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http://functions.wolfram.com/06.05.21.0007.01
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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))))
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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"]]]], ")"]]]], ")"]]]], ")"]]]], ")"]]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mo> ∫ </mo> <mrow> <mi> γ </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ∞ </mi> </mrow> </mrow> <mrow> <mi> γ </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ∞ </mi> </mrow> </mrow> </msubsup> <mrow> <mfrac> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> A </mi> </munderover> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> t </mi> <mo> + </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> B </mi> </munderover> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mi> k </mi> </msub> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> C </mi> </munderover> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> t </mi> <mo> + </mo> <msub> <mi> c </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> D </mi> </munderover> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> t </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mrow> <mi> B </mi> <mo> + </mo> <mi> C </mi> </mrow> <mo> , </mo> <mrow> <mi> A </mi> <mo> + </mo> <mi> D </mi> </mrow> </mrow> <mrow> <mi> A </mi> <mo> , </mo> <mi> B </mi> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mi> t </mi> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mi> B </mi> </msub> </mrow> <mo> , </mo> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> c </mi> <mi> C </mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mi> A </mi> </msub> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> d </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> d </mi> <mi> D </mi> </msub> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> 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]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Δ </mi> <mo> ⩵ </mo> <mrow> <mi> A </mi> <mo> - </mo> <mi> B </mi> <mo> - </mo> <mi> C </mi> <mo> + </mo> <mi> D </mi> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> Ε </mi> <mo> ⩵ </mo> <mrow> <mi> A </mi> <mo> + </mo> <mi> B </mi> <mo> - </mo> <mi> C </mi> <mo> - </mo> <mi> D </mi> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> ν </mi> <mo> ⩵ </mo> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> A </mi> </munderover> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> B </mi> </munderover> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> - </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> C </mi> </munderover> <msub> <mi> c </mi> <mi> k </mi> </msub> </mrow> <mo> - </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> D </mi> </munderover> <msub> <mi> d </mi> <mi> k </mi> </msub> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mi> min </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> a </mi> <mi> A </mi> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> < </mo> <mi> γ </mi> <mo> < </mo> <mrow> <mi> min </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> b </mi> <mi> B </mi> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> Ε </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ∧ </mo> <mrow> <mi> Ε </mi> <mo> > </mo> <mn> 0 </mn> </mrow> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> Ε </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ∧ </mo> <mrow> <mi> Ε </mi> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mrow> <mi> γ </mi> <mo> ⁢ </mo> <mi> Δ </mi> </mrow> <mo> + </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mfrac> <mi> Ε </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> < </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mi> z </mi> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> Ε </mi> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> Δ </mi> <mo> ≠ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mrow> <mi> γ </mi> <mo> ⁢ </mo> <mi> Δ </mi> </mrow> <mo> + </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> </mrow> <mo> < </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mi> z </mi> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> Ε </mi> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> Δ </mi> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> z </mi> <mo> ≠ </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> z </mi> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <msubsup> <mo> ∫ </mo> <mrow> <mi> γ </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ∞ </mi> </mrow> </mrow> <mrow> <mi> γ </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ∞ </mi> </mrow> </mrow> </msubsup> <mrow> <mfrac> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> A </mi> </munderover> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> t </mi> <mo> + </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> B </mi> </munderover> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mi> k </mi> </msub> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> C </mi> </munderover> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> t </mi> <mo> + </mo> <msub> <mi> c </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> D </mi> </munderover> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> t </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mrow> <mi> B </mi> <mo> + </mo> <mi> C </mi> </mrow> <mo> , </mo> <mrow> <mi> A </mi> <mo> + </mo> <mi> D </mi> </mrow> </mrow> <mrow> <mi> A </mi> <mo> , </mo> <mi> B </mi> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mi> t </mi> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mi> B </mi> </msub> </mrow> <mo> , </mo> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> c </mi> <mi> C </mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mi> A </mi> </msub> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> d </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> d </mi> <mi> D </mi> </msub> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> 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]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Δ </mi> <mo> ⩵ </mo> <mrow> <mi> A </mi> <mo> - </mo> <mi> B </mi> <mo> - </mo> <mi> C </mi> <mo> + </mo> <mi> D </mi> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> Ε </mi> <mo> ⩵ </mo> <mrow> <mi> A </mi> <mo> + </mo> <mi> B </mi> <mo> - </mo> <mi> C </mi> <mo> - </mo> <mi> D </mi> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> ν </mi> <mo> ⩵ </mo> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> A </mi> </munderover> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> B </mi> </munderover> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> - </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> C </mi> </munderover> <msub> <mi> c </mi> <mi> k </mi> </msub> </mrow> <mo> - </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> D </mi> </munderover> <msub> <mi> d </mi> <mi> k </mi> </msub> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> - 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</mo> <mfrac> <mi> Ε </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> < </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mi> z </mi> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> Ε </mi> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> Δ </mi> <mo> ≠ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mrow> <mi> γ </mi> <mo> ⁢ </mo> <mi> Δ </mi> </mrow> <mo> + </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> </mrow> <mo> < </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mi> z </mi> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> Ε </mi> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> Δ </mi> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> z </mi> <mo> ≠ </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> z </mi> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
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