Laplace transforms: what are they and how to evaluate them
Laplace transforms are used extensively in engineering and science to transform differential equations into algebraic ones, making them a lot easier to solve. They are also used in dynamics and control theory to obtain specific parameters that determine the behaviour of a complicated dynamical system, such as its poles and zeros. In this article, I will go through some of the main aspects of Laplace transforms and how to evaluate them.
We begin our discussion by defining the one-sided Laplace transform of a function f(t) as
If f(t) is a function of time (as is often the case in practice), then s is a variable that has units of inverse time, or Hertz, making it a frequency variable. Therefore, F(s) is some form of frequency representation of f(t), but this is not to be confused with the Fourier transform which tells you the content of all frequencies in the function f(t). We should also note that the variable s can be complex.
The inverse Laplace transform, sometimes known as the Bromnich integral, is expressed as
where
The inverse Laplace transform is computed as a contour integral in the complex plane, sometimes using Cauchy’s integral formula or the Residue theorem. In practice however, if we want to obtain the inverse transform of a…
