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; }

 HypergeometricPFQ

 http://functions.wolfram.com/07.22.13.0011.01

 Input Form

 z^3 Derivative[3][w][z] + (3 - 2 r - 3 s + r (Subscript[b, 1] + Subscript[b, 2])) z^2 Derivative[2][w][z] + (1 + 3 (-1 + s) s + r (-2 + 4 s) + r^2 (1 - a z^r) - r (-1 + r + 2 s) Subscript[b, 2] + r Subscript[b, 1] (1 - r - 2 s + r Subscript[b, 2])) z Derivative[1][w][z] + ((-a) r^2 z^r (-s + r Subscript[a, 1]) - s (r + s - r Subscript[b, 1]) (r + s - r Subscript[b, 2])) w[z] == 0 /; w[z] == Subscript[c, 1] z^s HypergeometricPFQRegularized[{Subscript[a, 1]}, {Subscript[b, 1], Subscript[b, 2]}, a z^r] + Subscript[c, 2] z^s (MeijerG[{{1 - Subscript[a, 1]}, {}}, {{0, 1 - Subscript[b, 1]}, {1 - Subscript[b, 2]}}, a z^r] + MeijerG[{{1 - Subscript[a, 1]}, {}}, {{0, 1 - Subscript[b, 2]}, {1 - Subscript[b, 1]}}, a z^r]) + Subscript[c, 3] z^s MeijerG[{{1 - Subscript[a, 1]}, {}}, {{0, 1 - Subscript[b, 1], 1 - Subscript[b, 2]}, {}}, (-a) z^r]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["z", "3"], RowBox[List[SuperscriptBox["w", TagBox[RowBox[List["(", "3", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "r"]], "-", RowBox[List["3", " ", "s"]], "+", RowBox[List["r", " ", RowBox[List["(", RowBox[List[SubscriptBox["b", "1"], "+", SubscriptBox["b", "2"]]], ")"]]]]]], " ", ")"]], SuperscriptBox["z", "2"], RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["3", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "s"]], ")"]], " ", "s"]], "+", RowBox[List["r", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["4", " ", "s"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["r", "2"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], ")"]]]], "-", RowBox[List["r", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "r", "+", RowBox[List["2", " ", "s"]]]], ")"]], " ", SubscriptBox["b", "2"]]], "+", RowBox[List["r", " ", SubscriptBox["b", "1"], " ", RowBox[List["(", RowBox[List["1", "-", "r", "-", RowBox[List["2", " ", "s"]], "+", RowBox[List["r", " ", SubscriptBox["b", "2"]]]]], ")"]]]]]], ")"]], " ", "z", " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["r", "2"], " ", SuperscriptBox["z", "r"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "s"]], "+", RowBox[List["r", " ", SubscriptBox["a", "1"]]]]], ")"]]]], "-", RowBox[List["s", " ", RowBox[List["(", RowBox[List["r", "+", "s", "-", RowBox[List["r", " ", SubscriptBox["b", "1"]]]]], ")"]], " ", RowBox[List["(", RowBox[List["r", "+", "s", "-", RowBox[List["r", " ", SubscriptBox["b", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["w", "[", "z", "]"]]]]]], "\[Equal]", "0"]], "/;", " ", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], SuperscriptBox["z", "s"], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", SubscriptBox["a", "1"], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"]]], "}"]], ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", SuperscriptBox["z", "s"], RowBox[List["(", RowBox[List[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "-", SubscriptBox["a", "1"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["1", "-", SubscriptBox["b", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", SubscriptBox["b", "2"]]], "}"]]]], "}"]], ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]], "+", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "-", SubscriptBox["a", "1"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["1", "-", SubscriptBox["b", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", SubscriptBox["b", "1"]]], "}"]]]], "}"]], ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]]]], ")"]]]], "+", RowBox[List[SubscriptBox["c", "3"], " ", SuperscriptBox["z", "s"], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "-", SubscriptBox["a", "1"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["1", "-", SubscriptBox["b", "1"]]], ",", RowBox[List["1", "-", SubscriptBox["b", "2"]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "r"]]]]], "]"]]]]]]]]]]]]

 MathML Form

 z 3 w ( 3 ) TagBox[RowBox[List["(", "3", ")"]], Derivative] ( z ) + ( ( b 1 + b 2 ) r - 2 r - 3 s + 3 ) z 2 w ′′ ( z ) + ( ( 1 - a z r ) r 2 + ( 4 s - 2 ) r - ( r + 2 s - 1 ) b 2 r + b 1 ( b 2 r - r - 2 s + 1 ) r + 3 ( s - 1 ) s + 1 ) z w ( z ) + ( - a r 2 ( r a 1 - s ) z r - s ( - b 1 r + r + s ) ( - b 2 r + r + s ) ) w ( z ) 0 /; w ( z ) c 1 z s 1 F ~ 2 ( a 1 ; b 1 , b 2 ; a z r ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox[OverscriptBox["F", "~"], "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[SubscriptBox["a", "1"], HypergeometricPFQRegularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List["a", " ", SuperscriptBox["z", "r"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] + c 2 z s ( G 1 , 3 2 , 1 ( a z r 1 - a 1 0 , 1 - b 1 , 1 - b 2 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["1", ",", "3"]], RowBox[List["2", ",", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["a", " ", SuperscriptBox["z", "r"]]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"]]], MeijerG, Rule[Editable, True]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "2"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] + G 1 , 3 2 , 1 ( a z r 1 - a 1 0 , 1 - b 2 , 1 - b 1 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["1", ",", "3"]], RowBox[List["2", ",", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["a", " ", SuperscriptBox["z", "r"]]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"]]], MeijerG, Rule[Editable, True]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] ) + c 3 z s G 1 , 3 3 , 1 ( - a z r 1 - a 1 0 , 1 - b 1 , 1 - b 2 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["1", ",", "3"]], RowBox[List["3", ",", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "r"]]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"]]], MeijerG, Rule[Editable, True]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "2"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] Condition z 3 z 3 w z Subscript b 1 Subscript b 2 r -1 2 r -1 3 s 3 z 2 z 2 w z 1 -1 a z r r 2 4 s -2 r -1 r 2 s -1 Subscript b 2 r Subscript b 1 Subscript b 2 r -1 r -1 2 s 1 r 3 s -1 s 1 z z w z -1 a r 2 r Subscript a 1 -1 s z r -1 s -1 Subscript b 1 r r s -1 Subscript b 2 r r s w z 0 w z Subscript c 1 z s HypergeometricPFQRegularized Subscript a 1 Subscript b 1 Subscript b 2 a z r Subscript c 2 z s MeijerG 1 -1 Subscript a 1 0 1 -1 Subscript b 1 1 -1 Subscript b 2 a z r MeijerG 1 -1 Subscript a 1 0 1 -1 Subscript b 2 1 -1 Subscript b 1 a z r Subscript c 3 z s MeijerG 1 -1 Subscript a 1 0 1 -1 Subscript b 1 1 -1 Subscript b 2 -1 a z r [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[SuperscriptBox["z_", "3"], " ", RowBox[List[SuperscriptBox["w", TagBox[RowBox[List["(", "3", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "r_"]], "-", RowBox[List["3", " ", "s_"]], "+", RowBox[List["r_", " ", RowBox[List["(", RowBox[List[SubscriptBox["b_", "1"], "+", SubscriptBox["b_", "2"]]], ")"]]]]]], ")"]], " ", SuperscriptBox["z_", "2"], " ", RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["3", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "s_"]], ")"]], " ", "s_"]], "+", RowBox[List["r_", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["4", " ", "s_"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["r_", "2"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]]]], ")"]]]], "-", RowBox[List["r_", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "r_", "+", RowBox[List["2", " ", "s_"]]]], ")"]], " ", SubscriptBox["b_", "2"]]], "+", RowBox[List["r_", " ", SubscriptBox["b_", "1"], " ", RowBox[List["(", RowBox[List["1", "-", "r_", "-", RowBox[List["2", " ", "s_"]], "+", RowBox[List["r_", " ", SubscriptBox["b_", "2"]]]]], ")"]]]]]], ")"]], " ", "z_", " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a_"]], " ", SuperscriptBox["r_", "2"], " ", SuperscriptBox["z_", "r_"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "s_"]], "+", RowBox[List["r_", " ", SubscriptBox["a_", "1"]]]]], ")"]]]], "-", RowBox[List["s_", " ", RowBox[List["(", RowBox[List["r_", "+", "s_", "-", RowBox[List["r_", " ", SubscriptBox["b_", "1"]]]]], ")"]], " ", RowBox[List["(", RowBox[List["r_", "+", "s_", "-", RowBox[List["r_", " ", SubscriptBox["b_", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["w", "[", "z_", "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], " ", SuperscriptBox["z", "s"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", SubscriptBox["a", "1"], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"]]], "}"]], ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", SuperscriptBox["z", "s"], " ", RowBox[List["(", RowBox[List[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "-", SubscriptBox["a", "1"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["1", "-", SubscriptBox["b", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", SubscriptBox["b", "2"]]], "}"]]]], "}"]], ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]], "+", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "-", SubscriptBox["a", "1"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["1", "-", SubscriptBox["b", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", SubscriptBox["b", "1"]]], "}"]]]], "}"]], ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]]]], ")"]]]], "+", RowBox[List[SubscriptBox["c", "3"], " ", SuperscriptBox["z", "s"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "-", SubscriptBox["a", "1"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["1", "-", SubscriptBox["b", "1"]]], ",", RowBox[List["1", "-", SubscriptBox["b", "2"]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "r"]]]]], "]"]]]]]]]]]]]]]]

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

 2007-05-02