Two-sided Laplace transform

Summary

The two-sided, or bilateral, Laplace transform of a function $f$ on the whole real line is $\int_{-\infty}^{\infty} f(t) e^{-s t} \, dt$. It treats past and future times alike, unlike the one-sided Laplace transform, which only sees $t \ge 0$. It converges in a vertical strip of the complex plane, and on the imaginary axis it becomes the Fourier transform. In probability it gives the moment generating function, with $s$ replaced by $-s$.