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 JacobiP

 http://functions.wolfram.com/05.06.13.0007.01

 Input Form

 h[z]^2 Derivative[2][w][z] + (h[z] (((a - b + (2 + a + b) g[z]) h[z] Derivative[1][g][z])/ (-1 + g[z]^2) - 2 Derivative[1][h][z]) - (h[z]^2 Derivative[2][g][z])/ Derivative[1][g][z]) Derivative[1][w][z] - ((1/(-1 + g[z]^2)) h[z] Derivative[1][g][z] (n (1 + a + b + n) h[z] Derivative[1][g][z] + (a - b + (2 + a + b) g[z]) Derivative[1][h][z]) - 2 Derivative[1][h][z]^2 - (h[z] Derivative[1][h][z] Derivative[2][g][z])/ Derivative[1][g][z] + h[z] Derivative[2][h][z]) w[z] == 0 /; w[z] == Subscript[c, 1] h[z] JacobiP[n, a, b, g[z]] + Subscript[c, 2] h[z] MeijerG[{{1 + n, -a - b - n}, {}}, {{0, -a}, {}}, (1 - g[z])/2]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["h", "[", "z", "]"]], "2"], " ", RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["h", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "a", "+", "b"]], ")"]], " ", RowBox[List["g", "[", "z", "]"]]]]]], ")"]], " ", RowBox[List["h", "[", "z", "]"]], " ", RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox[RowBox[List["g", "[", "z", "]"]], "2"]]]], "-", RowBox[List["2", " ", RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]]]], ")"]]]], "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["h", "[", "z", "]"]], "2"], " ", RowBox[List[SuperscriptBox["g", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]]], ")"]], RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], " ", "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox[RowBox[List["g", "[", "z", "]"]], "2"]]]], RowBox[List["h", "[", "z", "]"]], " ", RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["n", " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "+", "n"]], ")"]], " ", RowBox[List["h", "[", "z", "]"]], " ", RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "a", "+", "b"]], ")"]], " ", RowBox[List["g", "[", "z", "]"]]]]]], ")"]], " ", RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]]]], ")"]]]], "-", RowBox[List["2", " ", SuperscriptBox[RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], "2"]]], "-", FractionBox[RowBox[List[RowBox[List["h", "[", "z", "]"]], " ", RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], " ", RowBox[List[SuperscriptBox["g", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]], "+", RowBox[List[RowBox[List["h", "[", "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["JacobiP", "[", RowBox[List["n", ",", "a", ",", "b", ",", RowBox[List["g", "[", "z", "]"]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], RowBox[List["h", "[", "z", "]"]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", "n"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["-", "a"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", FractionBox[RowBox[List["1", "-", RowBox[List["g", "[", "z", "]"]]]], "2"]]], "]"]]]]]]]]]]]]

 MathML Form

 w ′′ ( z ) h ( z ) 2 + ( h ( z ) ( ( a - b + ( a + b + 2 ) g ( z ) ) h ( z ) g ( z ) g ( z ) 2 - 1 - 2 h ( z ) ) - h ( z ) 2 g ′′ ( z ) g ( z ) ) w ( z ) - ( - 2 h ( z ) 2 - h ( z ) g ′′ ( z ) h ( z ) g ( z ) + 1 g ( z ) 2 - 1 h ( z ) g ( z ) ( n ( a + b + n + 1 ) h ( z ) g ( z ) + ( a - b + ( a + b + 2 ) g ( z ) ) h ( z ) ) + h ( z ) h ′′ ( z ) ) w ( z ) 0 /; w ( z ) c 1 h ( z ) P n ( a , b ) ( g ( z ) ) + c 2 h ( z ) G 2 , 2 2 , 2 ( 1 - g ( z ) 2 n + 1 , - a - b - n 0 , - a ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "2"]], RowBox[List["2", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[FractionBox[RowBox[List["1", "-", RowBox[List["g", "(", "z", ")"]]]], "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["n", "+", "1"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", "n"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["-", "a"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] Condition z 2 w z h z 2 h z a -1 b a b 2 g z h z z g z g z 2 -1 -1 -1 2 z h z -1 h z 2 z 2 g z z g z -1 z w z -1 -2 z h z 2 -1 h z z 2 g z z h z z g z -1 1 g z 2 -1 -1 h z z g z n a b n 1 h z z g z a -1 b a b 2 g z z h z h z z 2 h z w z 0 w z Subscript c 1 h z JacobiP n a b g z Subscript c 2 h z MeijerG n 1 -1 a -1 b -1 n 0 -1 a 1 -1 g z 2 -1 [/itex]

 Rule Form

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

 2007-05-02