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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $f \\colon (0, \\infty)^n \\to \\C$ be locally integrable.\nFor $s = (s_1, \\dots, s_n) \\in \\C^n$ and $x \\in (0, \\infty)^n$ write $x^{s - 1} = x_1^{s_1 - 1} \\cdots x_n^{s_n - 1}$.\nThe *$n$-dimensional Mellin transform* of $f$ is the function\n$$(\\mathcal{M} f)(s) = \\int_{(0, \\infty)^n} x^{s - 1} f(x)\\, dx,$$\ndefined at those $s \\in \\C^n$ where the integral converges absolutely.\n\n## Remarks\n\nFor $n = 1$ this is the [[Mellin transform|N5PVGRJE1T]].\nAbsolute convergence at $s$ depends only on $(\\operatorname{Re} s_1, \\dots, \\operatorname{Re} s_n)$, and the set of such real parts is convex.\n\nExample: $f(x) = e^{-x_1 - \\cdots - x_n}$.\nThe integral splits into a product, and $(\\mathcal{M} f)(s) = \\Gamma(s_1) \\cdots \\Gamma(s_n)$ for $\\operatorname{Re} s_j \u003e 0$, by the [[gamma function|0H5SYBSED9]] integral.\n\nNon-example: $f = 1$ has no Mellin transform.\nFor every $s$, the factor $\\int_0^\\infty x_1^{\\operatorname{Re} s_1 - 1}\\, dx_1$ diverges, at $0$ or at $\\infty$.\n",
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