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; }

 HypergeometricU

 http://functions.wolfram.com/07.33.13.0009.01

 Input Form

 g[z] h[z]^2 Derivative[1][g][z] Derivative[2][w][z] - h[z] (2 g[z] Derivative[1][g][z] Derivative[1][h][z] + h[z] ((-b + g[z]) Derivative[1][g][z]^2 + g[z] Derivative[2][g][z])) Derivative[1][w][z] - (a h[z]^2 Derivative[1][g][z]^3 - 2 g[z] Derivative[1][g][z] Derivative[1][h][z]^2 + h[z] ((b - g[z]) Derivative[1][g][z]^2 Derivative[1][h][z] - g[z] Derivative[1][h][z] Derivative[2][g][z] + g[z] Derivative[1][g][z] Derivative[2][h][z])) w[z] == 0 /; w[z] == Subscript[c, 1] h[z] Hypergeometric1F1Regularized[a, b, g[z]] + Subscript[c, 2] h[z] HypergeometricU[a, b, g[z]]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["g", "[", "z", "]"]], " ", 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["h", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", 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["h", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["g", "[", "z", "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], "2"]]], "+", RowBox[List[RowBox[List["g", "[", "z", "]"]], " ", 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["a", " ", SuperscriptBox[RowBox[List["h", "[", "z", "]"]], "2"], " ", SuperscriptBox[RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], "3"]]], "-", RowBox[List["2", " ", RowBox[List["g", "[", "z", "]"]], " ", 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["(", RowBox[List["b", "-", RowBox[List["g", "[", "z", "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], "2"], " ", RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "-", RowBox[List[RowBox[List["g", "[", "z", "]"]], " ", RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], " ", RowBox[List[SuperscriptBox["g", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["g", "[", "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["Hypergeometric1F1Regularized", "[", RowBox[List["a", ",", "b", ",", RowBox[List["g", "[", "z", "]"]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], RowBox[List["h", "[", "z", "]"]], RowBox[List["HypergeometricU", "[", RowBox[List["a", ",", "b", ",", RowBox[List["g", "[", "z", "]"]]]], "]"]]]]]]]]]]]]

 MathML Form

 g ( z ) g ( z ) h ( z ) 2 w ′′ ( z ) - h ( z ) ( 2 g ( z ) g ( z ) h ( z ) + h ( z ) ( ( g ( z ) - b ) g ( z ) 2 + g ( z ) g ′′ ( z ) ) ) w ( z ) - ( a h ( z ) 2 g ( z ) 3 - 2 g ( z ) h ( z ) 2 g ( z ) + h ( z ) ( ( b - g ( z ) ) h ( z ) g ( z ) 2 + g ( z ) h ′′ ( z ) g ( z ) - g ( z ) h ( z ) g ′′ ( z ) ) ) w ( z ) 0 /; w ( z ) c 1 h ( z ) 1 F ~ 1 ( a ; b ; g ( z ) ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox[OverscriptBox["F", "~"], "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["a", Hypergeometric1F1Regularized, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1Regularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox["b", Hypergeometric1F1Regularized, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1Regularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["g", "(", "z", ")"]], Hypergeometric1F1Regularized, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric1F1Regularized] + c 2 h ( z ) U TagBox["U", HypergeometricU] ( a , b , g ( z ) ) Condition g z z g z h z 2 z 2 w z -1 h z 2 g z z g z z h z h z g z -1 b z g z 2 g z z 2 g z z w z -1 a h z 2 z g z 3 -1 2 g z z h z 2 z g z h z b -1 g z z h z z g z 2 g z z 2 h z z g z -1 g z z h z z 2 g z w z 0 w z Subscript c 1 h z Hypergeometric1F1Regularized a b g z Subscript c 2 h z HypergeometricU a b g z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[RowBox[List["g", "[", "z_", "]"]], " ", 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["h", "[", "z_", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", 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["h", "[", "z_", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b_"]], "+", RowBox[List["g", "[", "z_", "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]], "2"]]], "+", RowBox[List[RowBox[List["g", "[", "z_", "]"]], " ", 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["a_", " ", SuperscriptBox[RowBox[List["h", "[", "z_", "]"]], "2"], " ", SuperscriptBox[RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]], "3"]]], "-", RowBox[List["2", " ", RowBox[List["g", "[", "z_", "]"]], " ", 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["(", RowBox[List["b_", "-", RowBox[List["g", "[", "z_", "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]], "2"], " ", RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "-", RowBox[List[RowBox[List["g", "[", "z_", "]"]], " ", RowBox[List[SuperscriptBox["h", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]], " ", RowBox[List[SuperscriptBox["g", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List[RowBox[List["g", "[", "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["Hypergeometric1F1Regularized", "[", RowBox[List["a", ",", "b", ",", RowBox[List["g", "[", "z", "]"]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", RowBox[List["h", "[", "z", "]"]], " ", RowBox[List["HypergeometricU", "[", RowBox[List["a", ",", "b", ",", RowBox[List["g", "[", "z", "]"]]]], "]"]]]]]]]]]]]]]]

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

 2007-05-02