Built on Facts

Lots of times we might have a complicated function which we wish were less complicated. Take the differential equation describing the simple pendulum, for instance:

I can’t solve that one explicitly. Nobody can - the sin term prevents a closed-form solution and so nearly the only practical way to work with it is to use numerical methods. There’s nothing wrong with that, but it’s not necessarily what we’d prefer. A stack of floating point numbers is great but it doesn’t necessarily breed understanding. An approximation might be helpful. For small angles,

Take a look at their graphs. Near the origin they get closer and closer to being the same thing:

So we can just put θ in the differential equation where we used to have sin(θ) and solve it easily. The result will only be valid for small displacements, but most of time a pendulum isn’t swinging at very high angles anyway.

This is just one example of a power series approximation. Basically you assume the function can be written as

And you do some calculus voodoo to find the constants. Then you just keep the first few terms depending on how accurate you want your approximation to be. Talking about this in more detail will be a good topic for a future post, but for now I just want to show off one of my favorite pathological functions.

It’s continuous everywhere except the origin, and even there it’s only a point discontinuity so we can define it to be equal to its limit, which is 0 at that point. That means the function is continuous everywhere, and differentiable everywhere. Its derivatives of all orders are continuous as well.

You can do the calculations to figure out the power series yourself, but here’s the result: its first derivative at the origin is 0. So is its second derivative. So is its third derivative. In fact, all of its derivatives of all orders are 0 at the origin. It’s power series thus seems to be 0, though the function itself is clearly not the zero function.

Why is this? Well, it turns out that power series are intimately connected to the behavior of the function in the complex plane. Though our function is perfectly well-behaved at 0 on the real line, it is horribly behaved at 0 in the complex plane. It has an essential singularity at that point; the limit as you approach 0 depends on how you do the approaching. For example, f(0.1) = 3.7 x 10^-44, but f(0.1 i) = 2.7 x 10⁴³. Clearly they’re not approaching the same thing. In fact, Picard’s great theorem guarantees that the limit will approach any given value depending on which direction you approach from. As you might expect, this wrecks the possibility of expanding the function in a power series about the point of singularity.

When you next do a power series approximation, spend a little time being thankful that most functions in physics aren’t so badly behaved.