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Hypergeometric Functions
MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z]
Integration
Definite integration
Integrals for classical integral transforms
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http://functions.wolfram.com/07.34.21.0088.01
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Integrate[(\[Tau]^(\[Alpha] - 1) 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)])/E^(\[Sigma] \[Tau]), {\[Tau], 0, Infinity}] ==
((k^\[Mu] l^(\[Alpha] - 1/2))/(\[Sigma]^\[Alpha]
(2 Pi)^(SuperStar[c] (k - 1) + (l - 1)/2)))
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}}, {{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 l^l)/(\[Sigma]^l k^(k (q - p)))] /;
Element[k, Integers] && k > 0 && Element[l, Integers] && l > 0 &&
GCD[k, l] == 1 && (\[DoubleStruckCapitalC]\[DoubleStruckCapitalC] /;
s == v == 1 && t == u == Subscript[d, 1] == 0)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[SuperscriptBox["\[Tau]", RowBox[List["\[Alpha]", "-", "1"]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Sigma]"]], " ", "\[Tau]"]]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "n"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "m"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", RowBox[List["m", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]]]], "}"]], ",", RowBox[List["\[Omega]", " ", SuperscriptBox["\[Tau]", FractionBox["l", "k"]]]]]], "]"]], RowBox[List["\[DifferentialD]", "\[Tau]"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["k", "\[Mu]"], SuperscriptBox["l", RowBox[List["\[Alpha]", "-", FractionBox["1", "2"]]]], SuperscriptBox["\[Sigma]", RowBox[List["-", "\[Alpha]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["2", "\[Pi]"]], ")"]], RowBox[List[RowBox[List[SuperscriptBox["c", "*"], RowBox[List["(", RowBox[List["k", "-", "1"]], ")"]]]], "+", FractionBox[RowBox[List["l", "-", "1"]], "2"]]]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Alpha]"]], "l"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["l", "-", "\[Alpha]"]], "l"], ",", FractionBox[SubscriptBox["a", "1"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "k", "-", "1"]], "k"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["a", "n"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "n"], "+", "k", "-", "1"]], "k"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "+", "k", "-", "1"]], "k"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["a", "p"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "p"], "+", "k", "-", "1"]], "k"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[SubscriptBox["b", "1"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "1"], "+", "k", "-", "1"]], "k"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["b", "m"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "m"], "+", "k", "-", "1"]], "k"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[SubscriptBox["b", RowBox[List["m", "+", "1"]]], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", RowBox[List["m", "+", "1"]]], "+", "k", "-", "1"]], "k"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["b", "q"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "q"], "+", "k", "-", "1"]], "k"]]], "}"]]]], "}"]], ",", FractionBox[RowBox[List[SuperscriptBox["\[Omega]", "k"], SuperscriptBox["l", "l"]]], RowBox[List[SuperscriptBox["\[Sigma]", "l"], SuperscriptBox["k", RowBox[List["k", RowBox[List["(", RowBox[List["q", "-", "p"]], ")"]]]]]]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["k", ">", "0"]], "\[And]", RowBox[List["l", "\[Element]", "Integers"]], "\[And]", RowBox[List["l", ">", "0"]], "\[And]", RowBox[List[RowBox[List["GCD", "[", RowBox[List["k", ",", "l"]], "]"]], "\[Equal]", "1"]], "\[And]", RowBox[List["(", RowBox[List["\[DoubleStruckCapitalC]\[DoubleStruckCapitalC]", "/;", RowBox[List[RowBox[List["s", "\[Equal]", "v", "\[Equal]", "1"]], "\[And]", RowBox[List["t", "\[Equal]", "u", "\[Equal]", SubscriptBox["d", "1"], "\[Equal]", "0"]]]]]], ")"]]]]]]]]
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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> <msup> <mi> τ </mi> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> σ </mi> </mrow> <mo> ⁢ </mo> <mi> τ </mi> </mrow> </msup> <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> ω </mi> <mo> ⁢ </mo> <msup> <mi> τ </mi> <mrow> <mi> l </mi> <mo> / </mo> <mi> k </mi> </mrow> </msup> </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["\[Omega]", " ", SuperscriptBox["\[Tau]", RowBox[List["l", "/", "k"]]]]], 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, False]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> τ </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mi> k </mi> <mi> μ </mi> </msup> <mo> ⁢ </mo> <msup> <mi> l </mi> <mrow> <mi> α </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> σ </mi> <mrow> <mo> - </mo> <mi> α </mi> </mrow> </msup> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mfrac> <mrow> <mi> l </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> c </mi> <mo> * </mo> </msup> </mrow> </mrow> </msup> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mrow> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> p </mi> </mrow> <mo> + </mo> <mi> l </mi> </mrow> <mo> , </mo> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> q </mi> </mrow> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> , </mo> <mrow> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mtext> </mtext> <mi> n </mi> </mrow> <mo> + </mo> <mi> l </mi> </mrow> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mi> ω </mi> <mi> k </mi> </msup> <mo> ⁢ </mo> <msup> <mi> l </mi> <mi> l </mi> </msup> </mrow> <mrow> <msup> <mi> σ </mi> <mi> l </mi> </msup> <mo> ⁢ </mo> <msup> <mi> k </mi> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> </mfrac> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> α </mi> </mrow> <mi> l </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> l </mi> <mo> - </mo> <mi> α </mi> </mrow> <mi> l </mi> </mfrac> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mi> n </mi> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> a </mi> <mi> n </mi> </msub> <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> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <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}
