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/07.25.13.0011.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2007-05-02