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; }

 HypergeometricPFQRegularized

 http://functions.wolfram.com/07.32.06.0036.01

 Input Form

 HypergeometricPFQRegularized[{}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, z] \[Proportional] (1/(4 Sqrt[2] Pi^(3/2))) E^(4 z^(1/4)) z^((1/4) (3/2 - Subscript[b, 1] - Subscript[b, 2] - Subscript[b, 3])) (1 + (-7 - 12 Subscript[b, 1]^2 - 12 Subscript[b, 2]^2 + 8 Subscript[b, 3] - 12 Subscript[b, 3]^2 + 8 Subscript[b, 2] (1 + Subscript[b, 3]) + 8 Subscript[b, 1] (1 + Subscript[b, 2] + Subscript[b, 3]))/(32 z^(1/4)) + (1/(2048 Sqrt[z])) (121 + 144 Subscript[b, 1]^4 + 144 Subscript[b, 2]^4 - 176 Subscript[b, 3] + 8 Subscript[b, 3]^2 - 64 Subscript[b, 3]^3 + 144 Subscript[b, 3]^4 - 64 Subscript[b, 2]^3 (1 + 3 Subscript[b, 3]) - 64 Subscript[b, 1]^3 (1 + 3 Subscript[b, 2] + 3 Subscript[b, 3]) + 8 Subscript[b, 2]^2 (1 - 24 Subscript[b, 3] + 44 Subscript[b, 3]^2) - 16 Subscript[b, 2] (11 - 21 Subscript[b, 3] + 12 Subscript[b, 3]^2 + 12 Subscript[b, 3]^3) + 8 Subscript[b, 1]^2 (1 + 44 Subscript[b, 2]^2 - 24 Subscript[b, 3] + 44 Subscript[b, 3]^2 - 8 Subscript[b, 2] (3 + Subscript[b, 3])) - 16 Subscript[b, 1] (11 + 12 Subscript[b, 2]^3 - 21 Subscript[b, 3] + 12 Subscript[b, 3]^2 + 12 Subscript[b, 3]^3 + 4 Subscript[b, 2]^2 (3 + Subscript[b, 3]) + Subscript[b, 2] (-21 - 40 Subscript[b, 3] + 4 Subscript[b, 3]^2))) + \[Ellipsis]) /; (Abs[z] -> Infinity)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", SubscriptBox["b", "3"]]], "}"]], ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox["1", RowBox[List["4", " ", SqrtBox["2"], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["4", " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]]], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[FractionBox["3", "2"], "-", SubscriptBox["b", "1"], "-", SubscriptBox["b", "2"], "-", SubscriptBox["b", "3"]]], ")"]]]]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["-", "7"]], "-", RowBox[List["12", " ", SubsuperscriptBox["b", "1", "2"]]], "-", RowBox[List["12", " ", SubsuperscriptBox["b", "2", "2"]]], "+", RowBox[List["8", " ", SubscriptBox["b", "3"]]], "-", RowBox[List["12", " ", SubsuperscriptBox["b", "3", "2"]]], "+", RowBox[List["8", " ", SubscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["b", "3"]]], ")"]]]], "+", RowBox[List["8", " ", SubscriptBox["b", "1"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["b", "2"], "+", SubscriptBox["b", "3"]]], ")"]]]]]], RowBox[List["32", " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]]], "+", RowBox[List[FractionBox["1", RowBox[List["2048", " ", SqrtBox["z"]]]], RowBox[List["(", RowBox[List["121", "+", RowBox[List["144", " ", SubsuperscriptBox["b", "1", "4"]]], "+", RowBox[List["144", " ", SubsuperscriptBox["b", "2", "4"]]], "-", RowBox[List["176", " ", SubscriptBox["b", "3"]]], "+", RowBox[List["8", " ", SubsuperscriptBox["b", "3", "2"]]], "-", RowBox[List["64", " ", SubsuperscriptBox["b", "3", "3"]]], "+", RowBox[List["144", " ", SubsuperscriptBox["b", "3", "4"]]], "-", RowBox[List["64", " ", SubsuperscriptBox["b", "2", "3"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["3", " ", SubscriptBox["b", "3"]]]]], ")"]]]], "-", RowBox[List["64", " ", SubsuperscriptBox["b", "1", "3"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["3", " ", SubscriptBox["b", "2"]]], "+", RowBox[List["3", " ", SubscriptBox["b", "3"]]]]], ")"]]]], "+", RowBox[List["8", " ", SubsuperscriptBox["b", "2", "2"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["24", " ", SubscriptBox["b", "3"]]], "+", RowBox[List["44", " ", SubsuperscriptBox["b", "3", "2"]]]]], ")"]]]], "-", RowBox[List["16", " ", SubscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List["11", "-", RowBox[List["21", " ", SubscriptBox["b", "3"]]], "+", RowBox[List["12", " ", SubsuperscriptBox["b", "3", "2"]]], "+", RowBox[List["12", " ", SubsuperscriptBox["b", "3", "3"]]]]], ")"]]]], "+", RowBox[List["8", " ", SubsuperscriptBox["b", "1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["44", " ", SubsuperscriptBox["b", "2", "2"]]], "-", RowBox[List["24", " ", SubscriptBox["b", "3"]]], "+", RowBox[List["44", " ", SubsuperscriptBox["b", "3", "2"]]], "-", RowBox[List["8", " ", SubscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List["3", "+", SubscriptBox["b", "3"]]], ")"]]]]]], ")"]]]], "-", RowBox[List["16", " ", SubscriptBox["b", "1"], " ", RowBox[List["(", RowBox[List["11", "+", RowBox[List["12", " ", SubsuperscriptBox["b", "2", "3"]]], "-", RowBox[List["21", " ", SubscriptBox["b", "3"]]], "+", RowBox[List["12", " ", SubsuperscriptBox["b", "3", "2"]]], "+", RowBox[List["12", " ", SubsuperscriptBox["b", "3", "3"]]], "+", RowBox[List["4", " ", SubsuperscriptBox["b", "2", "2"], " ", RowBox[List["(", RowBox[List["3", "+", SubscriptBox["b", "3"]]], ")"]]]], "+", RowBox[List[SubscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "21"]], "-", RowBox[List["40", " ", SubscriptBox["b", "3"]]], "+", RowBox[List["4", " ", SubsuperscriptBox["b", "3", "2"]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]

 MathML Form

 0 F ~ 3 ( ; b 1 , b 2 , b 3 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["0", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox["\[Null]", 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]], ",", TagBox[SubscriptBox["b", "3"], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox["z", HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] 1 4 2 π 3 / 2 4 z 4 z 1 4 ( 3 2 - b 1 - b 2 - b 3 ) ( 1 + - 12 b 1 2 + 8 ( b 2 + b 3 + 1 ) b 1 - 12 b 2 2 - 12 b 3 2 + 8 b 3 + 8 b 2 ( b 3 + 1 ) - 7 32 z 4 + 1 2048 z ( 144 b 1 4 - 64 ( 3 b 2 + 3 b 3 + 1 ) b 1 3 + 8 ( 44 b 2 2 - 8 ( b 3 + 3 ) b 2 + 44 b 3 2 - 24 b 3 + 1 ) b 1 2 - 16 ( 12 b 2 3 + 4 ( b 3 + 3 ) b 2 2 + ( 4 b 3 2 - 40 b 3 - 21 ) b 2 + 12 b 3 3 + 12 b 3 2 - 21 b 3 + 11 ) b 1 + 144 b 2 4 + 144 b 3 4 - 64 b 3 3 + 8 b 3 2 - 176 b 3 - 64 b 2 3 ( 3 b 3 + 1 ) + 8 b 2 2 ( 44 b 3 2 - 24 b 3 + 1 ) - 16 b 2 ( 12 b 3 3 + 12 b 3 2 - 21 b 3 + 11 ) + 121 ) + ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) Condition Proportional HypergeometricPFQRegularized Subscript b 1 Subscript b 2 Subscript b 3 z 1 4 2 1 2 3 2 -1 4 z 1 4 z 1 4 3 2 -1 Subscript b 1 -1 Subscript b 2 -1 Subscript b 3 1 -12 Subscript b 1 2 8 Subscript b 2 Subscript b 3 1 Subscript b 1 -1 12 Subscript b 2 2 -1 12 Subscript b 3 2 8 Subscript b 3 8 Subscript b 2 Subscript b 3 1 -7 32 z 1 4 -1 1 2048 z 1 2 -1 144 Subscript b 1 4 -1 64 3 Subscript b 2 3 Subscript b 3 1 Subscript b 1 3 8 44 Subscript b 2 2 -1 8 Subscript b 3 3 Subscript b 2 44 Subscript b 3 2 -1 24 Subscript b 3 1 Subscript b 1 2 -1 16 12 Subscript b 2 3 4 Subscript b 3 3 Subscript b 2 2 4 Subscript b 3 2 -1 40 Subscript b 3 -21 Subscript b 2 12 Subscript b 3 3 12 Subscript b 3 2 -1 21 Subscript b 3 11 Subscript b 1 144 Subscript b 2 4 144 Subscript b 3 4 -1 64 Subscript b 3 3 8 Subscript b 3 2 -1 176 Subscript b 3 -1 64 Subscript b 2 3 3 Subscript b 3 1 8 Subscript b 2 2 44 Subscript b 3 2 -1 24 Subscript b 3 1 -1 16 Subscript b 2 12 Subscript b 3 3 12 Subscript b 3 2 -1 21 Subscript b 3 11 121 Rule z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", SubscriptBox["b_", "2"], ",", SubscriptBox["b_", "3"]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["4", " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]]], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[FractionBox["3", "2"], "-", SubscriptBox["bb", "1"], "-", SubscriptBox["bb", "2"], "-", SubscriptBox["bb", "3"]]], ")"]]]]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["-", "7"]], "-", RowBox[List["12", " ", SubsuperscriptBox["bb", "1", "2"]]], "-", RowBox[List["12", " ", SubsuperscriptBox["bb", "2", "2"]]], "+", RowBox[List["8", " ", SubscriptBox["bb", "3"]]], "-", RowBox[List["12", " ", SubsuperscriptBox["bb", "3", "2"]]], "+", RowBox[List["8", " ", SubscriptBox["bb", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["bb", "3"]]], ")"]]]], "+", RowBox[List["8", " ", SubscriptBox["bb", "1"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["bb", "2"], "+", SubscriptBox["bb", "3"]]], ")"]]]]]], RowBox[List["32", " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]]], "+", FractionBox[RowBox[List["121", "+", RowBox[List["144", " ", SubsuperscriptBox["bb", "1", "4"]]], "+", RowBox[List["144", " ", SubsuperscriptBox["bb", "2", "4"]]], "-", RowBox[List["176", " ", SubscriptBox["bb", "3"]]], "+", RowBox[List["8", " ", SubsuperscriptBox["bb", "3", "2"]]], "-", RowBox[List["64", " ", SubsuperscriptBox["bb", "3", "3"]]], "+", RowBox[List["144", " ", SubsuperscriptBox["bb", "3", "4"]]], "-", RowBox[List["64", " ", SubsuperscriptBox["bb", "2", "3"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["3", " ", SubscriptBox["bb", "3"]]]]], ")"]]]], "-", RowBox[List["64", " ", SubsuperscriptBox["bb", "1", "3"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["3", " ", SubscriptBox["bb", "2"]]], "+", RowBox[List["3", " ", SubscriptBox["bb", "3"]]]]], ")"]]]], "+", RowBox[List["8", " ", SubsuperscriptBox["bb", "2", "2"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["24", " ", SubscriptBox["bb", "3"]]], "+", RowBox[List["44", " ", SubsuperscriptBox["bb", "3", "2"]]]]], ")"]]]], "-", RowBox[List["16", " ", SubscriptBox["bb", "2"], " ", RowBox[List["(", RowBox[List["11", "-", RowBox[List["21", " ", SubscriptBox["bb", "3"]]], "+", RowBox[List["12", " ", SubsuperscriptBox["bb", "3", "2"]]], "+", RowBox[List["12", " ", SubsuperscriptBox["bb", "3", "3"]]]]], ")"]]]], "+", RowBox[List["8", " ", SubsuperscriptBox["bb", "1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["44", " ", SubsuperscriptBox["bb", "2", "2"]]], "-", RowBox[List["24", " ", SubscriptBox["bb", "3"]]], "+", RowBox[List["44", " ", SubsuperscriptBox["bb", "3", "2"]]], "-", RowBox[List["8", " ", SubscriptBox["bb", "2"], " ", RowBox[List["(", RowBox[List["3", "+", SubscriptBox["bb", "3"]]], ")"]]]]]], ")"]]]], "-", RowBox[List["16", " ", SubscriptBox["bb", "1"], " ", RowBox[List["(", RowBox[List["11", "+", RowBox[List["12", " ", SubsuperscriptBox["bb", "2", "3"]]], "-", RowBox[List["21", " ", SubscriptBox["bb", "3"]]], "+", RowBox[List["12", " ", SubsuperscriptBox["bb", "3", "2"]]], "+", RowBox[List["12", " ", SubsuperscriptBox["bb", "3", "3"]]], "+", RowBox[List["4", " ", SubsuperscriptBox["bb", "2", "2"], " ", RowBox[List["(", RowBox[List["3", "+", SubscriptBox["bb", "3"]]], ")"]]]], "+", RowBox[List[SubscriptBox["bb", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "21"]], "-", RowBox[List["40", " ", SubscriptBox["bb", "3"]]], "+", RowBox[List["4", " ", SubsuperscriptBox["bb", "3", "2"]]]]], ")"]]]]]], ")"]]]]]], RowBox[List["2048", " ", SqrtBox["z"]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List["4", " ", SqrtBox["2"], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

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

 2001-10-29