## Monday Math 101

The Laplace Transform
Part 2: Convergence

When considering the Laplace transform
,
we must consider the region of convergence; those values of the complex number s for which the above integral converges.

A function f(t) is of exponential order at infinity if and only if there exist real numbers M and a, and positive real number T such that  for all t>T; note then that
 for .
Thus, we see that if the function f(t), t≥0, is piecewise continuous and has exponential order at infinity with , then the Laplace transform is defined for .
In fact, it can be shown that if the Laplace transform converges for some , then it converges for any s with , and thus the region of convergence is (if not the entire complex plane) a half-plane of the form , with all or some of the boundary  possibly included.