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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 for classical integral transforms





http://functions.wolfram.com/07.34.21.0086.01









  


  










Input Form





Integrate[(\[Tau]^(\[Alpha] - 1)/(\[Tau] + z)^\[Lambda]) 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]^(l/k)], {\[Tau], 0, Infinity}] == ((k^\[Mu] l^(\[Lambda] - 1) z^(\[Alpha] - \[Lambda]))/ ((2 Pi)^(SuperStar[c] (k - 1) + l - 1) Gamma[\[Lambda]])) MeijerG[{{(1 - \[Alpha])/l, \[Ellipsis], (l - \[Alpha])/l, Subscript[a, 1]/k, \[Ellipsis], (Subscript[a, 1] + k - 1)/k, \[Ellipsis], Subscript[a, n]/k, \[Ellipsis], (Subscript[a, n] + k - 1)/ k}, {Subscript[a, n + 1]/k, \[Ellipsis], (Subscript[a, n + 1] + k - 1)/ k, \[Ellipsis], Subscript[a, p]/k, \[Ellipsis], (Subscript[a, p] + k - 1)/k}}, {{(\[Lambda] - \[Alpha])/l, \[Ellipsis], (\[Lambda] - \[Alpha] + l - 1)/l, Subscript[b, 1]/k, \[Ellipsis], (Subscript[b, 1] + k - 1)/k, \[Ellipsis], Subscript[b, m]/k, \[Ellipsis], (Subscript[b, m] + k - 1)/k}, {Subscript[b, m + 1]/k, \[Ellipsis], (Subscript[b, m + 1] + k - 1)/k, \[Ellipsis], Subscript[b, q]/k, \[Ellipsis], (Subscript[b, q] + k - 1)/k}}, (\[Omega]^k z^l)/k^(k (q - p))] /; Element[k, Integers] && k > 0 && Element[l, Integers] && l > 0 && GCD[k, l] == 1 && (\[DoubleStruckCapitalC]\[DoubleStruckCapitalC] /; \[Sigma] == z^(-1) && s == t == u == v == 1 && Subscript[c, 1] == 1 - \[Lambda] && Subscript[d, 1] == 0)










Standard Form





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MathML Form







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</mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <mo> ) </mo> </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> &#955; </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> <mrow> <mi> &#969; </mi> <mo> &#8290; </mo> <msup> <mi> &#964; </mi> <mrow> <mi> l </mi> <mo> / </mo> <mi> k </mi> </mrow> </msup> </mrow> <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; 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</mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mi> n </mi> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mi> n </mi> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mi> p </mi> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mi> &#955; </mi> <mo> - </mo> <mi> &#945; </mi> </mrow> <mi> l </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <mrow> <mi> &#955; </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mi> l </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> l </mi> </mfrac> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mi> m </mi> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mi> m </mi> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mi> q </mi> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> </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[RowBox[List[&quot;k&quot;, &quot; &quot;, &quot;p&quot;]], &quot;+&quot;, &quot;l&quot;]], &quot;,&quot;, RowBox[List[RowBox[List[&quot;k&quot;, &quot; &quot;, &quot;q&quot;]], &quot;+&quot;, &quot;l&quot;]]]], RowBox[List[RowBox[List[RowBox[List[&quot;k&quot;, &quot; &quot;, &quot;m&quot;]], &quot;+&quot;, &quot;l&quot;]], &quot;,&quot;, RowBox[List[RowBox[List[&quot;k&quot;, &quot; &quot;, &quot;n&quot;]], &quot;+&quot;, &quot;l&quot;]]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[FractionBox[RowBox[List[SuperscriptBox[&quot;\[Omega]&quot;, &quot;k&quot;], &quot; 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</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> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> , </mo> <mi> l </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#8450;&#8450; </mi> <mo> /; </mo> <mrow> <mrow> <mi> &#963; </mi> <mo> &#10869; </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <mi> s </mi> <mo> &#10869; </mo> <mi> t </mi> <mo> &#10869; </mo> <mi> u </mi> <mo> &#10869; </mo> <mi> v </mi> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> &#10869; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#955; </mi> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> d </mi> <mn> 1 </mn> </msub> <mo> &#10869; </mo> <mn> 0 </mn> </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





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