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/07.31.06.0040.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z] \[Proportional] (Product[Gamma[Subscript[b, j]], {j, 1, q}]/Product[Gamma[Subscript[a, j]], {j, 1, p}]) (Sum[Residue[(Gamma[s] Product[Gamma[Subscript[a, j] - s], {j, 1, p}])/ Product[Gamma[Subscript[b, j] - s], {j, 1, q}]/(-z)^s, {s, Subscript[a, k]}] (1 + O[1/z]), {k, 1, p}] + KroneckerDelta[q, p + 1] Subscript[d, 1] (-z)^\[Chi] Cos[2 Sqrt[-z] + \[Chi] Pi] (1 + O[1/Sqrt[-z]]) + (UnitStep[q - p] - KroneckerDelta[q, p + 1]) Subscript[d, 2] z^\[Chi] Exp[\[Beta] z^(1/\[Beta])] (1 + O[1/z^(1/\[Beta])])) /; (Abs[z] -> Infinity) && \[Beta] == q - p + 1 && \[Chi] == (1/\[Beta]) ((\[Beta] - 1)/2 + Sum[Subscript[a, k], {k, 1, p}] - Sum[Subscript[b, k], {k, 1, q}]) && 2 Subscript[d, 2] == Subscript[d, 1] == (2 (2 Pi)^((1 - \[Beta])/2))/Sqrt[\[Beta]]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "q"], RowBox[List["Gamma", "[", SubscriptBox["b", "j"], "]"]]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List["Gamma", "[", SubscriptBox["a", "j"], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "p"], RowBox[List[RowBox[List["Residue", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "j"], "-", "s"]], "]"]], " "]]]], RowBox[List[" ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "q"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "j"], "-", "s"]], "]"]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "s"]]]]], ",", RowBox[List["{", RowBox[List["s", ",", SubscriptBox["a", "k"]]], "}"]]]], "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]]]], "+", RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List["q", ",", RowBox[List["p", "+", "1"]]]], "]"]], SubscriptBox["d", "1"], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "\[Chi]"], RowBox[List["Cos", "[", RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]], "+", RowBox[List["\[Chi]", " ", "\[Pi]"]]]], "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", SqrtBox[RowBox[List["-", "z"]]]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["q", "-", "p"]], "]"]], "-", RowBox[List["KroneckerDelta", "[", RowBox[List["q", ",", RowBox[List["p", "+", "1"]]]], "]"]]]], ")"]], SubscriptBox["d", "2"], " ", SuperscriptBox["z", "\[Chi]"], " ", RowBox[List["Exp", "[", RowBox[List["\[Beta]", " ", SuperscriptBox["z", FractionBox["1", "\[Beta]"]]]], "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", RowBox[List["1", "/", "\[Beta]"]]]], "]"]]]], ")"]]]]]], ")"]]]]]], "/;", "\[InvisibleSpace]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List["\[Beta]", "\[Equal]", RowBox[List["q", "-", "p", "+", "1"]]]], "\[And]", RowBox[List["\[Chi]", "\[Equal]", RowBox[List[FractionBox["1", "\[Beta]"], RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[Beta]", "-", "1"]], "2"], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "p"], SubscriptBox["a", "k"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "q"], SubscriptBox["b", "k"]]]]], ")"]]]]]], "\[And]", RowBox[List[RowBox[List["2", SubscriptBox["d", "2"]]], "\[Equal]", SubscriptBox["d", "1"], "\[Equal]", FractionBox[RowBox[List["2", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], FractionBox[RowBox[List["1", "-", "\[Beta]"]], "2"]]]], SqrtBox["\[Beta]"]]]]]]]]]]

 MathML Form

 p F q ( a 1 , , a p ; b 1 , , b q ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", TraditionalForm]], SubscriptBox["F", FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "p"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox["z", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] j = 1 q Γ ( b j ) j = 1 p Γ ( a j ) ( k = 1 p res s ( Γ ( s ) j = 1 p Γ ( a j - s ) j = 1 q Γ ( b j - s ) ( - z ) - s ) ( a k ) ( 1 + O ( 1 z ) ) + δ KroneckerDelta q , p + 1 d 1 ( - z ) χ cos ( π χ + 2 - z ) ( 1 + O ( 1 - z ) ) + ( θ UnitStep ( q - p ) - δ KroneckerDelta q , p + 1 ) d 2 z χ β z 1 / β ( 1 + O ( 1 z 1 / β ) ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) β q - p + 1 χ 1 β ( β - 1 2 + k = 1 p a k - k = 1 q b k ) 2 d 2 d 1 2 ( 2 π ) 1 - β 2 β Condition Proportional HypergeometricPFQ Subscript a 1 Subscript a p Subscript b 1 Subscript b q z j 1 q Gamma Subscript b j j 1 p Gamma Subscript a j -1 k 1 p Subscript res s Gamma s j 1 p Gamma Subscript a j -1 s j 1 q Gamma Subscript b j -1 s -1 -1 z -1 s Subscript a k 1 O 1 z -1 KroneckerDelta q p 1 Subscript d 1 -1 z χ χ 2 -1 z 1 2 1 O 1 -1 z 1 2 -1 UnitStep q -1 p -1 KroneckerDelta q p 1 Subscript d 2 z χ β z 1 β -1 1 O 1 z 1 β -1 -1 Rule z β q -1 p 1 χ 1 β -1 β -1 2 -1 k 1 p Subscript a k -1 k 1 q Subscript b k 2 Subscript d 2 Subscript d 1 2 2 1 -1 β 2 -1 β 1 2 -1 [/itex]

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

 2001-10-29

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