What you're looking at is a complicated diagram that we'll build up piece by piece. This is the Laplace transform—one of the most powerful tools for studying differential equations. As a student, you can learn how to use it to solve equations while remaining completely in the dark about what it's actually doing.
Think about learning to drive a car versus learning how an internal combustion engine works. Both are worthy pursuits. Driving is probably more practical. But there's something deeply satisfying about popping open the hood and understanding the mechanism inside.
Similarly, our main goal with this video is to pop the hood and show you some of the beautiful math that awaits us inside this object. Strict speaking, a lot of what Grant wants to show isn't necessary if your only goal is to solve equations. That said, for the differential equation students among you, this content should make what you have to memorize a lot more memorable.
The Problem It Solves
Before just plopping down the definition, let's talk about what problem the Laplace transform is trying to solve.
Two primary ideas are worth restating. First, you need to understand exponential functions—written as e to the s times t. Here, t represents time, and s is a number determining which specific exponential we're talking about. But very importantly for this topic, we're going to give s the freedom to take on complex number values.
The quick summary: if s has an imaginary part, the output of your function rotates in the complex plane as time ticks forward. When the real part of s is negative, the magnitude decays toward zero over time. But if it happened to be positive, that would mean the magnitude grows—growing exponentially.
Why do we care about these functions? It's because a lot of functions, especially those arising in physics, can be expressed as combinations of exponential pieces.
Cosine as Exponential Pieces
There's one very friendly example that's going to be helpful to return to repeatedly throughout this lesson: the cosine of t. This undulating function can be broken down as a sum of two purely imaginary exponentials. Basically, if you take e to the i times t, that gives you rotation counterclockwise, and e to the negative i t rotates the other way.
When you add these two together—perhaps imagining adding two rotating vectors tipping over—the imaginary parts cancel each other out, and what you're left with is something whose output remains locked to the real number line, oscillating back and forth. This sum goes between -2 and positive 2, but a cosine only goes between -1 and 1. So finally you multiply everything by one-half.
More complicated functions might break down into more exponential pieces. For example, later in this lesson, you'll dig into something called the driven harmonic oscillator—a mass on a spring that's influenced by some external force that oscillates over time. As a spoiler, the solution to the equation describing that ends up looking like a sum of four exponential pieces.
Two of those pieces oscillate and decay in a way that matches the natural resonant frequency of the spring, and then the other two oscillate in a way that matches the external force.
What We Want
For cases like this and many others, what we would like is some sort of tool—a mathematical machine where you can pump in a function or even a differential equation describing that function, and that machine will somehow reveal for us what specific exponential pieces that function breaks down into. That is, it exposes what these values s in the exponent all are, as well as what the corresponding coefficients are.
You might wonder: why exponential pieces? Why not focus on something else?
The short answer is that for these functions, when you take a derivative, it looks precisely the same as multiplying by some number—namely s. What you'll see by the end is how this means that same machine that lets us dissect functions into exponential pieces also allows us to turn differential equations into algebra. Essentially, because everywhere you see a derivative, it turns into multiplication by s.
The S-Plane
In this context where we are so focused on complex valued exponentials, engineers have a special name for this complex plane representing all possible values for that term s in the exponent. They call it the s-plane.
A helpful mental image: think of each individual point on the s-plane as encoding the entire exponential function e to the s times t. For this picture, the graphs shown only depict the real component of the output. But keep in mind, each point is representing the full complex valued function.
These graphs are enough for intuition. You'll notice how bigger imaginary parts correspond to faster oscillation. And then as your eyes scan from left to right, the real part reflects either decay or growth.
The Laplace Transform
With this as our goal, the machine that exposes how a function breaks into exponential pieces is—you've guessed it—the Laplace transform.
The word transform is a little funny. In the same way that a function takes in a number and spits out a new number, we often use the word transform in math for a more meta operation that takes in an entire function and spits out a new function. In the case of the Laplace transform, a typical convention is to name our new function with a capitalized version of whatever you use to name that original function.
And in this setting where our original function takes in time as an input, the new transformed function has a new different kind of input: a complex value s.
It's helpful to quickly preview what this new function actually does before pulling up the definition and start dissecting it. Suppose your original function f(t) really can be broken down as a sum of several exponential pieces. When you apply this Laplace transform, giving you a new function of this new variable s.
If you were to plot this new function over the s-plane—what you'll see are these sharp spikes above each value of s that corresponds to one of those exponential pieces. These spikes have a fancy name: they are called the poles of your function.
So even if you didn't know ahead of time how your function could be broken down as a sum of exponentials, if you understand this transformed version—and specifically if you understand the poles—that can reveal for you what those exponential pieces are.
The Definition
At this point, you're probably itching to see how this thing is actually defined. In its full glory, this is what that definition looks like, which we can think of as two separate steps. First, you multiply your function by the expression e to the s times t. And then second, you integrate that result over time from t = 0 to infinity.
We'll talk all about that integral and its nuances in just a minute, but for the moment, focus on that inner expression. This term s is the newly introduced parameter—the one that is the input of our new transformed function. Every time you see the letter s in this video, it's a complex number.
And the way to think about it: you might imagine freely moving around this value s on the s-plane and it's kind of sniffing around to find which specific exponential functions line up closely with our function f(t).
For example, let's suppose our function is the cosine of t—which as we discussed just a minute ago, we already know can be broken down as a sum of two exponentials, e to the i times t and e to the negative i times t.
I've already previewed the idea that when you plot the final result, you should see these spikes over the key values of s—which in this example would mean poles above plus i and negative i. And you can already get a little intuition by taking this example and substituting in the expanded expression for the cosine of t.
Looking at this expansion, I want you to take a moment to think about the product of these two terms. When you multiply two exponentials, the contents of those exponents add together. So what you're left with is e raised to the (i minus s) times t.
Let me go ahead and plot that value on the lower right, keeping the s-plane on the upper right. Just like the exponentials we've already seen, as time goes from 0 to infinity, this oscillates and decays—or maybe it grows. The specific shape depends on how we set that value s.
Now in this case as you move around that value s, there is one specific value causing uniquely boring behavior. If you set s equal to i, then that term in the exponent becomes zero. So the entire function just looks like e to the zero—which is stuck at the constant 1.
And this is a key idea: for almost all values of s, an exponential like this is going to change with time typically looking like some kind of spiral. But at the special value that we're hunting for—in this case, s equals i—things get stuck at a constant instead.
Detecting Constant Behavior
Stepping back, if you were a mathematician trying to invent some kind of machine that will detect the exponential pieces lurking inside a function—for example, detecting that a cosine has an e to the i times t and an e to the negative i times t lurking inside it—you might ask yourself: is there something I can do that can detect when one of the terms in a sum like this is secretly just a constant?
In essence, this is the role played by the integral that wraps everything up in the full definition—integrating as time goes from 0 to infinity. And on the one hand, if you're comfortable with calculus, you might be able to anticipate why this would result in some kind of sharp spike over the desired values of s.
If you integrate a constant from zero all the way up to infinity, it blows up. But there's actually a fair bit of nuance here.
To start, that function inside the integral takes on complex number values. And this raises a natural question for those of us curious to interpret things in a satisfying way: how do you think about integrating a complex valued function?
If you're up for it, what we'll do with the next ten minutes or so is really dig in and dissect what an integral like this really means, how to visualize it, and explain why the plots shown represent something slightly distinct from the literal meaning of this integral.
Visualizing the Integral
As with any new piece of math, it's best to start easy and work our way up in complexity. So to kick things off, let's just ignore this function f(t) itself and only focus on integrating the e to the s times t part.
And to warm up, before jumping into the complexity of it all, let's just suppose that s is a real number—meaning this is a friendly real valued function that we can plot with a graph like normal. Typically, when you first learn calculus, you learn to interpret an integral as telling you the area under a graph like this.
For example, if you set the value s equal to 1 and you go through the procedure for actually calculating this integral—you know, you take an anti-derivative, you take the difference at the two different bounds, or because there's an infinity, you could say you take the limiting value at that bound—the way it all works out is that this expression equals 1.
And you can interpret that as telling you that the area under this graph is 1—which is kind of fun. I guess that means that all the area under this infinite tail is exactly enough to fill in the rest of this unit square.
Then if we reintroduce that value s and set it equal to something that's not one, the effect is to squish the graph in the horizontal direction. That's always the effect if you multiply the input of a function by some constant. And so the area under this graph, which started out as 1, must now be 1/s.
The key thing to remember here is how if we let s get smaller and smaller and approach the value at zero, then that area gets bigger and bigger and bigger—actually approaching infinity and approaching it quite quickly too.
But graphs are not the only way to visualize functions, and area is not the only way to understand integrals. You should get in the habit of flexing your mind a little bit more.
A Pool of Water
Let me show you another way to think about this that will generalize more easily once we let that function take on complex number values.
Think about this integral just between 0 and 1—something with a unit length. And I want you to imagine all this area under the graph as a pool of water, which you let kind of slush down until it becomes level. The height of this pool is telling you the average value of that function between 0 and 1.
And then because the width of this pool is just 1, then its area is the same thing as its height. So this integral up top when it's over a unit interval is telling you the average value of the function on that interval.
Similarly, the integral from 1 to 2 would be telling you the average value the function takes over that interval from 1 to 2. And then same deal as you keep integrating along a bunch of other unit intervals.
So then if you want the integral from zero out to infinity, what you can think about is taking all of these average values on those intervals and adding them all together. This is something we can work with.
Now let's look at the complex case. S is going to be some complex number. And then the function e to the s times t cycles and decays around the complex plane as you let time go from zero up to infinity. The specific way that it cycles and decays or grows depends on that value of s, and we'll get a distinct path through the complex plane for each one.
Now if you want to integrate this function on a unit interval—let's say from the values t = 0 to t = 1—imagine taking a sample of all of the outputs in this range and then finding the average, the center of mass for all those points. That average value is the meaning of this integral—which I'll represent with a little arrow.
And actually, it'll be helpful if we put this integral in its own context.