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 GegenbauerC

 http://functions.wolfram.com/07.14.13.0009.01

 Input Form

 Derivative[2][w][z] - (Log[r] + (a^2 r^(2 z) (1 + 2 \[Lambda]) Log[r])/ (1 - a^2 r^(2 z)) + 2 Log[s]) Derivative[1][w][z] + ((1/(-1 + a^2 r^(2 z))) ((-Log[s]) (Log[r] + Log[s]) + a^2 r^(2 z) (\[Nu] Log[r] + Log[s]) ((-(2 \[Lambda] + \[Nu])) Log[r] + Log[s]))) w[z] == 0 /; w[z] == Subscript[c, 1] s^z GegenbauerC[\[Nu], \[Lambda], a r^z] + Subscript[c, 2] s^z (1 - a^2 r^(2 z))^((1 - 2 \[Lambda])/4) LegendreQ[\[Nu] + \[Lambda] - 1/2, 1/2 - \[Lambda], 2, a r^z]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "r", "]"]], "+", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Lambda]"]]]], ")"]], " ", RowBox[List["Log", "[", "r", "]"]]]], RowBox[List["1", "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]]]]]]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", "s", "]"]]]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[RowBox[List[FractionBox["1", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]]]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "s", "]"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "r", "]"]], "+", RowBox[List["Log", "[", "s", "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Nu]", " ", RowBox[List["Log", "[", "r", "]"]]]], "+", RowBox[List["Log", "[", "s", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], ")"]]]], " ", RowBox[List["Log", "[", "r", "]"]]]], "+", RowBox[List["Log", "[", "s", "]"]]]], ")"]]]]]], ")"]]]], " ", RowBox[List["w", "[", "z", "]"]]]]]], "\[Equal]", "0"]], "/;", " ", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], SuperscriptBox["s", "z"], RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", RowBox[List["a", " ", SuperscriptBox["r", "z"]]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], SuperscriptBox["s", "z"], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List[SuperscriptBox["a", "2"], SuperscriptBox["r", RowBox[List["2", "z"]]]]]]], ")"]], FractionBox[RowBox[List["1", "-", RowBox[List["2", "\[Lambda]"]]]], "4"]], " ", RowBox[List["LegendreQ", "[", RowBox[List[RowBox[List["\[Nu]", "+", "\[Lambda]", "-", FractionBox["1", "2"]]], ",", RowBox[List[FractionBox["1", "2"], " ", "-", "\[Lambda]"]], ",", "2", ",", RowBox[List["a", " ", SuperscriptBox["r", "z"]]]]], "]"]]]]]]]]]]]]

 MathML Form

 w ′′ ( z ) - ( a 2 ( 2 λ + 1 ) log ( r ) r 2 z 1 - a 2 r 2 z + log ( r ) + 2 log ( s ) ) w ( z ) + a 2 r 2 z ( ν log ( r ) + log ( s ) ) ( log ( s ) - ( 2 λ + ν ) log ( r ) ) - log ( s ) ( log ( r ) + log ( s ) ) a 2 r 2 z - 1 w ( z ) 0 /; w ( z ) c 1 s z C ν λ ( a r z ) + c 2 s z ( 1 - a 2 r 2 z ) 1 - 2 λ 4 Q TagBox["Q", LegendreQ] λ + ν - 1 2 1 2 - λ ( a r z TagBox[RowBox[List["a", " ", SuperscriptBox["r", "z"]]], HoldComplete[LegendreQ, 2]] ) Condition z 2 w z -1 a 2 2 λ 1 r r 2 z 1 -1 a 2 r 2 z -1 r 2 s z w z a 2 r 2 z ν r s s -1 2 λ ν r -1 s r s a 2 r 2 z -1 -1 w z 0 w z Subscript c 1 s z Subscript C ν λ a r z Subscript c 2 s z 1 -1 a 2 r 2 z 1 -1 2 λ 4 -1 LegendreQ λ ν -1 1 2 1 2 -1 λ 2 a r z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "r_", "]"]], "+", FractionBox[RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["r_", RowBox[List["2", " ", "z_"]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Lambda]_"]]]], ")"]], " ", RowBox[List["Log", "[", "r_", "]"]]]], RowBox[List["1", "-", RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["r_", RowBox[List["2", " ", "z_"]]]]]]]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", "s_", "]"]]]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "s_", "]"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "r_", "]"]], "+", RowBox[List["Log", "[", "s_", "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["r_", RowBox[List["2", " ", "z_"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Nu]_", " ", RowBox[List["Log", "[", "r_", "]"]]]], "+", RowBox[List["Log", "[", "s_", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Lambda]_"]], "+", "\[Nu]_"]], ")"]]]], " ", RowBox[List["Log", "[", "r_", "]"]]]], "+", RowBox[List["Log", "[", "s_", "]"]]]], ")"]]]]]], ")"]], " ", RowBox[List["w", "[", "z_", "]"]]]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["r_", RowBox[List["2", " ", "z_"]]]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], " ", SuperscriptBox["s", "z"], " ", RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", RowBox[List["a", " ", SuperscriptBox["r", "z"]]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", SuperscriptBox["s", "z"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]]]]]], ")"]], RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[Lambda]"]]]], ")"]]]]], " ", RowBox[List["LegendreQ", "[", RowBox[List[RowBox[List["\[Nu]", "+", "\[Lambda]", "-", FractionBox["1", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]], ",", "2", ",", RowBox[List["a", " ", SuperscriptBox["r", "z"]]]]], "]"]]]]]]]]]]]]]]

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

 2007-05-02