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Hypergeometric Functions
MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z]
Integration
Definite integration
Classical Meijer's integral from two G functions
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http://functions.wolfram.com/07.34.21.0011.01
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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] w]
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]}}, \[Tau] z],
{\[Tau], 0, Infinity}] ==
MeijerG[{{Subscript[a, 1], \[Ellipsis], Subscript[a, n],
1 - \[Alpha] - Subscript[d, 1], \[Ellipsis],
1 - \[Alpha] - Subscript[d, s]}, {1 - \[Alpha] - Subscript[d, s + 1],
\[Ellipsis], 1 - \[Alpha] - Subscript[d, v], Subscript[a, n + 1],
\[Ellipsis], Subscript[a, p]}}, {{Subscript[b, 1], \[Ellipsis],
Subscript[b, m], 1 - \[Alpha] - Subscript[c, 1], \[Ellipsis],
1 - \[Alpha] - Subscript[c, t]}, {1 - \[Alpha] - Subscript[c, t + 1],
\[Ellipsis], 1 - \[Alpha] - Subscript[c, u], Subscript[b, m + 1],
\[Ellipsis], Subscript[b, q]}}, z/w]/w^\[Alpha] /;
(\[DoubleStruckCapitalC]\[DoubleStruckCapitalC] /; k == l == 1)
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mi> ∞ </mi> </msubsup> <mrow> <mrow> <msup> <mi> τ </mi> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mi> u </mi> <mo> , </mo> <mi> v </mi> </mrow> <mrow> <mi> s </mi> <mo> , </mo> <mi> t </mi> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> τ </mi> <mo> ⁢ </mo> <mi> w </mi> </mrow> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> c </mi> <mi> t </mi> </msub> <mo> , </mo> <msub> <mi> c </mi> <mrow> <mi> t </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> c </mi> <mi> u </mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi> d </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </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["\[Tau]", " ", "w"]], 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]], ",", TagBox[SubscriptBox["c", "u"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[SubscriptBox["d", "1"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["d", "s"], MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["d", RowBox[List["s", "+", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["d", "v"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> ⁢ </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> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> τ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </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> … </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> <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'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["p", ",", "q"]], RowBox[List["m", ",", "n"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["\[Tau]", " ", "z"]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[SubscriptBox["a", "1"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "n"], MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", RowBox[List["n", "+", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "p"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[SubscriptBox["b", "1"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "m"], MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", RowBox[List["m", "+", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> τ </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mi> w </mi> <mrow> <mo> - 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Date Added to functions.wolfram.com (modification date)
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| MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z,r] | | |
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){kind=small}
