Meijer G-function: Integration (formula 07.34.21.0013)

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

Generalization of classical Meijer's integral from two G functions

Main formulas

http://functions.wolfram.com/07.34.21.0013.01

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]}}, \[Sigma] \[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]^(l/k)], {\[Tau], 0, Infinity}] == (((k^\[Mu] l^(\[Rho] + \[Alpha] (v - u) - 1))/ (2 Pi)^(SuperStar[b] (l - 1) + SuperStar[c] (k - 1))) MeijerG[{{Subscript[a, 1]/k, \[Ellipsis], (Subscript[a, 1] + k - 1)/k, \[Ellipsis], Subscript[a, n]/k, \[Ellipsis], (Subscript[a, n] + k - 1)/ k, (1 - \[Alpha] - Subscript[d, 1])/l, \[Ellipsis], (l - \[Alpha] - Subscript[d, 1])/l, \[Ellipsis], (1 - \[Alpha] - Subscript[d, s])/l, \[Ellipsis], (l - \[Alpha] - Subscript[d, s])/l}, {(1 - \[Alpha] - Subscript[d, s + 1])/l, \[Ellipsis], (l - \[Alpha] - Subscript[d, s + 1])/l, \[Ellipsis], (1 - \[Alpha] - Subscript[d, v])/l, \[Ellipsis], (l - \[Alpha] - Subscript[d, v])/l, 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, (1 - \[Alpha] - Subscript[c, 1])/l, \[Ellipsis], (l - \[Alpha] - Subscript[c, 1])/l, \[Ellipsis], (1 - \[Alpha] - Subscript[c, t])/l, \[Ellipsis], (l - \[Alpha] - Subscript[c, t])/l}, {(1 - \[Alpha] - Subscript[c, t + 1])/l, \[Ellipsis], (l - \[Alpha] - Subscript[c, t + 1])/l, \[Ellipsis], (1 - \[Alpha] - Subscript[c, u])/l, \[Ellipsis], (l - \[Alpha] - Subscript[c, u])/l, 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 k^(k (p - q)))/(\[Sigma]^l l^(l (u - v)))])/ \[Sigma]^\[Alpha] /; Element[k, Integers] && k > 0 && Element[l, Integers] && l > 0 && GCD[k, l] == 1 && \[DoubleStruckCapitalC]\[DoubleStruckCapitalC]

Standard Form

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MathML Form

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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]