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  "body": "\n## Definition\n\nLet $\\Gamma \\subseteq \\R^n$ be a closed [[convex cone|6VE4KA12VD]] with nonempty interior, and let $\\ip{\\cdot}{\\cdot}$ be the standard inner product on $\\R^n$.\nLet $\\Gamma^* = \\set{y \\in \\R^n : \\ip{y}{x} \\ge 0 \\text{ for all } x \\in \\Gamma}$ be its [[dual cone|QFY1JR1TK4]], and let $\\Omega$ be the interior of $\\Gamma^*$.\nLet $f \\colon \\Gamma \\to \\C$ be locally integrable.\nThe *Laplace transform* of $f$ on $\\Gamma$ is the function\n$$(\\mathcal{L}_\\Gamma f)(z) = \\int_\\Gamma e^{-\\ip{z}{x}} f(x)\\, dx, \\qquad z = y + i\\eta \\in \\C^n,$$\ndefined at those $z$ where the integral converges absolutely; here $\\ip{z}{x} = \\sum_j z_j x_j$.\n\n## Remarks\n\nIf $y \\in \\Omega$, there is $c \u003e 0$ with $\\ip{y}{x} \\ge c \\abs{x}$ for all $x \\in \\Gamma$, by compactness of $\\Gamma \\cap \\set{\\abs{x} = 1}$.\nSo if $\\abs{f(x)} \\le C (1 + \\abs{x})^N$, the transform converges absolutely on the tube $\\Omega + i \\R^n$ and is holomorphic there.\n\nExample: $\\Gamma = [0, \\infty)^n$ is self-dual, and $\\mathcal{L}_\\Gamma$ is the [[multidimensional Laplace transform|QT3M720TCV]].\nFor $f = 1$ one gets $\\mathcal{L}_\\Gamma f(z) = 1/(z_1 \\cdots z_n)$ on $\\operatorname{Re} z_j \u003e 0$.\n\nExample: let $\\Gamma$ be the cone of positive semidefinite real symmetric $m \\times m$ matrices, with $\\ip{y}{x} = \\tr(yx)$.\nLet $dx$ be the Lebesgue measure in the entries $x_{ij}$, $i \\le j$.\nThe cone is self-dual, and $\\Omega$ is the cone of [[positive definite matrices|0KX66CZMBB]].\nFor real $s \u003e (m - 1)/2$ and $y \\in \\Omega$,\n$$\\int_\\Omega e^{-\\tr(yx)} (\\det x)^{s - (m+1)/2}\\, dx = \\Gamma_m(s)\\, (\\det y)^{-s}, \\qquad \\Gamma_m(s) = \\pi^{m(m-1)/4} \\prod_{j=0}^{m-1} \\Gamma\\left(s - \\frac{j}{2}\\right).$$\n\nNon-example: let $\\Gamma$ contain a line $\\R v$.\nThen $\\Gamma^*$ lies in the hyperplane $v^\\perp$ and $\\Omega$ is empty.\nSince $\\Gamma + \\R v = \\Gamma$, Fubini's theorem along the direction $v$ shows that even $f = 1$ has a divergent transform at every $z$.\n",
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