html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 LegendreP

 http://functions.wolfram.com/05.07.13.0005.01

 Input Form

 h[z]^2 Derivative[1][g][z] Derivative[2][w][z] - (2 h[z] Derivative[1][g][z] Derivative[1][h][z] + h[z]^2 ((2 g[z] Derivative[1][g][z]^2)/(1 - g[z]^2) + Derivative[2][g][z])) Derivative[1][w][z] + (-(((\[Mu]^2 - n (1 + n) (1 - g[z]^2)) h[z]^2 Derivative[1][g][z]^3)/ (1 - g[z]^2)^2) + 2 Derivative[1][g][z] Derivative[1][h][z]^2 + h[z] (Derivative[1][h][z] ((2 g[z] Derivative[1][g][z]^2)/(1 - g[z]^2) + Derivative[2][g][z]) - Derivative[1][g][z] Derivative[2][h][z])) w[z] == 0 /; w[z] == Subscript[c, 1] h[z] LegendreP[n, \[Mu], 2, g[z]] + Subscript[c, 2] h[z] LegendreQ[n, \[Mu], 2, g[z]]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["h", "[", "z", "]"]], "2"], " ", RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["h", "[", "z", "]"]], " ", RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], " ", RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["h", "[", "z", "]"]], "2"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["g", "[", "z", "]"]], " ", SuperscriptBox[RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], "2"]]], RowBox[List["1", "-", SuperscriptBox[RowBox[List["g", "[", "z", "]"]], "2"]]]], "+", " ", RowBox[List[SuperscriptBox["g", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], ")"]]]]]], ")"]], RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], " ", "+", " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[Mu]", "2"], "-", RowBox[List["n", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["g", "[", "z", "]"]], "2"]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["h", "[", "z", "]"]], "2"], " ", SuperscriptBox[RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], "3"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["g", "[", "z", "]"]], "2"]]], ")"]], "2"]]]], "+", RowBox[List["2", " ", RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], " ", SuperscriptBox[RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], "2"]]], "+", " ", RowBox[List[RowBox[List["h", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["g", "[", "z", "]"]], " ", SuperscriptBox[RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], "2"]]], RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["g", "[", "z", "]"]], "2"]]], ")"]]], "+", RowBox[List[SuperscriptBox["g", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], ")"]]]], "-", RowBox[List[RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], " ", RowBox[List[SuperscriptBox["h", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]]]], ")"]]]]]], ")"]], RowBox[List["w", "[", "z", "]"]]]]]], "\[Equal]", "0"]], "/;", " ", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], RowBox[List["h", "[", "z", "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List["n", ",", "\[Mu]", ",", "2", ",", RowBox[List["g", "[", "z", "]"]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], RowBox[List["h", "[", "z", "]"]], " ", RowBox[List["LegendreQ", "[", RowBox[List["n", ",", "\[Mu]", ",", "2", ",", RowBox[List["g", "[", "z", "]"]]]], "]"]]]]]]]]]]]]

 MathML Form

 g ( z ) h ( z ) 2 w ′′ ( z ) - ( ( 2 g ( z ) g ( z ) 2 1 - g ( z ) 2 + g ′′ ( z ) ) h ( z ) 2 + 2 g ( z ) h ( z ) h ( z ) ) w ( z ) + ( - μ 2 - n ( n + 1 ) ( 1 - g ( z ) 2 ) ( 1 - g ( z ) 2 ) 2 h ( z ) 2 g ( z ) 3 + 2 h ( z ) 2 g ( z ) + h ( z ) ( h ( z ) ( 2 g ( z ) g ( z ) 2 1 - g ( z ) 2 + g ′′ ( z ) ) - g ( z ) h ′′ ( z ) ) ) w ( z ) 0 /; w ( z ) c 1 h ( z ) P TagBox["P", LegendreP] n μ ( g ( z ) TagBox[RowBox[List["g", "(", "z", ")"]], HoldComplete[LegendreP, 2]] ) + c 2 h ( z ) Q TagBox["Q", LegendreQ] n μ ( g ( z ) TagBox[RowBox[List["g", "(", "z", ")"]], HoldComplete[LegendreQ, 2]] ) Condition z g z h z 2 z 2 w z -1 2 g z z g z 2 1 -1 g z 2 -1 z 2 g z h z 2 2 z g z z h z h z z w z -1 μ 2 -1 n n 1 1 -1 g z 2 1 -1 g z 2 2 -1 h z 2 z g z 3 2 z h z 2 z g z h z z h z 2 g z z g z 2 1 -1 g z 2 -1 z 2 g z -1 z g z z 2 h z w z 0 w z Subscript c 1 h z LegendreP n μ 2 g z Subscript c 2 h z LegendreQ n μ 2 g z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["h", "[", "z_", "]"]], "2"], " ", RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["h", "[", "z_", "]"]], " ", RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]], " ", RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["h", "[", "z_", "]"]], "2"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["g", "[", "z_", "]"]], " ", SuperscriptBox[RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]], "2"]]], RowBox[List["1", "-", SuperscriptBox[RowBox[List["g", "[", "z_", "]"]], "2"]]]], "+", RowBox[List[SuperscriptBox["g", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], ")"]]]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[Mu]_", "2"], "-", RowBox[List["n_", " ", RowBox[List["(", RowBox[List["1", "+", "n_"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["g", "[", "z_", "]"]], "2"]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["h", "[", "z_", "]"]], "2"], " ", SuperscriptBox[RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]], "3"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["g", "[", "z_", "]"]], "2"]]], ")"]], "2"]]]], "+", RowBox[List["2", " ", RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]], " ", SuperscriptBox[RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]], "2"]]], "+", RowBox[List[RowBox[List["h", "[", "z_", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["g", "[", "z_", "]"]], " ", SuperscriptBox[RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]], "2"]]], RowBox[List["1", "-", SuperscriptBox[RowBox[List["g", "[", "z_", "]"]], "2"]]]], "+", RowBox[List[SuperscriptBox["g", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], ")"]]]], "-", RowBox[List[RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]], " ", RowBox[List[SuperscriptBox["h", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["w", "[", "z_", "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], " ", RowBox[List["h", "[", "z", "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List["n", ",", "\[Mu]", ",", "2", ",", RowBox[List["g", "[", "z", "]"]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", RowBox[List["h", "[", "z", "]"]], " ", RowBox[List["LegendreQ", "[", RowBox[List["n", ",", "\[Mu]", ",", "2", ",", RowBox[List["g", "[", "z", "]"]]]], "]"]]]]]]]]]]]]]]

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

 2007-05-02