Definition and Elementary Properties

Abstract

Let f(t) be an arbitrary function defined on the interval 0 ≤ t < ∞ then

$$ F(p) = \int_0^\infty {{e^{ - pt}}f(t)dt} $$

(1)

is the Laplace transform of f(t), provided that the integral exists. We shall confine our attention to functions f(t) which are absolutely integrable on any interval 0 ≤ t ≤ a, and for which F(α) exists for some real α. It may readily be shown that for such a function F(p) is an analytic function of p for Re(p) > α, as follows.