Function of exponential order

Summary

A function $f$ on $[0, \infty)$ is of exponential order when it grows no faster than some exponential $M e^{at}$ for large $t$. This is the usual sufficient condition for the Laplace transform of $f$ to exist. For a piecewise continuous $f$ of exponential order $a$, the integral $\int_0^\infty f(t) e^{-st}\, dt$ converges absolutely for $\operatorname{Re} s > a$. Most functions met in linear differential equations, such as polynomials, exponentials, sines and cosines, are of exponential order.