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variants of this functions
MeijerG






Mathematica Notation

Traditional Notation









Hypergeometric Functions > MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z] > Integration > Definite integration > Integral including G functions with shifted arguments





http://functions.wolfram.com/07.34.21.0082.01









  


  










Input Form





Integrate[\[Tau]^(\[Alpha] - 1) MeijerG[{{Subscript[c, 1], \[Ellipsis], Subscript[c, t]}, {Subscript[c, t + 1], \[Ellipsis], Subscript[c, u]}}, {{Subscript[d, 1], \[Ellipsis], Subscript[d, s]}, {Subscript[d, s + 1], \[Ellipsis], Subscript[d, v]}}, \[Tau] + \[Sigma]] MeijerG[{{Subscript[a, 1], \[Ellipsis], Subscript[a, n]}, {Subscript[a, n + 1], \[Ellipsis], Subscript[a, p]}}, {{Subscript[b, 1], \[Ellipsis], Subscript[b, m]}, {Subscript[b, m + 1], \[Ellipsis], Subscript[b, q]}}, \[Omega] \[Tau]], {\[Tau], 0, Infinity}] == Sum[((-\[Sigma])^k/k!) MeijerG[{{1 - \[Alpha], Subscript[a, 1], \[Ellipsis], Subscript[a, n], 1 + k - \[Alpha] - Subscript[d, 1], \[Ellipsis], 1 + k - \[Alpha] - Subscript[d, s]}, {1 + k - \[Alpha] - Subscript[d, s + 1], \[Ellipsis], 1 + k - \[Alpha] - Subscript[d, v], Subscript[a, n + 1], \[Ellipsis], Subscript[a, p]}}, {{Subscript[b, 1], \[Ellipsis], Subscript[b, m], 1 + k - \[Alpha] - Subscript[c, 1], \[Ellipsis], 1 + k - \[Alpha] - Subscript[c, t]}, {1 + k - \[Alpha] - Subscript[c, t + 1], \[Ellipsis], 1 + k - \[Alpha] - Subscript[c, u], 1 + k - \[Alpha], Subscript[b, m + 1], \[Ellipsis], Subscript[b, q]}}, \[Omega]], {k, 0, Infinity}] /; SuperStar[b] > 0 && SuperStar[c] > 0 && Abs[Arg[\[Sigma]]] < Pi && Abs[Arg[\[Omega]]] < SuperStar[c] Pi && -Min[Re[Subscript[b, 1]], \[Ellipsis], Re[Subscript[b, m]]] < Re[\[Alpha]] < 2 - Max[Re[Subscript[a, 1]], \[Ellipsis], Re[Subscript[a, n]]] - Max[Re[Subscript[c, 1]], \[Ellipsis], Re[Subscript[c, t]]]










Standard Form





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MathML Form







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&quot;-&quot;, &quot;\[Alpha]&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, 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;n&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;k&quot;, &quot;-&quot;, &quot;\[Alpha]&quot;, &quot;-&quot;, SubscriptBox[&quot;d&quot;, &quot;1&quot;], &quot;+&quot;, &quot;1&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;k&quot;, &quot;-&quot;, &quot;\[Alpha]&quot;, &quot;-&quot;, SubscriptBox[&quot;d&quot;, &quot;s&quot;], &quot;+&quot;, &quot;1&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;k&quot;, &quot;-&quot;, &quot;\[Alpha]&quot;, &quot;-&quot;, SubscriptBox[&quot;d&quot;, RowBox[List[&quot;s&quot;, &quot;+&quot;, &quot;1&quot;]]], &quot;+&quot;, &quot;1&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;k&quot;, &quot;-&quot;, &quot;\[Alpha]&quot;, &quot;-&quot;, SubscriptBox[&quot;d&quot;, &quot;v&quot;], &quot;+&quot;, &quot;1&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;a&quot;, RowBox[List[&quot;n&quot;, &quot;+&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;p&quot;], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[SubscriptBox[&quot;b&quot;, &quot;1&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;b&quot;, &quot;m&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;k&quot;, &quot;-&quot;, &quot;\[Alpha]&quot;, &quot;-&quot;, SubscriptBox[&quot;c&quot;, &quot;1&quot;], &quot;+&quot;, &quot;1&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;k&quot;, &quot;-&quot;, &quot;\[Alpha]&quot;, &quot;-&quot;, SubscriptBox[&quot;c&quot;, &quot;t&quot;], &quot;+&quot;, &quot;1&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;k&quot;, &quot;-&quot;, &quot;\[Alpha]&quot;, &quot;-&quot;, SubscriptBox[&quot;c&quot;, RowBox[List[&quot;t&quot;, &quot;+&quot;, &quot;1&quot;]]], &quot;+&quot;, &quot;1&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;k&quot;, &quot;-&quot;, &quot;\[Alpha]&quot;, &quot;-&quot;, SubscriptBox[&quot;c&quot;, &quot;u&quot;], &quot;+&quot;, &quot;1&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;k&quot;, &quot;-&quot;, &quot;\[Alpha]&quot;, &quot;+&quot;, &quot;1&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;b&quot;, RowBox[List[&quot;m&quot;, &quot;+&quot;, &quot;1&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;b&quot;, &quot;q&quot;], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msup> <mi> b </mi> <mo> * </mo> </msup> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msup> <mi> c </mi> <mo> * </mo> </msup> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#963; </mi> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mi> &#960; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#969; </mi> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mrow> <msup> <mi> c </mi> <mo> * </mo> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> </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> 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> m </mi> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &lt; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mrow> <mi> max </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> n </mi> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> max </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> c </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> c </mi> <mi> t </mi> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29





© 1998- Wolfram Research, Inc.