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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/07.34.21.0092.01
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Integrate[\[Tau]^(\[Alpha] - 1) BesselY[\[Nu], b \[Tau]]
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]^((2 l)/k)], {\[Tau], 0, Infinity}] ==
((k^\[Mu] (2 l)^(\[Alpha] - 1))/(b^\[Alpha] (2 Pi)^(SuperStar[c] (k - 1))))
MeijerG[{{(1 - (\[Alpha] + \[Nu])/2)/l, \[Ellipsis],
(l - (\[Alpha] + \[Nu])/2)/l, (1 - (\[Alpha] - \[Nu])/2)/l,
\[Ellipsis], (l - (\[Alpha] - \[Nu])/2)/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, (3 - \[Alpha] + \[Nu])/(2 l), \[Ellipsis],
((1 - \[Alpha] + \[Nu])/2 + l)/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, (3 - \[Alpha] + \[Nu])/(2 l), \[Ellipsis],
((1 - \[Alpha] + \[Nu])/2 + l)/l}}, (\[Omega]^k (2 l)^(2 l))/
(b^(2 l) k^(k (q - p)))] /; Element[k, Integers] && k > 0 &&
Element[l, Integers] && l > 0 && GCD[k, l] == 1 &&
(\[DoubleStruckCapitalC]\[DoubleStruckCapitalC] /; \[Alpha] ->
\[Alpha]/2 && \[Sigma] == b^2/4 && s == 2 && t == 0 && u == 1 && v == 3 &&
Subscript[c, 1] == Subscript[d, 3] == (1 - \[Nu])/2 &&
Subscript[d, 1] == -(\[Nu]/2) && Subscript[d, 2] == \[Nu]/2)
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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> <mrow> <msub> <mi> Y </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> τ </mi> </mrow> <mo> ) </mo> </mrow> <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> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> l </mi> </mrow> <mi> k </mi> </mfrac> </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]", FractionBox[RowBox[List["2", " ", "l"]], "k"]]]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", 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</mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mi> k </mi> <mi> μ </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> l </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> b </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> <msup> <mi> c </mi> <mo> * </mo> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </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> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> l </mi> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> q </mi> </mrow> <mo> + </mo> <mi> l </mi> </mrow> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> , </mo> <mrow> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> l </mi> </mrow> </mrow> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mi> ω </mi> <mi> k </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> l </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> l </mi> </mrow> </msup> </mrow> <mrow> <msup> <mi> b </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> l </mi> </mrow> </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> <mfrac> <mrow> <mi> α </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mi> l </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> 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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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