r_", "2"], " ", SuperscriptBox["z_", RowBox[List["2", " ", "r_"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Mu]_", "2"], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["z_", RowBox[List["2", " ", "r_"]]]]]]], ")"]], " ", "n_", " ", RowBox[List["(", RowBox[List["1", "+", "n_"]], ")"]]]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["z_", RowBox[List["2", " ", "r_"]]]]]]], ")"]], "2"]]]], ")"]], " ", 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", ",", "\[Mu]", ",", "2", ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", SuperscriptBox["z", "s"], " ", RowBox[List["LegendreQ", "[", RowBox[List["n", ",", "\[Mu]", ",", "2", ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]]]]]]]]]]]]]]

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

 2007-05-02