The Laplace Transform

Abstract

The main applications of the Laplace transform are directed toward problems in which the time t is the independent variable. We shall therefore use this variable in this chapter. Let f(t) be a complex-valued function of the real variable t such that f(t)e ^-ct is abolutely integrable over 0 < t < ∞, where c is a real number. Then the Laplace transform of f(t), t ≥ 0, is defined as

$$ \tilde f(s) = \mathcal{L}\left\{ {f(t)} \right\} = \int_0^\infty {f(t)e^{ - st} dt, Re s > c,} $$

(1)

where s = σ + iω. The Laplace transform defined by (1) has the following basic properties.