Laplace Transformation

Abstract

Let f(t) be a real or complex-valued function defined on the positive part R₊ of the real axis. The Laplace transform of f(t) is defined as the function F(s)

$$F\left( s \right): = \int_0^\infty {e^{ - st} } f\left( t \right)dt,$$

((1.1))

provided that the integral exists. We also say, f(t) is the original function or for short the original and F(s) is its image function or its image. For a given image function F(s), we call f(t) the inverse Laplace transform of F(s). It is not necessary but we prefer to extend always the domain of an original function f(t) to the left by defining f(t) = 0 for t < 0.