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variants of this functions
Gamma




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Gamma[z] > Integral transforms > Inverse Mellin transforms





http://functions.wolfram.com/06.05.22.0005.01









  


  








Input Form





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








Standard Form





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MathML Form







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</mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </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> s </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mfrac> <mo> ] </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </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> <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> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <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> t </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> t </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> <mo> - </mo> <mfrac> <mi> &#917; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> + </mo> <mrow> <mi> &#916; </mi> <mo> &#8290; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> </mrow> <mo> &lt; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mi> t </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> &#916; </mi> <mo> &#8290; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> </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> t </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> t </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> t </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





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





2001-10-29