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  "body": "\n## Definition\n\nLet $a_1, \\dots, a_m$, $b_1, \\dots, b_n$, $c_1, \\dots, c_p$ and $d_1, \\dots, d_q$ be complex numbers, and let $\\Gamma$ be the [[gamma function|0H5SYBSED9]].\nAssume that $a_j + b_k$ is not in $\\set{0, -1, -2, \\dots}$ for all $j$ and $k$.\nThen the poles $s = -a_j - \\ell$ of the factors $\\Gamma(a_j + s)$ are disjoint from the poles $s = b_k + \\ell$ of the factors $\\Gamma(b_k - s)$, where $\\ell \\in \\N$.\nLet $L$ be a path from $-i\\infty$ to $+i\\infty$.\nAssume that all poles of the first family lie on its left and all poles of the second family on its right.\nFix $z \\ne 0$ and a branch of $\\log z$, and put $z^{-s} = e^{-s \\log z}$.\nA *Barnes integral* is a [[contour integral|06ZGPYTWK8]] of the form\n$$\\frac{1}{2\\pi i} \\int_L \\frac{\\prod_{j=1}^{m} \\Gamma(a_j + s) \\prod_{k=1}^{n} \\Gamma(b_k - s)}{\\prod_{j=1}^{p} \\Gamma(c_j + s) \\prod_{k=1}^{q} \\Gamma(d_k - s)}\\, z^{-s}\\, ds,$$\nfor those $z$ for which it converges.\n\n## Remarks\n\nExample: take $m = 1$, $a_1 = 0$, no other factors, $x \u003e 0$, and $L$ the line $\\operatorname{Re} s = c$ with $c \u003e 0$.\nAll poles $s = 0, -1, -2, \\dots$ of $\\Gamma(s)$ lie to the left of $L$, so $L$ is admissible.\nShifting the line to the left and summing the residues $(-1)^\\ell x^\\ell / \\ell!$ at $s = -\\ell$ gives\n$$\\frac{1}{2\\pi i} \\int_{c - i\\infty}^{c + i\\infty} \\Gamma(s)\\, x^{-s}\\, ds = e^{-x}.$$\nNon-example: the same integrand on the line $\\operatorname{Re} s = -\\tfrac12$ is not a Barnes integral in this sense, since the pole $s = 0$ lies on the wrong side.\nIts value is $e^{-x} - 1$, because the residue $1$ at $s = 0$ is lost.\nBarnes' representation of the [[hypergeometric function|463X5V31D9]] reads, for $\\abs{\\arg(-z)} \u003c \\pi$ and $a, b$ not in $\\set{0, -1, -2, \\dots}$,\n$$\\frac{\\Gamma(a)\\Gamma(b)}{\\Gamma(c)}\\, {}_2F_1(a, b; c; z) = \\frac{1}{2\\pi i} \\int_{-i\\infty}^{i\\infty} \\frac{\\Gamma(a + s)\\Gamma(b + s)\\Gamma(-s)}{\\Gamma(c + s)} (-z)^s\\, ds,$$\nwith the path separating the poles of $\\Gamma(a + s)\\Gamma(b + s)$ from those of $\\Gamma(-s)$.\nThe Meijer G-function and the Fox H-function are defined by integrals of this type.\n",
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