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  "body": "\n## Summary\n\nThe Laplace transform on a convex cone integrates a function on a closed [[convex cone|6VE4KA12VD]] $\\Gamma \\subseteq \\R^n$ against $e^{-\\ip{y}{x}}$.\nHere $y$ runs over the interior of the [[dual cone|QFY1JR1TK4]].\nFor the orthant $\\Gamma = [0, \\infty)^n$ it is the [[multidimensional Laplace transform|QT3M720TCV]].\nFor a function of moderate growth, the transform extends to a holomorphic function on a tube domain over the interior of the dual cone.\nOn the cone of positive definite matrices it gives the multivariate gamma function, and it is a basic tool of analysis on symmetric cones.\n\n## Explanations\n\n### L1 human\n\nThe ordinary Laplace transform lives on the half-line $[0, \\infty)$, and the exponential $e^{-yx}$ decays there when $y \u003e 0$.\nIn several variables the half-line is replaced by a cone, such as a quarter plane or the set of positive definite matrices.\nThe exponential $e^{-\\ip{y}{x}}$ decays along the whole cone exactly when $y$ points into the dual cone.\nFor example, on the quarter plane with $f = 1$ the transform is $1/(y_1 y_2)$ for $y_1, y_2 \u003e 0$.\n",
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