Meijer G-function: Integration (formula 07.34.21.0089)

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

Integrals for classical integral transforms

http://functions.wolfram.com/07.34.21.0089.01

Input Form

Integrate[\[Tau]^(\[Alpha] - 1) Sin[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] - 1/2))/(b^\[Alpha] (2 (2 Pi)^(SuperStar[c] (k - 1) - 1/2)))) MeijerG[{{(1 - \[Alpha])/(2 l), \[Ellipsis], ((-1 - \[Alpha])/2 + l)/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, (1 - \[Alpha]/2)/l, \[Ellipsis], (l - \[Alpha]/2)/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 (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 == 1 && t == u == Subscript[d, 2] == 0 && v == 2 && Subscript[d, 1] == 1/2)

Standard Form

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MathML Form

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</mi> <mo> ⩵ </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> </mrow> <mo> ∧ </mo> <mrow> <mi> s </mi> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> t </mi> <mo> ⩵ </mo> <mi> u </mi> <mo> ⩵ </mo> <msub> <mi> d </mi> <mn> 2 </mn> </msub> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> v </mi> <mo> ⩵ </mo> <mn> 2 </mn> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> d </mi> <mn> 1 </mn> </msub> <mo> ⩵ </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </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

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