Built on Facts

I first met this function sometime in the year 2001 in the manual for a graphing calculator. The manual said that the function had no "closed-form analytic antiderivative" but nonetheless the calculator could integrate it numerically. At the time I had no idea what any of that meant, but upon taking a high school calculus class I met the function again as a demonstration of the concept of a limit. In my freshman calculus class in college I met the it yet again and learned that while this function and all continuous functions have an antiderivative, the antiderivative can't always be expressed in a closed form - i.e., in a finite number of terms involving elementary functions. Which explained what the calculator manual meant.

Then I met the function again when learning about the Fourier transform of a rectangular function. And again when learning about Bessel functions. And again while learning contour integration and the residue theorem. I'm not sure I'd say this function is one of the most common functions in day-to-day calculations, but it's got to be one of the most versatile pedagogical examples in all of mathematics.

It's called the sinc function, and abbreviating its name to "f" as usual, it's defined like this:

The graph of the function looks like this:

If you're careful you'll have noticed that this function must have a single point missing. At x = 0, the definition requires a division by zero, which is impossible. Thus the graph properly ought to have an infinitely tiny single point missing at the apex of the central peak. But as you can also see, as x gets close to zero the function itself gets close to 1. Thus there's no problem with scrapping the definition above for that exceptional case, and just defining it as equal to 1 at that point. The demonstration that the limit of the function is 1 when x approaches 0 is usually the first time a student will meet this function as a teaching example.

Now we've met the function. Soon we'll get to know it a little better. While I'm already seriously backlogged on Sunday Function ideas (and the upcoming "Coolest Functions" series), I'm going to try to go through a little bit of detail over some of the particular things this function is used to demonstrate. In particular the way to get the integral of this function over the whole real line from the contour integral in the complex plane is just too interesting to pass up.

Ok, ok, my definition of interesting is a little atypical. But I'm pretty sure I can avoid boring y'all to the point of pitching yourselves off the nearest cliff. I think you'll even enjoy it, as the method is both powerful and deep without being particularly technical.