Earlier this month I promised to show you a simple example of regularization. Well, here it is. This is a particular combination of integrals I’ve been working with a lot lately:

A quick look at the formulas shows you that both integrands have singularities at $\mathbf{x} = 0$ and $\mathbf{y} = 0$.

OK, well, that’s why we have two integrands. We can change variables from $\xi\mathbf{y}\to\mathbf{x}$, subtract them, and the singularities will cancel out, right? You can do this integral by hand in polar coordinates, or just pop it into Mathematica:

Just one problem: that’s not the right answer! You can tell because the value of this integral had better depend on $\xi$, but this expression doesn’t. This isn’t anything so mundane …

Last year I posted about an infinite sum involving the mean of the harmonic numbers,

The method I used to evaluate this was to approximate the sum by an integral. This is a technique that is used in various places in physics, such as in the computation of the Fermi surface in metals.

In this particular case, we only cared about the large-$n$ limiting behavior of the sum, namely that it grows sublinearly for large $n$. But suppose you wanted to know whether there was, say, a constant term as well. Here’s one way to figure that out. Instead of converting the entire sum to an integral, you choose some cutoff value $a$, and keep the first $a$ terms in the sum explicitly.

Now you have a value, $a$, that represents a “break” in your sequence, but it’s an arbitrary …