Meijer G-function: Integration (formula 07.34.21.0082)

MeijerG

$Mathematica Notation$

Hypergeometric Functions

MeijerG[{{a₁,...,a_n},{a_n+1,...,a_p}},{{b₁,...,b_m},{b_m+1,...,b_q}},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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</mo> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mi> Re </mi> <mo> ⁡ </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> ⁡ </mo> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mrow> <mi> max </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mi> Re </mi> <mo> ⁡ </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> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> c </mi> <mi> t </mi> </msub> <mo> ) </mo> 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</mo> <mo> , </mo> <msub> <mi> d </mi> <mi> s </mi> </msub> <mo> , </mo> <msub> <mi> d </mi> <mrow> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> d </mi> <mi> v </mi> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["u", ",", "v"]], RowBox[List["s", ",", "t"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["\[Sigma]", "+", "\[Tau]"]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[SubscriptBox["c", "1"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["c", "t"], MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["c", RowBox[List["t", "+", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", 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<mi> d </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> α </mi> <mo> - </mo> <msub> <mi> d </mi> <mi> s </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> α </mi> <mo> - </mo> <msub> <mi> d </mi> <mrow> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> α </mi> <mo> - </mo> <msub> <mi> d </mi> <mi> v </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </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> … </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> m </mi> </msub> <mo> , </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> α </mi> <mo> - </mo> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> α </mi> <mo> - </mo> <msub> <mi> c </mi> <mi> t </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> α </mi> <mo> - </mo> <msub> <mi> c </mi> <mrow> <mi> t </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> α </mi> <mo> - </mo> <msub> <mi> c </mi> <mi> u </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> 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</mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>

Rule Form

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

2001-10-29

MeijerG[{{a₁,...,a_n},{a_n+1,...,a_p}},{{b₁,...,b_m},{b_m+1,...,b_q}},z,r]