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 HypergeometricPFQRegularized

 http://functions.wolfram.com/07.32.07.0004.01

 Input Form

 HypergeometricPFQRegularized[{Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z] == (1/(2 Pi I)) (1/Product[Gamma[Subscript[a, k]], {k, 1, p}]) Integrate[(Gamma[s] Product[Gamma[Subscript[a, k] - s], {k, 1, p}])/ Product[Gamma[Subscript[b, k] - s], {k, 1, q}]/(-z)^s, {s, \[Gamma] - I Infinity, \[Gamma] + I Infinity}] /; 0 < \[Gamma] < Min[Re[Subscript[a, 1]], \[Ellipsis], Re[Subscript[a, p]]] && ((p == q + 1 && Abs[Arg[-z]] < Pi) || (p == q && Abs[Arg[-z]] < Pi/2) || (p == q - 1 && \[Gamma] < 1/4 + (1/2) Re[Sum[Subscript[b, j], {j, 1, q}] - Sum[Subscript[a, j], {j, 1, p}]]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", 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"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]], FractionBox["1", RowBox[List[" ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "p"], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]]]]], RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["\[Gamma]", "-", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]], RowBox[List["\[Gamma]", "+", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "p"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "k"], "-", "s"]], "]"]]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "q"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "k"], "-", "s"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "s"]]], RowBox[List["\[DifferentialD]", "s"]]]]]]]]]], "/;", RowBox[List[RowBox[List["0", "<", "\[Gamma]", "<", RowBox[List["Min", "[", RowBox[List[RowBox[List["Re", "[", SubscriptBox["a", "1"], "]"]], ",", "\[Ellipsis]", ",", RowBox[List["Re", "[", SubscriptBox["a", "p"], "]"]]]], "]"]]]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["p", "\[Equal]", RowBox[List["q", "+", "1"]]]], "\[And]", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", RowBox[List["-", "z"]], "]"]], "]"]], "<", "\[Pi]"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["p", "\[Equal]", "q"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", RowBox[List["-", "z"]], "]"]], "]"]], "<", FractionBox["\[Pi]", "2"]]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["p", "\[Equal]", RowBox[List["q", "-", "1"]]]], "\[And]", RowBox[List["\[Gamma]", "<", RowBox[List[FractionBox["1", "4"], "+", RowBox[List[FractionBox["1", "2"], RowBox[List["Re", "[", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "q"], SubscriptBox["b", "j"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "p"], SubscriptBox["a", "j"]]]]], "]"]]]]]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 p F ~ q ( a 1 , , a q + 1 ; b 1 , , b q ; z ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", TraditionalForm]], SubscriptBox[OverscriptBox["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", RowBox[List["q", "+", "1"]]], 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]], ";", "z"]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]] 1 2 π k = 1 p Γ ( a k ) γ - γ + Γ ( s ) k = 1 p Γ ( a k - s ) ( - z ) - s k = 1 q Γ ( b k - s ) s /; 0 < γ < min ( Re ( a 1 ) , , Re ( a p ) ) ( ( p q + 1 "\[LeftBracketingBar]" arg ( - z ) "\[RightBracketingBar]" < π ) ( p q "\[LeftBracketingBar]" arg ( - z ) "\[RightBracketingBar]" < π 2 ) ( p q - 1 γ < 1 2 Re ( j = 1 q b j - j = 1 p a j ) + 1 4 ) ) p F ~ q ( a 1 , , a q + 1 ; b 1 , , b q ; z ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", TraditionalForm]], SubscriptBox[OverscriptBox["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", RowBox[List["q", "+", "1"]]], 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]], ";", "z"]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]] 1 2 π k = 1 p Γ ( a k ) γ - γ + Γ ( s ) k = 1 p Γ ( a k - s ) ( - z ) - s k = 1 q Γ ( b k - s ) s /; 0 < γ < min ( Re ( a 1 ) , , Re ( a p ) ) ( ( p q + 1 "\[LeftBracketingBar]" arg ( - z ) "\[RightBracketingBar]" < π ) ( p q "\[LeftBracketingBar]" arg ( - z ) "\[RightBracketingBar]" < π 2 ) ( p q - 1 γ < 1 2 Re ( j = 1 q b j - j = 1 p a j ) + 1 4 ) ) [/itex]

 Rule Form

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

 2001-10-29