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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 > Integrals including G functions with rational arguments





http://functions.wolfram.com/07.34.21.0096.01









  


  










Input Form





Integrate[(\[Tau]^(\[Alpha] - 1)/(\[Tau] + z)^\[Beta]) 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] + z)^l)/\[Tau]^k], {\[Tau], 0, Infinity}] == ((Sqrt[2 Pi] k^(\[Alpha] - 1/2) l^(1/2 - \[Beta]) z^(\[Alpha] - \[Beta]))/ (k - l)^(\[Alpha] - \[Beta] + 1/2)) MeijerG[{{(\[Alpha] - \[Beta] + 1)/(k - l), \[Ellipsis], (\[Alpha] - \[Beta] + k - l)/(k - l), Subscript[a, 1], \[Ellipsis], Subscript[a, n]}, {Subscript[a, n + 1], \[Ellipsis], Subscript[a, p], \[Beta]/l, \[Ellipsis], (\[Beta] + l - 1)/l}}, {{\[Alpha]/k, \[Ellipsis], (\[Alpha] + k - 1)/k, Subscript[b, 1], \[Ellipsis], Subscript[b, m]}, {Subscript[b, m + 1], \[Ellipsis], Subscript[b, q]}}, ((\[Omega] l^l)/k^k) (z/(k - l))^(l - k)] /; Element[k, Integers] && k > 0 && Element[l, Integers] && l > 0 && k > l && SuperStar[c] > 0 && Abs[Arg[\[Omega] z^(l - k)]] < Pi SuperStar[c] && (k - l) Min[Re[Subscript[b, 1]], \[Ellipsis], Re[Subscript[b, m]]] + Re[\[Beta] - \[Alpha]] > 0 && k - k Max[Re[Subscript[a, 1]], \[Ellipsis], Re[Subscript[a, n]]] + Re[\[Alpha]] > 0










Standard Form





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MathML Form







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</mo> <mo> ( </mo> <mrow> <mi> &#946; </mi> <mo> - </mo> <mi> &#945; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> k </mi> <mo> - </mo> <mrow> <mi> k </mi> <mo> &#8290; </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> </mrow> <mo> + </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mi> &#8734; </mi> </msubsup> <mrow> <mfrac> <mrow> <msup> <mi> &#964; </mi> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mtext> </mtext> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mi> &#964; </mi> </mrow> <mo> ) </mo> </mrow> <mi> &#946; </mi> </msup> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mi> p </mi> <mo> , </mo> <mi> q </mi> </mrow> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> </msubsup> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> &#969; </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mi> &#964; </mi> </mrow> <mo> ) </mo> </mrow> <mi> l </mi> </msup> </mrow> <msup> <mi> &#964; </mi> <mi> k </mi> </msup> </mfrac> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> a </mi> <mi> n </mi> </msub> <mo> , </mo> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> a </mi> <mi> p </mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> m </mi> </msub> <mo> , </mo> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[&quot;p&quot;, &quot;,&quot;, &quot;q&quot;]], RowBox[List[&quot;m&quot;, &quot;,&quot;, &quot;n&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[FractionBox[RowBox[List[&quot;\[Omega]&quot;, &quot; &quot;, SuperscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;z&quot;, &quot;+&quot;, &quot;\[Tau]&quot;]], &quot;)&quot;]], &quot;l&quot;]]], SuperscriptBox[&quot;\[Tau]&quot;, &quot;k&quot;]], MeijerG, Rule[Editable, True]], &quot;\[VerticalSeparator]&quot;, GridBox[List[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;n&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[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, False]] </annotation> </semantics> <mo> &#8290; 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</mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mfrac> <mrow> <mi> &#969; </mi> <mo> &#8290; </mo> <msup> <mi> l </mi> <mi> l </mi> </msup> </mrow> <msup> <mi> k </mi> <mi> k </mi> </msup> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mrow> <mi> k </mi> <mo> - </mo> <mi> l </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> l </mi> <mo> - </mo> <mi> k </mi> </mrow> </msup> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[FractionBox[RowBox[List[&quot;\[Omega]&quot;, &quot; &quot;, SuperscriptBox[&quot;l&quot;, &quot;l&quot;]]], SuperscriptBox[&quot;k&quot;, &quot;k&quot;]], SuperscriptBox[RowBox[List[&quot;(&quot;, FractionBox[&quot;z&quot;, RowBox[List[&quot;k&quot;, &quot;-&quot;, &quot;l&quot;]]], &quot;)&quot;]], RowBox[List[&quot;l&quot;, &quot;-&quot;, &quot;k&quot;]]]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mrow> <semantics> <mfrac> <mrow> <mi> &#945; </mi> <mo> - </mo> <mi> &#946; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mi> l </mi> </mrow> </mfrac> <annotation encoding='Mathematica'> TagBox[FractionBox[RowBox[List[&quot;\[Alpha]&quot;, &quot;-&quot;, &quot;\[Beta]&quot;, &quot;+&quot;, &quot;1&quot;]], RowBox[List[&quot;k&quot;, &quot;-&quot;, &quot;l&quot;]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mo> &#8230; </mo> <annotation encoding='Mathematica'> TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mfrac> <mrow> <mi> &#945; </mi> <mo> - </mo> <mi> &#946; </mi> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mi> l </mi> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mi> l </mi> </mrow> </mfrac> <annotation encoding='Mathematica'> TagBox[FractionBox[RowBox[List[&quot;\[Alpha]&quot;, &quot;-&quot;, &quot;\[Beta]&quot;, &quot;+&quot;, &quot;k&quot;, &quot;-&quot;, &quot;l&quot;]], RowBox[List[&quot;k&quot;, &quot;-&quot;, &quot;l&quot;]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[&quot;a&quot;, &quot;1&quot;], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mo> &#8230; </mo> <annotation encoding='Mathematica'> TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <msub> <mi> a </mi> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[&quot;a&quot;, &quot;n&quot;], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[&quot;a&quot;, RowBox[List[&quot;n&quot;, &quot;+&quot;, &quot;1&quot;]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mo> &#8230; </mo> <annotation encoding='Mathematica'> TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <msub> <mi> a </mi> <mi> p </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[&quot;a&quot;, &quot;p&quot;], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mfrac> <mi> &#946; 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</mo> <annotation encoding='Mathematica'> TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mfrac> <mrow> <mi> &#945; </mi> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <annotation encoding='Mathematica'> TagBox[FractionBox[RowBox[List[&quot;\[Alpha]&quot;, &quot;+&quot;, &quot;k&quot;, &quot;-&quot;, &quot;1&quot;]], &quot;k&quot;], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[&quot;b&quot;, &quot;1&quot;], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mo> &#8230; </mo> <annotation encoding='Mathematica'> TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <msub> <mi> b </mi> <mi> m </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[&quot;b&quot;, &quot;m&quot;], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[&quot;b&quot;, RowBox[List[&quot;m&quot;, &quot;+&quot;, &quot;1&quot;]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mo> &#8230; </mo> <annotation encoding='Mathematica'> TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <msub> <mi> b </mi> <mi> q </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[&quot;b&quot;, &quot;q&quot;], MeijerG, Rule[Editable, True]] </annotation> </semantics> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> l </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> k </mi> <mo> &gt; </mo> <mi> l </mi> </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> <mrow> <mi> &#969; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mi> l </mi> <mo> - </mo> <mi> k </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <msup> <mi> c </mi> <mo> * </mo> </msup> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> l </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </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> + </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#946; </mi> <mo> - </mo> <mi> &#945; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> k </mi> <mo> - </mo> <mrow> <mi> k </mi> <mo> &#8290; </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> </mrow> <mo> + </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29