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 Hypergeometric2F1

 http://functions.wolfram.com/07.23.13.0036.01

 Input Form

 h[z]^2 Derivative[2][w][z] - (2 Derivative[1][h][z] + h[z] (((c - (1 + a + b) g[z]) Derivative[1][g][z])/((-1 + g[z]) g[z]) + Derivative[2][g][z]/Derivative[1][g][z])) h[z] Derivative[1][w][z] + ((a b h[z]^2 Derivative[1][g][z]^2)/((-1 + g[z]) g[z]) + 2 Derivative[1][h][z]^2 + h[z] (((c - (1 + a + b) g[z]) Derivative[1][g][z] Derivative[1][h][z])/ ((-1 + g[z]) g[z]) + (Derivative[1][h][z] Derivative[2][g][z])/ Derivative[1][g][z] - Derivative[2][h][z])) w[z] == 0 /; w[z] == Subscript[c, 1] h[z] Hypergeometric2F1Regularized[a, b, c, g[z]] + Subscript[c, 2] h[z] MeijerG[{{1 - a, 1 - b}, {}}, {{0, 1 - c}, {}}, g[z]]

 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["2", " ", RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["h", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["c", "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a", "+", "b"]], ")"]], " ", RowBox[List["g", "[", "z", "]"]]]]]], ")"]], " ", RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["g", "[", "z", "]"]]]], ")"]], " ", RowBox[List["g", "[", "z", "]"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["g", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]]], ")"]]]]]], ")"]], RowBox[List["h", "[", "z", "]"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", " ", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["a", " ", "b", " ", SuperscriptBox[RowBox[List["h", "[", "z", "]"]], "2"], " ", SuperscriptBox[RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], "2"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["g", "[", "z", "]"]]]], ")"]], " ", RowBox[List["g", "[", "z", "]"]]]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], "2"]]], "+", RowBox[List[RowBox[List["h", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["c", "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a", "+", "b"]], ")"]], " ", RowBox[List["g", "[", "z", "]"]]]]]], ")"]], " ", RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], " ", RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["g", "[", "z", "]"]]]], ")"]], " ", RowBox[List["g", "[", "z", "]"]]]]], "+", FractionBox[RowBox[List[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[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["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", "b", ",", "c", ",", 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", "-", "a"]], ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["1", "-", "c"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["g", "[", "z", "]"]]]], "]"]]]]]]]]]]]]

 MathML Form

 h ( z ) 2 w ′′ ( z ) - h ( z ) ( h ( z ) ( ( c - ( a + b + 1 ) g ( z ) ) g ( z ) ( g ( z ) - 1 ) g ( z ) + g ′′ ( z ) g ( z ) ) + 2 h ( z ) ) w ( z ) + ( a b h ( z ) 2 g ( z ) 2 ( g ( z ) - 1 ) g ( z ) + 2 h ( z ) 2 + h ( z ) ( ( c - ( a + b + 1 ) g ( z ) ) g ( z ) h ( z ) ( g ( z ) - 1 ) g ( z ) + g ′′ ( z ) h ( z ) g ( z ) - h ′′ ( z ) ) ) w ( z ) 0 /; w ( z ) c 1 h ( z ) 2 F ~ 1 ( a , b ; c ; g ( z ) ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox[OverscriptBox["F", "~"], "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox["b", Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["c", Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[RowBox[List["g", "(", "z", ")"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] + c 2 h ( z ) G 2 , 2 2 , 2 ( g ( z ) 1 - a , 1 - b 0 , 1 - c ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "2"]], RowBox[List["2", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["g", "(", "z", ")"]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["1", "-", "a"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "b"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "c"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] Condition h z 2 z 2 w z -1 h z h z c -1 a b 1 g z z g z g z -1 g z -1 z 2 g z z g z -1 2 z h z z w z a b h z 2 z g z 2 g z -1 g z -1 2 z h z 2 h z c -1 a b 1 g z z g z z h z g z -1 g z -1 z 2 g z z h z z g z -1 -1 z 2 h z w z 0 w z Subscript c 1 h z Hypergeometric2F1Regularized a b c g z Subscript c 2 h z MeijerG 1 -1 a 1 -1 b 0 1 -1 c 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["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List[RowBox[List["h", "[", "z_", "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["c_", "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a_", "+", "b_"]], ")"]], " ", RowBox[List["g", "[", "z_", "]"]]]]]], ")"]], " ", RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["g", "[", "z_", "]"]]]], ")"]], " ", RowBox[List["g", "[", "z_", "]"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["g", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]], RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["h", "[", "z_", "]"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["a_", " ", "b_", " ", SuperscriptBox[RowBox[List["h", "[", "z_", "]"]], "2"], " ", SuperscriptBox[RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]], "2"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["g", "[", "z_", "]"]]]], ")"]], " ", RowBox[List["g", "[", "z_", "]"]]]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]], "2"]]], "+", RowBox[List[RowBox[List["h", "[", "z_", "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["c_", "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a_", "+", "b_"]], ")"]], " ", RowBox[List["g", "[", "z_", "]"]]]]]], ")"]], " ", RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]], " ", RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["g", "[", "z_", "]"]]]], ")"]], " ", RowBox[List["g", "[", "z_", "]"]]]]], "+", FractionBox[RowBox[List[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[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["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", "b", ",", "c", ",", 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", "-", "a"]], ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["1", "-", "c"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["g", "[", "z", "]"]]]], "]"]]]]]]]]]]]]]]

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

 2007-05-02