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  "body": "\n## Summary\n\nA Barnes integral, also called a Mellin–Barnes integral, is a kind of [[contour integral|06ZGPYTWK8]].\nIts integrand is a quotient of products of values of the [[gamma function|0H5SYBSED9]], times a power $z^{-s}$.\nThe path runs from $-i\\infty$ to $+i\\infty$ and separates two families of poles of the integrand.\nMany special functions, such as the [[hypergeometric function|463X5V31D9]], have such an integral as a representation.\nClosing the path to the left or to the right and summing residues turns the integral into a series.\nThis gives convergent series and asymptotic expansions.\n\n## Explanations\n\n### L1 human\n\nThe gamma function has poles at $0, -1, -2, \\dots$, and a Barnes integral is built so that these poles sit on known sides of a vertical path.\nWhen one pushes the path far to one side, the [[residue theorem|032HKYS329]] collects one term of a series at each pole that the path crosses.\nSo a single integral can stand for a power series near $z = 0$ and for another expansion near $z = \\infty$.\nThe smallest example is $\\frac{1}{2\\pi i} \\int_{c - i\\infty}^{c + i\\infty} \\Gamma(s)\\, x^{-s}\\, ds$ with $c \u003e 0$ and $x \u003e 0$.\nThe poles of $\\Gamma(s)$ at $s = -n$ have residues $(-1)^n / n!$, and their sum gives $e^{-x}$.\n",
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