Most textbooks, especially ones not aimed at college math majors, give a definition of “function” that seems quite intuitive. They’ll say something along the lines of: a function is a rule that takes an input x and turns it into an output f(x). Formally this isn’t quite right – the essence of a function is in the set of ordered pairs {x, f(x)} and not in the specific rule that connects them. There doesn’t even have to be such a rule.

But the idea of function as a machine is such a powerful and intuitive one that it tends to be used pretty universally until you have a good reason to abandon it. Non-mathematicians rarely encounter such reasons, even in the more mathematically demanding disciplines like physics, computer science, and engineering. In fact, most of the time we tend to double down and promiscuously apply the “function as machine” picture to *operators*. If a function is a machine that turns numbers into other numbers, and operator is a machine that turns functions into other functions. One such operator is called the Laplace transform, after the french mathematician Pierre-Simon Laplace. But I think we’ll stick to calling these posts Sunday Functions, even if we take the occasional look at operators.

So pick a function f(x) – you can pick just about any f(x) you want – and multiply it by e^(-sx). That’ll give you a new function. For instance, if you started off with f(x) = x^2, which looks like this:

You’ll end up with (x^2)*e^(-sx), which looks like this for s = 1:

Note the different y-axis scales, and of course note that the precise shape would be a little different if we had picked a different s. Now we’ll construct the Laplace transform, which we’ll call F(s) or L(f(x)), where the latter denotes applying the transform L to our original function f(x). It creates a function of s based on an integral over x:

After doing the integral, we’ll find that the Laplace transform f(x) = x^2 is:

All right, that’s really lovely, but what does that do practically? Well, not much. What’s so interesting about the Laplace transform is what happens if you take a look at how the transform relates functions to their derivatives:

Where in the second line we used integration by parts. Notice what happened. We discovered that the Laplace transform of a derivative doesn’t involve the derivative of the Laplace transform. It just involves the Laplace transform multiplied by s. This is a truly remarkable property. If you have a linear differential equation in f(x) and you take the Laplace transform, you’ll just have an algebraic function in F(s). Solve that using 9th grade algebra and do the inverse Laplace transform and you’ve solved a differential equation without doing any real work at all.

Well, other than finding the inverse Laplace transform. That’s actually pretty difficult to do from scratch. Fortunately large tables of functions and their inverse transforms have been compiled and for many cases of interest all you have to do is look up the one you need.

You can go on and do more creative things like letting s be a complex number, and then you’ll discover a relationship between the Laplace and Fourier transforms. That’s interesting in itself and also useful for things like the mathematics of electromagnetic field propagation in a dispersive medium.

All in all, a nice little mathematical machine. Or whatever it is!