Multidimensional Laplace transform

Summary

The 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 $e^{-s_1 x_1 - \cdots - s_n x_n}$. It is the Laplace transform applied in each variable in turn. It turns partial derivatives into multiplication by the variables $s_j$, up to boundary terms. So it is used for linear partial differential equations with constant coefficients. In probability it gives the joint Laplace transform of nonnegative random vectors.