68 Chapter 3 Laplace transforms

(i) Suppose that $f:(0,\mathrm{\infty })\to 𝐂$ is a continuous function such that

for some $M>0$ and $\beta \in 𝐑.$ Here $\beta$ is called the exponential type or growth rate. Then we say that $f$ is of exponential type, or satisfies (E).

(ii) We then define the Laplace transform by

Sometimes $ℒ(f)(s)$ is written as $\widehat{f}(s).$ Here $x,t$ are time variables; whereas $s$ is the transform variable.