Multidimensional Laplace transform

Remarks

For $n = 1$ this is the Laplace transform. If the integral converges absolutely at $s$, it converges absolutely at every $s'$ with $\operatorname{Re} s'_j \ge \operatorname{Re} s_j$ for all $j$.

Example: $f(x_1, x_2) = e^{a x_1 + b x_2}$ with real $a, b$. For $\operatorname{Re} s_1 > a$ and $\operatorname{Re} s_2 > b$ the integral splits into a product, and $F(s_1, s_2) = \frac{1}{(s_1 - a)(s_2 - b)}$.

Non-example: $f(x_1, x_2) = e^{x_1 x_2}$ has no Laplace transform. For any $s$, let $c = \abs{s_1} + \abs{s_2}$. If $x_1, x_2 \ge c$, then $x_1 x_2 - \operatorname{Re}(s_1) x_1 - \operatorname{Re}(s_2) x_2 \ge x_1 x_2 - c x_1 - c x_2 = (x_1 - c)(x_2 - c) - c^2$. So $\abs{e^{-s \cdot x} f(x)} \ge 1$ on the set where $(x_1 - c)(x_2 - c) \ge c^2$, which has infinite area, and the integral does not converge absolutely.

The transform is often written $\mathcal{L}_n f$, and the variables are kept real in many applications.