The True Power of the Laplace Transform

A magical bridge between two parallel worlds

Suppose you’re facing a complex question or a very hard problem, difficult to solve due to limited tools or knowledge. Imagine now that you have the ability to carefully reformulate this problem into a new and easier one. Not only can you solve this transformed problem, but you can also convert its solution back into a solution for the original, challenging issue. That is the heart of the Laplace transform.

It is literally a shortcut that cuts about 3–4 years of advanced mathematics from a problem and in some instances transforms it from a university-level problem to a high-school-level problem. Specifically, it transforms calculus into algebra and some VERY difficult equations to not as difficult equations. We will see some examples of this a little later.

What is not as well-known about it is that the Laplace transform is very useful in pure mathematics as well. The transform is mostly taught to electrical and mechanical engineers, people studying physical systems or control theory but this mathematical powerhouse can do much more than that!

The point that I really want you to take away from this article is that there exist two parallel worlds and every time you do something to a function, you are actually doing two different things in parallel. More about that a little later.

Definition, examples, and properties

Before moving on, I think a definition is in order. I realize that I might lose some readers here because it might look a little scary at first, but the beauty of this theory is that you can get very far by “just” understanding the overarching principle and some specific examples. So I encourage the reader to stay a little longer before quitting. It gets a lot easier a little later.

In the following, I won’t go into convergence issues as it is out of scope of this article. I will simply state formulas that are correct whenever they converge. I will also not talk a lot about the time and complex frequency domains (sometimes called s-domain) even though this is the classical approach.

The Laplace transform is defined as follows:

Say you have a function f of a variable t, then the Laplace transform ℒ(f) is another function F of a variable s defined by:

The reason behind the variable names is that when interpreting this transformation in terms of physics, we take a signal as a function of time and output a signal in terms of (complex) frequency. We will not discuss exactly what this correspondence is but I can tell you that it is related to its sister — the Fourier transform.

Note that F is not an antiderivative of f which is standard notation for some authors. The infinity symbol in the upper limit of the integral should be interpreted as taking a limit i.e.

assuming that the limit exists.

Let’s quickly go through a thoroughly worked-out example so that you don’t lose all hope about getting through this alive.

Let’s choose one of the simplest functions: f(t) = t.

We can use integration by parts to get the following:

Here, we have used L’Hôpital’s rule along the way, but you don’t need to understand this to follow along — simply note that when f(t) = t, then F(s) = 1/s². By an even simpler use of the above technique, we can find that when f(t) = 1 then F(s) = 1/s.

From now on, we will not be as precise as above. I encourage the reader to take the unproven results in this article as exercises. Let us see what multiplying by an exponential in the time domain looks like in the “parallel” world.

So multiplying by the exponential function shifts the argument in the s-domain. This is a stronger result than it seems and it is used again and again in the literature.

An important example is the Laplace transform of the exponential function. Specifically, we have

In the same way, we can transform the trigonometric functions. When taking the Laplace transform of the function f(t) = sin(t), for example, we use integration by parts a couple of times and solve an equation. When the dust has settled, we arrive at the following very useful results:

The Laplace transform has some amazingly useful properties. The most basic and in some sense most important one is that it is a linear operator, meaning the following always holds:

where F and G are the Laplace transforms of f and g respectively.

Another extremely important property is that the Laplace transform is a one-to-one mapping, meaning that it has a unique inverse transform. That is, whatever we do in one world has a parallel action in the other world.

For the technical reader:

The details surrounding this are a bit technical and are out of scope of this article. For the interested reader, I can say that two integrable functions have the same Laplace transform only if they differ on a set of Lebesgue measure zero and that the exact formula for the inverse transform requires complex integration theory (contour integration).

Proving “the most beautiful equation in the world” by Laplace transform

The following has been called the most beautiful result in all of mathematics. It is due to Leonhard Euler who showed us that the exponential function is related to the trigonometric functions:

where i has the property that i² = -1.

Normally we prove this by expanding out the infinite power series for the exponential function and using the infinite version of the distributive law etc. This however requires some technical arguments that many authors leave out for simplicity.

What is not that well-known is that we can prove Euler’s result above using Laplace transforms.

and now we can take the inverse Laplace transform on both sides to get

where we have used the linearity of the inverse transform. We recognize the above as the trigonometric functions cosine and sine:

as promised.

From calculus to algebra and back again

One of the strongest properties the Laplace transform possesses is that it turns derivatives into polynomials. Specifically, we have

that is multiplying by s in the s-domain corresponds to differentiating in the time space. We have higher analogs of this as well. For example:

and this pattern continues. In the same way, dividing by s corresponds to integration in the time domain.

These formulas might seem simple, but they have the power to transform differential equations into polynomial equations which are much easier to solve. In some instances, they even transform partial differential equations into ordinary differential equations.

Example

Let’s say we want to solve the differential equation:

with initial conditions f(0) = 0 and f '(0) = 0. Let’s take the Laplace transform on both sides.

and by substituting in the initial conditions and isolating F(s), we get

Let us pause here for a second and appreciate a few things. First of all, the complicated equation has turned into a simple function definition! Second, the initial conditions are naturally encoded into the solution in the s-domain so instead of having several extra conditions, they are all compatified into a single equation.

Now we simply take the inverse Laplace transform on both sides to find f. In practice, we often use partial fraction decomposition but we can also use a formula.

This is only the beginning of the fruitful use of the Laplace transform to DEs but unfortunately, this article won't go into more detail than this.

More advanced and exotic use cases

That multiplying by s in the s-domain corresponds to differentiation and dividing by s corresponds to integration turns out to be special examples of a more general correspondence.

Convolutions

The product of two functions F and G in the s-domain defines an operation of two functions f and g in the time domain called convolution denoted f * g. According to Wikipedia:

Convolution has applications that include probability, statistics, acoustics, spectroscopy, signal processing and image processing, geophysics, engineering, physics, computer vision and differential equations.

Integrals over the positive reals and the Dirichlet integral

An immensely useful property of the Laplace transform is the following:

Let’s try our formula on a notoriously difficult integral known as the Dirichlet integral. The problem asks to find the following:

which is kind of a “hello world” of powerful integral techniques. We have

By using that the antiderivative of 1/(x²+1) is arctan(x) + c, we get

Laplace series

I have been looking forward to sharing the following with you all. As always, we assume that all general series and integrals converge.

Say we have a periodic function f with period P and Fourier series

then we can obtain a series for the Laplace transform of f by linearity:

If P = 2π, then the above simplifies to

where the as and bs are the even and odd Fourier coefficients respectively.

I don’t know if this has a name because I haven’t seen this before so I call it the Laplace series. Let me know if it already has a name.

Let’s try to apply this to an example. Let f(t) = t in the interval [0, 2π] and then extend periodically. The graph of the function looks like the following:

The Fourier series for this function is given by:

and we know that this function is identical to the sawtooth function defined above except at the points of discontinuity i.e. at t ∈ 2πℤ where it is exactly equal to π.

I went through the painful process of calculating the Laplace transform of f by hand before using our formula. After summing an infinite series of integrals we get

Since the Fourier series of f only differs from f on a set of measure 0, we know that the above is exactly equal to the Laplace series of f. That is:

The essence of this is that even the Fourier series has a parallel series in the s-domain. That means that each (relatively well-behaved) periodic function has a Fourier series representing it in the time domain and a Laplace series representing it in the complex frequency domain.

We need however to be a little careful since the Laplace transform only transforms the function on the positive real numbers. It doesn’t know anything about the function at the negative axis.

This theory opens the doors to the field of analytic number theory where series such as the one on the left-hand side are very important to study.

The Laplace transform was my first true mathematical love.

In fact, in my parent’s house (in my old room) there is still a whiteboard on the wall and there is only one piece of mathematics on it: The Laplace transform.

Note that I got the notion slightly wrong. Instead of correcting it, however, I leave it as a reminder of a principle almost as powerful as the transformation itself. Namely that we are all on the same path towards understanding. Some are further than others but we have all taken our first steps and walked the same terrain and likely stumbled a thousand times in the process.