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; }

 Gamma

 http://functions.wolfram.com/06.05.21.0007.01

 Input Form

 Integrate[Product[Gamma[Subscript[a, k] + t] Product[Gamma[Subscript[b, k] - t], {k, 1, B}], {k, 1, A}]/ Product[Gamma[Subscript[c, k] + t] Product[Gamma[Subscript[d, k] - t], {k, 1, D}], {k, 1, C}]/z^t, {t, \[Gamma] - I Infinity, \[Gamma] + I Infinity}] == 2 Pi I MeijerG[{{1 - Subscript[b, 1], \[Ellipsis], 1 - Subscript[b, B]}, {Subscript[c, 1], \[Ellipsis], Subscript[c, C]}}, {{Subscript[a, 1], \[Ellipsis], Subscript[a, A]}, {1 - Subscript[d, 1], \[Ellipsis], 1 - Subscript[d, D]}}, z] /; \[CapitalDelta] == A + D - B - C && \[CapitalEpsilon] == A + B - C - D && \[Nu] == Sum[Subscript[a, k], {k, 1, A}] + Sum[Subscript[b, k], {k, 1, B}] - Sum[Subscript[c, k], {k, 1, C}] - Sum[Subscript[d, k], {k, 1, D}] && -Min[Re[Subscript[a, 1]], \[Ellipsis], Re[Subscript[a, A]]] < \[Gamma] < Min[Re[Subscript[b, 1]], \[Ellipsis], Re[Subscript[b, B]]] && ((Abs[Arg[z]] < (Pi \[CapitalEpsilon])/2 && \[CapitalEpsilon] > 0) || (Abs[Arg[z]] == (Pi \[CapitalEpsilon])/2 && \[CapitalEpsilon] > 0 && \[CapitalDelta] \[Gamma] + Re[\[Nu]] - \[CapitalEpsilon]/2 < -1) || (z > 0 && \[CapitalEpsilon] == 0 && \[CapitalDelta] != 0 && \[CapitalDelta] \[Gamma] + Re[\[Nu]] < 1/2) || (z > 0 && \[CapitalEpsilon] == 0 && \[CapitalDelta] == 0 && ((Re[\[Nu]] < 0 && z != 1) || (Re[\[Nu]] < -1 && z == 1))))

 Standard Form

 Cell[BoxData[RowBox[List[" ", RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["\[Gamma]", "-", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]], RowBox[List["\[Gamma]", "+", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]]], RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "A"], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "k"], "+", "t"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "B"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "k"], "-", "t"]], "]"]], " "]]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "C"], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["c", "k"], "+", "t"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "D"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["d", "k"], "-", "t"]], "]"]]]]]]]]], SuperscriptBox["z", RowBox[List["-", "t"]]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List["2", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["b", "B"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["c", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["c", "C"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "A"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", SubscriptBox["d", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["d", "D"]]]]], "}"]]]], "}"]], ",", "z"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["\[CapitalDelta]", "\[Equal]", RowBox[List["A", "+", "D", "-", "B", "-", "C"]]]], "\[And]", RowBox[List["\[CapitalEpsilon]", "\[Equal]", RowBox[List["A", "+", "B", "-", "C", "-", "D"]]]], "\[And]", RowBox[List["\[Nu]", "\[Equal]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "A"], SubscriptBox["a", "k"]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "B"], SubscriptBox["b", "k"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "C"], SubscriptBox["c", "k"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "D"], SubscriptBox["d", "k"]]]]]]], "\[And]", RowBox[List[RowBox[List["-", RowBox[List["Min", "[", RowBox[List[RowBox[List["Re", "[", SubscriptBox["a", "1"], "]"]], ",", "\[Ellipsis]", ",", RowBox[List["Re", "[", SubscriptBox["a", "A"], "]"]]]], "]"]]]], "<", "\[Gamma]", "<", RowBox[List["Min", "[", RowBox[List[RowBox[List["Re", "[", SubscriptBox["b", "1"], "]"]], ",", "\[Ellipsis]", ",", RowBox[List["Re", "[", SubscriptBox["b", "B"], "]"]]]], "]"]]]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "z", "]"]], "]"]], "<", FractionBox[RowBox[List["\[Pi]", " ", "\[CapitalEpsilon]"]], "2"]]], "\[And]", RowBox[List["\[CapitalEpsilon]", ">", "0"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "z", "]"]], "]"]], "\[Equal]", FractionBox[RowBox[List["\[Pi]", " ", "\[CapitalEpsilon]"]], "2"]]], "\[And]", RowBox[List["\[CapitalEpsilon]", ">", "0"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["\[CapitalDelta]", " ", "\[Gamma]"]], "+", RowBox[List["Re", "[", "\[Nu]", "]"]], "-", FractionBox["\[CapitalEpsilon]", "2"]]], "<", RowBox[List["-", "1"]]]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["z", ">", "0"]], "\[And]", RowBox[List["\[CapitalEpsilon]", "\[Equal]", "0"]], "\[And]", RowBox[List["\[CapitalDelta]", "\[NotEqual]", "0"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["\[CapitalDelta]", " ", "\[Gamma]"]], "+", RowBox[List["Re", "[", "\[Nu]", "]"]]]], "<", FractionBox["1", "2"]]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["z", ">", "0"]], "\[And]", RowBox[List["\[CapitalEpsilon]", "\[Equal]", "0"]], "\[And]", RowBox[List["\[CapitalDelta]", "\[Equal]", "0"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], "<", "0"]], "\[And]", RowBox[List["z", "\[NotEqual]", "1"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], "<", RowBox[List["-", "1"]]]], "\[And]", RowBox[List["z", "\[Equal]", "1"]]]], ")"]]]], ")"]]]], ")"]]]], ")"]]]]]]]]]]

 MathML Form

 γ - γ + k = 1 A Γ ( t + a k ) k = 1 B Γ ( b k - t ) k = 1 C Γ ( t + c k ) k = 1 D Γ ( d k - t ) z - t t 2 π G B + C , A + D A , B ( t 1 - b 1 , , 1 - b B , c 1 , , c C a 1 , , a A , 1 - d 1 , , 1 - d D ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List[RowBox[List["B", "+", "C"]], ",", RowBox[List["A", "+", "D"]]]], RowBox[List["A", ",", "B"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["t", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "B"]]], MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["c", "1"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["c", "C"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[SubscriptBox["a", "1"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "A"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["d", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["d", "D"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]] /; Δ A - B - C + D Ε A + B - C - D ν k = 1 A a k + k = 1 B b k - k = 1 C c k - k = 1 D d k - min ( Re ( a 1 ) , , Re ( a A ) ) < γ < min ( Re ( b 1 ) , , Re ( b B ) ) ( "\[LeftBracketingBar]" arg ( z ) "\[RightBracketingBar]" < π Ε 2 Ε > 0 "\[LeftBracketingBar]" arg ( z ) "\[RightBracketingBar]" π Ε 2 Ε > 0 γ Δ + Re ( ν ) - Ε 2 < - 1 z > 0 Ε 0 Δ 0 γ Δ + Re ( ν ) < 1 2 z > 0 Ε 0 Δ 0 ( Re ( ν ) < 0 z 1 Re ( ν ) < - 1 z 1 ) ) γ - γ + k = 1 A Γ ( t + a k ) k = 1 B Γ ( b k - t ) k = 1 C Γ ( t + c k ) k = 1 D Γ ( d k - t ) z - t t 2 π G B + C , A + D A , B ( t 1 - b 1 , , 1 - b B , c 1 , , c C a 1 , , a A , 1 - d 1 , , 1 - d D ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List[RowBox[List["B", "+", "C"]], ",", RowBox[List["A", "+", "D"]]]], RowBox[List["A", ",", "B"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["t", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "B"]]], MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["c", "1"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["c", "C"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[SubscriptBox["a", "1"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "A"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["d", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["d", "D"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]] /; Δ A - B - C + D Ε A + B - C - D ν k = 1 A a k + k = 1 B b k - k = 1 C c k - k = 1 D d k - min ( Re ( a 1 ) , , Re ( a A ) ) < γ < min ( Re ( b 1 ) , , Re ( b B ) ) ( "\[LeftBracketingBar]" arg ( z ) "\[RightBracketingBar]" < π Ε 2 Ε > 0 "\[LeftBracketingBar]" arg ( z ) "\[RightBracketingBar]" π Ε 2 Ε > 0 γ Δ + Re ( ν ) - Ε 2 < - 1 z > 0 Ε 0 Δ 0 γ Δ + Re ( ν ) < 1 2 z > 0 Ε 0 Δ 0 ( Re ( ν ) < 0 z 1 Re ( ν ) < - 1 z 1 ) ) [/itex]

 Rule Form

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

 2001-10-29

© 1998-2013 Wolfram Research, Inc.