Definition

Assume that f(t) is a piecewise continuous function. The Laplace transform of f(t) is denoted L{ f (t )} and defined by,

There is an optional notation for Laplace transforms. For the sake of convenience we will frequently denote Laplace transforms by,

L { f (t )} = F (s)

With this optional notation, note that the transform is actually a function of a new variable, s, and which all the t's will drop out into the integration process.

Here, the integral in the definition of the transform is termed as an improper integral and this would probably be best to recall how these types of integrals work before we in fact jump in calculating some transforms.