Math

The Intuition behind the Laplace Transform

The Laplace transform (LT) is arguably the king of applied mathematics. Every engineer, physicist, and mathematician is bound to have encountered the Laplace transform at some point. From turning calculus into algebra to transforming a system from the time to the s-domain, this mathematical tool has made our lives a lot easier. But what exactly is the intuition behind the Laplace transform and how can we use this intuition to understand its applications?

Definition and Connection with the Fourier Transform

The Laplace and the Fourier Transform (FT) are by far the most popular and widespread transforms in all of mathematics. For anyone who is not already familiar with the Fourier transform I recommend reading up on it by clicking the link above as it will make everything we are about to say a lot easier.

The Laplace transform of a function x(t) is defined by the following integral

At first, it looks very similar to the integral of the Fourier Transform.

In fact, there are only two differences.

For once, we see that the integral in the LT starts from 0 and not from minus infinity. This is actually sort of a convention because, in engineering, we use the LT most of the time to…