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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 $s \\cdot x = s_1 x_1 + \\cdots + s_n x_n$.\nThe *$n$-dimensional Laplace transform* of $f$ is the function\n$$F(s) = \\int_{[0, \\infty)^n} e^{-s \\cdot x} f(x)\\, dx,$$\ndefined at those $s \\in \\C^n$ where the integral converges absolutely.\n\n## Remarks\n\nFor $n = 1$ this is the [[Laplace transform|0V49JEZW06]].\nIf the integral converges absolutely at $s$, it converges absolutely at every $s'$ with $\\operatorname{Re} s'_j \\ge \\operatorname{Re} s_j$ for all $j$.\n\nExample: $f(x_1, x_2) = e^{a x_1 + b x_2}$ with real $a, b$.\nFor $\\operatorname{Re} s_1 \u003e a$ and $\\operatorname{Re} s_2 \u003e b$ the integral splits into a product, and $F(s_1, s_2) = \\frac{1}{(s_1 - a)(s_2 - b)}$.\n\nNon-example: $f(x_1, x_2) = e^{x_1 x_2}$ has no Laplace transform.\nFor any $s$, let $c = \\abs{s_1} + \\abs{s_2}$.\nIf $x_1, x_2 \\ge c$, then $x_1 x_2 - \\operatorname{Re}(s_1) x_1 - \\operatorname{Re}(s_2) x_2 \\ge x_1 x_2 - c x_1 - c x_2 = (x_1 - c)(x_2 - c) - c^2$.\nSo $\\abs{e^{-s \\cdot x} f(x)} \\ge 1$ on the set where $(x_1 - c)(x_2 - c) \\ge c^2$, which has infinite area, and the integral does not converge absolutely.\n\nThe transform is often written $\\mathcal{L}_n f$, and the variables are kept real in many applications.\n",
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