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/07.27.13.0011.01

 Input Form

 Derivative[3][w][z] + (-((3 (-1 + r + s))/z) + (a r z^(-1 + r) (3 + Subscript[a, 1] + Subscript[a, 2] + Subscript[a, 3]))/(-1 + a z^r) - (r (1 + Subscript[b, 1] + Subscript[b, 2]))/(z (-1 + a z^r))) Derivative[2][w][z] + ((1 - 3 r + 2 r^2 - 3 s + 6 r s + 3 s^2)/z^2 + (r (-1 + r + 2 s))/(z^2 (-1 + a z^r)) + (a r z^(-2 + r) (3 - 2 r - 6 s + Subscript[a, 3] - 2 s Subscript[a, 3] + Subscript[a, 2] (1 - 2 s + r Subscript[a, 3]) + Subscript[a, 1] (1 - 2 s + r Subscript[a, 2] + r Subscript[a, 3])))/ (-1 + a z^r) - (r ((-(-1 + r + 2 s)) Subscript[b, 2] + Subscript[b, 1] (1 - r - 2 s + r Subscript[b, 2])))/ (z^2 (-1 + a z^r))) Derivative[1][w][z] + (-((s (2 r^2 + 3 r s + s^2))/z^3) - (r s (r + s))/(z^3 (-1 + a z^r)) + (a r z^(-3 + r) (Subscript[a, 1] (s - r Subscript[a, 2]) (s - r Subscript[a, 3]) + s (2 r + 3 s + s Subscript[a, 3] + Subscript[a, 2] (s - r Subscript[a, 3]))))/(-1 + a z^r) + (r s ((-(r + s)) Subscript[b, 2] + Subscript[b, 1] (-r - s + r Subscript[b, 2])))/(z^3 (-1 + a z^r))) w[z] == 0 /; w[z] == Subscript[c, 1] z^s HypergeometricPFQRegularized[ {Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, a z^r] + Subscript[c, 2] z^s (MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2], 1 - Subscript[a, 3]}, {}}, {{0, 1 - Subscript[b, 1]}, {1 - Subscript[b, 2]}}, a z^r] + MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2], 1 - Subscript[a, 3]}, {}}, {{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], 1 - Subscript[a, 3]}, {}}, {{0, 1 - Subscript[b, 1], 1 - Subscript[b, 2]}, {}}, (-a) z^r]

 Standard Form

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 MathML Form

 w ( 3 ) TagBox[RowBox[List["(", "3", ")"]], Derivative] ( z ) + ( a r ( a 1 + a 2 + a 3 + 3 ) z r - 1 a z r - 1 - r ( b 1 + b 2 + 1 ) z ( a z r - 1 ) - 3 ( r + s - 1 ) z ) w ′′ ( z ) + ( a r ( - 2 r - 6 s - 2 s a 3 + a 3 + a 2 ( - 2 s + r a 3 + 1 ) + a 1 ( - 2 s + r a 2 + r a 3 + 1 ) + 3 ) z r - 2 a z r - 1 + 2 r 2 + 6 s r - 3 r + 3 s 2 - 3 s + 1 z 2 + r ( r + 2 s - 1 ) z 2 ( a z r - 1 ) - r ( b 1 ( b 2 r - r - 2 s + 1 ) - ( r + 2 s - 1 ) b 2 ) z 2 ( a z r - 1 ) ) w ( z ) + ( a r ( a 1 ( s - r a 2 ) ( s - r a 3 ) + s ( 2 r + 3 s + s a 3 + a 2 ( s - r a 3 ) ) ) z r - 3 a z r - 1 + r s ( b 1 ( b 2 r - r - s ) - ( r + s ) b 2 ) z 3 ( a z r - 1 ) - s ( 2 r 2 + 3 s r + s 2 ) z 3 - r s ( r + s ) z 3 ( a z r - 1 ) ) w ( z ) 0 /; w ( z ) c 1 z s 3 F ~ 2 ( a 1 , a 2 , a 3 ; b 1 , b 2 ; a z r ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox[OverscriptBox["F", "~"], "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[SubscriptBox["a", "3"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["a", " ", SuperscriptBox["z", "r"]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQRegularized] + c 2 z s ( G 3 , 3 2 , 3 ( a z r 1 - 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a 1 , 1 - a 2 , 1 - a 3 0 , 1 - b 2 , 1 - b 1 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["3", ",", "3"]], RowBox[List["2", ",", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["a", " ", SuperscriptBox["z", "r"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "2"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "3"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "2"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] ) + c 3 z s G 3 , 3 3 , 3 ( - a z r 1 - a 1 , 1 - a 2 , 1 - a 3 0 , 1 - b 1 , 1 - b 2 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["3", ",", "3"]], RowBox[List["3", ",", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "r"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "2"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "3"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "2"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] Condition z 3 w z a r Subscript a 1 Subscript a 2 Subscript a 3 3 z r -1 a z r -1 -1 -1 r Subscript b 1 Subscript b 2 1 z a z r -1 -1 -1 3 r s -1 z -1 z 2 w z a r -2 r -1 6 s -1 2 s Subscript a 3 Subscript a 3 Subscript a 2 -2 s r Subscript a 3 1 Subscript a 1 -2 s r Subscript a 2 r Subscript a 3 1 3 z r -2 a z r -1 -1 2 r 2 6 s r -1 3 r 3 s 2 -1 3 s 1 z 2 -1 r r 2 s -1 z 2 a z r -1 -1 -1 r Subscript b 1 Subscript b 2 r -1 r -1 2 s 1 -1 r 2 s -1 Subscript b 2 z 2 a z r -1 -1 z w z a r Subscript a 1 s -1 r Subscript a 2 s -1 r Subscript a 3 s 2 r 3 s s Subscript a 3 Subscript a 2 s -1 r Subscript a 3 z r -3 a z r -1 -1 r s Subscript b 1 Subscript b 2 r -1 r -1 s -1 r s Subscript b 2 z 3 a z r -1 -1 -1 s 2 r 2 3 s r s 2 z 3 -1 -1 r s r s z 3 a z r -1 -1 w z 0 w z Subscript c 1 z s HypergeometricPFQRegularized Subscript a 1 Subscript a 2 Subscript a 3 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 1 -1 Subscript a 3 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 1 -1 Subscript a 3 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 1 -1 Subscript a 3 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