1.59 Laplace transforms

Let $f:(0,\infty)\rightarrow{\mathbb{R}}$ be a continuous function.

The Laplace transform is defined for suitable values of the parameter $s$, and is often denoted ${\cal L}[f](s)$. Such integrals are widely used to solve differential equations; see §6. The term transfer function is also used by engineers. Laplace transforms are also used in Probability Theory, where they are called the moment generating functions.