Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
Gamma






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Gamma[z] > Integration > Definite integration > Involving the direct function





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







<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> &#8747; </mo> <mrow> <mi> &#947; </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> <mrow> <mi> &#947; </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> </msubsup> <mrow> <mfrac> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> A </mi> </munderover> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> t </mi> <mo> + </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> B </mi> </munderover> <mrow> <mi> &#915; </mi> <mo> &#8289; </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> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> C </mi> </munderover> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> t </mi> <mo> + </mo> <msub> <mi> c </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> D </mi> </munderover> <mrow> <mi> &#915; </mi> <mo> &#8289; </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> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> t </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </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> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mi> t </mi> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> &#8230; </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> &#8230; </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> &#8230; </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> &#8230; </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[&quot;G&quot;, MeijerG], RowBox[List[RowBox[List[&quot;B&quot;, &quot;+&quot;, &quot;C&quot;]], &quot;,&quot;, RowBox[List[&quot;A&quot;, &quot;+&quot;, &quot;D&quot;]]]], RowBox[List[&quot;A&quot;, &quot;,&quot;, &quot;B&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[&quot;t&quot;, MeijerG, Rule[Editable, True]], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[RowBox[List[TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;1&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;B&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;c&quot;, &quot;1&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;c&quot;, &quot;C&quot;], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[SubscriptBox[&quot;a&quot;, &quot;1&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;a&quot;, &quot;A&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;d&quot;, &quot;1&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;d&quot;, &quot;D&quot;]]], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> &#916; </mi> <mo> &#10869; </mo> <mrow> <mi> A </mi> <mo> - </mo> <mi> B </mi> <mo> - </mo> <mi> C </mi> <mo> + </mo> <mi> D </mi> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#917; </mi> <mo> &#10869; </mo> <mrow> <mi> A </mi> <mo> + </mo> <mi> B </mi> <mo> - </mo> <mi> C </mi> <mo> - </mo> <mi> D </mi> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#957; </mi> <mo> &#10869; </mo> <mrow> <mrow> <munderover> <mo> &#8721; </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> &#8721; </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> &#8721; </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> &#8721; </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> &#8743; </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mi> min </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> a </mi> <mi> A </mi> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &lt; </mo> <mi> &#947; </mi> <mo> &lt; </mo> <mrow> <mi> min </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> b </mi> <mi> B </mi> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#917; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#917; </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#917; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#917; </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mrow> <mi> &#947; </mi> <mo> &#8290; </mo> <mi> &#916; </mi> </mrow> <mo> + </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mfrac> <mi> &#917; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> &lt; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mi> z </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#917; </mi> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#916; </mi> <mo> &#8800; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mrow> <mi> &#947; </mi> <mo> &#8290; </mo> <mi> &#916; </mi> </mrow> <mo> + </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> </mrow> <mo> &lt; </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mi> z </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#917; </mi> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#916; </mi> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> z </mi> <mo> &#8800; </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> z </mi> <mo> &#10869; </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> &#8747; </mo> <mrow> <mi> &#947; </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> <mrow> <mi> &#947; </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> </msubsup> <mrow> <mfrac> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> A </mi> </munderover> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> t </mi> <mo> + </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> B </mi> </munderover> <mrow> <mi> &#915; </mi> <mo> &#8289; </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> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> C </mi> </munderover> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> t </mi> <mo> + </mo> <msub> <mi> c </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> D </mi> </munderover> <mrow> <mi> &#915; </mi> <mo> &#8289; </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> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> t </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </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> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mi> t </mi> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> &#8230; </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> &#8230; </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> &#8230; </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> &#8230; </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[&quot;G&quot;, MeijerG], RowBox[List[RowBox[List[&quot;B&quot;, &quot;+&quot;, &quot;C&quot;]], &quot;,&quot;, RowBox[List[&quot;A&quot;, &quot;+&quot;, &quot;D&quot;]]]], RowBox[List[&quot;A&quot;, &quot;,&quot;, &quot;B&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[&quot;t&quot;, MeijerG, Rule[Editable, True]], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[RowBox[List[TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;1&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;B&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;c&quot;, &quot;1&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;c&quot;, &quot;C&quot;], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[SubscriptBox[&quot;a&quot;, &quot;1&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;a&quot;, &quot;A&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;d&quot;, &quot;1&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;d&quot;, &quot;D&quot;]]], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> &#916; </mi> <mo> &#10869; </mo> <mrow> <mi> A </mi> <mo> - </mo> <mi> B </mi> <mo> - </mo> <mi> C </mi> <mo> + </mo> <mi> D </mi> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#917; </mi> <mo> &#10869; </mo> <mrow> <mi> A </mi> <mo> + </mo> <mi> B </mi> <mo> - </mo> <mi> C </mi> <mo> - </mo> <mi> D </mi> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#957; </mi> <mo> &#10869; </mo> <mrow> <mrow> <munderover> <mo> &#8721; </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> &#8721; </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> &#8721; </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> &#8721; </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> &#8743; </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mi> min </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> a </mi> <mi> A </mi> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &lt; </mo> <mi> &#947; </mi> <mo> &lt; </mo> <mrow> <mi> min </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> b </mi> <mi> B </mi> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#917; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#917; </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#917; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#917; </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mrow> <mi> &#947; </mi> <mo> &#8290; </mo> <mi> &#916; </mi> </mrow> <mo> + </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mfrac> <mi> &#917; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> &lt; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mi> z </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#917; </mi> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#916; </mi> <mo> &#8800; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mrow> <mi> &#947; </mi> <mo> &#8290; </mo> <mi> &#916; </mi> </mrow> <mo> + </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> </mrow> <mo> &lt; </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mi> z </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#917; </mi> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#916; </mi> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> z </mi> <mo> &#8800; </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> z </mi> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", 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["(", 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_"]], "]"]]]]]]]], ")"]], " ", SuperscriptBox["z_", RowBox[List["-", "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_"]], "]"]]]]]]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", SubscriptBox["bb", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["bb", "B"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["cc", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["cc", "C"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["aa", "A"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", SubscriptBox["dd", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["dd", "D"]]]]], "}"]]]], "}"]], ",", "z"]], "]"]]]], "/;", RowBox[List[RowBox[List["\[CapitalDelta]", "\[Equal]", RowBox[List["A", "+", "D", "-", "B", "-", "C"]]]], "&&", RowBox[List["\[CapitalEpsilon]", "\[Equal]", RowBox[List["A", "+", "B", "-", "C", "-", "D"]]]], "&&", 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"]]]]]]], "&&", RowBox[List[RowBox[List["-", RowBox[List["Min", "[", RowBox[List[RowBox[List["Re", "[", SubscriptBox["aa", "1"], "]"]], ",", "\[Ellipsis]", ",", RowBox[List["Re", "[", SubscriptBox["aa", "A"], "]"]]]], "]"]]]], "<", "\[Gamma]", "<", RowBox[List["Min", "[", RowBox[List[RowBox[List["Re", "[", SubscriptBox["bb", "1"], "]"]], ",", "\[Ellipsis]", ",", RowBox[List["Re", "[", SubscriptBox["bb", "B"], "]"]]]], "]"]]]], "&&", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "z", "]"]], "]"]], "<", FractionBox[RowBox[List["\[Pi]", " ", "\[CapitalEpsilon]"]], "2"]]], "&&", RowBox[List["\[CapitalEpsilon]", ">", "0"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "z", "]"]], "]"]], "\[Equal]", FractionBox[RowBox[List["\[Pi]", " ", "\[CapitalEpsilon]"]], "2"]]], "&&", RowBox[List["\[CapitalEpsilon]", ">", "0"]], "&&", RowBox[List[RowBox[List[RowBox[List["\[CapitalDelta]", " ", "\[Gamma]"]], "+", RowBox[List["Re", "[", "\[Nu]", "]"]], "-", FractionBox["\[CapitalEpsilon]", "2"]]], "<", RowBox[List["-", "1"]]]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["z", ">", "0"]], "&&", RowBox[List["\[CapitalEpsilon]", "\[Equal]", "0"]], "&&", RowBox[List["\[CapitalDelta]", "\[NotEqual]", "0"]], "&&", RowBox[List[RowBox[List[RowBox[List["\[CapitalDelta]", " ", "\[Gamma]"]], "+", RowBox[List["Re", "[", "\[Nu]", "]"]]]], "<", FractionBox["1", "2"]]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["z", ">", "0"]], "&&", RowBox[List["\[CapitalEpsilon]", "\[Equal]", "0"]], "&&", RowBox[List["\[CapitalDelta]", "\[Equal]", "0"]], "&&", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], "<", "0"]], "&&", RowBox[List["z", "\[NotEqual]", "1"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], "<", RowBox[List["-", "1"]]]], "&&", RowBox[List["z", "\[Equal]", "1"]]]], ")"]]]], ")"]]]], ")"]]]], ")"]]]]]]]]]]










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





2001-10-29





© 1998-2014 Wolfram Research, Inc.