A Review

exponential asymptotics

Exponential Asymptotics then, is the bridge between God’s picture-perfect, but wholly unrealizable representations, and the Devil’s ambiguous, but entirely necessary evils.

Last time, we left off talking about asymptotic approximations. Nature — we explained — is often too complicated to describe exactly, and so we turn to these approximate methods for help. Unfortunately, such approximations are divergent and mathematically badly-behaved, even though they provide excellent results.

Now, we will see that there are indeed consequences to using these devilish and divergent approximations.

Dancing with the Devil

Maybe the best thing we can do is look at a concrete example.

For this, we’re going to imagine the flow of water with speed U over a step which is relatively well submerged. To solve this problem, we propose the following asymptotic expansion for the surface of the water:

In this case, the perturbation parameter, ε corresponds to a measure of the inertial and gravitational forces. The idealized state, or first approximation corresponds to the case that ε = 0, or the case for which the force of gravity is infinite.

What that means is that our first order asymptotic approximation tells us the free surface should look like this:

That’s right. Completely flat.

Sure, this does make some sense: Imagine the step as being infinitely deep. Then in this ideal case, free surface wouldn’t feel the bump at all. Hence it would lie completely flat.

Unfortunately, this isn’t quite what happens.

If we use a computer and calculate the solution, it gives us something like this:

Somewhere on that free surface, something is happening.

The asymptotic approximation we calculated does a wonderful job initially, but then out of nowhere, a light flickers on and Pop!.

Waves appear where there were none.

Into the Mist

Maybe the first person who documented the first “WTF?!” moment upon encountering this strange phenomenon was none other than George Gabriel Stokes, one of the greats of British applied maths. He described the phenomenon as a new term, entering into view from behind a mist:

How do we clear the mist?

The argument goes like this: First, we have an approximation that works on the far left. Now, somewhere, along the surface, a switch is turned on, and an exponential (or a sinusoidal, if you will) is turned on.

Thus, there must be something that’s going on the complexified free surface.

Whoa there, Missy! What the heck does that mean?

By complexified free surface, we’re referring to extending the water surface (a real number) to the complex (or `imaginary’) numbers. In other words, if x measures the distance along the water surface (say, 1 meter, 2 meters, etc.), then we allow x to be an imaginary number (say, 1 + i or 2 - 2i)

Physically (and philosophically), it’s pretty nonsensical. We picture it like this:

So we’ve extended the surface by adding an extra dimension.

This, while baffling to outsiders, is a well known trick by mathematicians. Maybe there’s an easier way to explain it.

The Imaginary Route to School

Let’s suppose that Billy walks from his house (x = -10) to school (x = 10) every day.

Monday, however, he wakes up and hears on the radio that due to faulty engineering and a freak division-by-zero accident, the bridge (x = 0) on the way to school has collapsed. What can Billy (determined and diligent student that he is) do?

The answer is that Billy needs to find another way to school. He needs to complexify his usual route, enter into another dimension, and find a new path.

Billy’s adventure is the essence behind the mathematician’s method: Often, we will have some sort of singularity between two points. To deal with it, we seek a new path. It turns out that useful paths can be taken in the ‘imaginary’ plane. So while this is definitely unphysical and unintuitive, take a lesson from Billy.

If he can warp his mind to change his usual route, so can you.

Singularities and Stokes Lines

Let’s return to our complexified water surface.

Well, it turns out that on this extended surface, there are singularities: points where our asymptotic approximation (remember that?) goes terribly awry. This is the reason why the asymptotic expansions diverge — because of these singularities.

With a lot of work and advanced maths, it can be proven that there exists lines (called Stokes Lines) emanating from these singularities, across which a small exponential term is switched on.

Like into the mist, remember?

And so following the above picture, you can see that the asymptotic approximation (the flat surface) works well at the start. But then it hits the edge of the Stokes Line, and at that point a teensy exponential is turned on.

So actually, we should have,

Exponential Asymptotics is the name we give to the branch of mathematics which studies these small exponentials which seem to fly under the radar of ordinary methods. It’s these hidden exponentials which emerge from the divergent tails of asymptotic expansions.

From Heaven to Hell

It’s time to wrap up.

In each previous part, I ended by asking a crucial question. They were:

Part I: Sometimes, as mere mortals with a capacity for only understanding the finite, we’re forced to cut corners and make approximations. Most of the time, it won’t matter. When will it matter?

Part II: In order to describe nature’s oceans and lakes, mathematicians have had to use divergent (badly-behaved) series approximations. Although these give superb results, we are nonetheless representing perfect phenomenons using a far-from perfect representations. “Inventions of the devil”, said Niels Abel. But this is a practical necessity. How else are we to describe God’s infinity with our finite minds? The real question is, what’s being lost in translation?

In this part, we showed that while divergence is bad, there’s no need to disregard it. By using certain mathematical tools, we can unravel the divergence, and reveal what details were lost using the approximations.

Exponential Asymptotics then, is the bridge between God’s picture-perfect, but wholly unrealizable representations, and the Devil’s ambiguous, but entirely necessary evils.

This bridge is where I stand and this bridge is what I study.