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  "body": "\n## Summary\n\nThe multidimensional Laplace transform takes a function of $n$ variables on the orthant $[0, \\infty)^n$ to a function of $n$ complex variables, by integrating against $e^{-s_1 x_1 - \\cdots - s_n x_n}$.\nIt is the [[Laplace transform|0V49JEZW06]] applied in each variable in turn.\nIt turns partial derivatives into multiplication by the variables $s_j$, up to boundary terms.\nSo it is used for linear partial differential equations with constant coefficients.\nIn probability it gives the joint Laplace transform of nonnegative random vectors.\n\n## Explanations\n\n### L1 human\n\nThe ordinary Laplace transform weighs a function of time by a decaying exponential.\nIn several variables one uses one decay rate for each variable.\nFor example, for $f(x_1, x_2) = 1$ on the quarter plane the transform is $\\int_0^\\infty \\int_0^\\infty e^{-s_1 x_1 - s_2 x_2}\\, dx_1\\, dx_2 = \\frac{1}{s_1 s_2}$ for $s_1, s_2 \u003e 0$.\nA product of functions of single variables becomes a product of one-variable Laplace transforms.\n",
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