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.

Comments

#1 andy.s
July 5, 2009

A neat thing about this is that it’s the Fourier transform of an Ideal Low-Pass filter: an electrical filter that will only pass low frequency signals.

The sinc function is its impulse response – the way such a circuit would react to a single pulse at time 0.

The reason this is weird is that the hypothetical circuit starts reacting at t<0, before the impulse is received – an ideal filter would be a-causal.

I guess that puts this circuit in the Acme Physics catalog – the same one that sells massless springs, frictionless pulleys and point masses for undergraduate experiments.
#2 Comrade PhysioProf
July 5, 2009

I love that function!
#3 6EQUJ5
July 5, 2009

I’ve always thought of it as a negative half and a positive half separated by a gap of zero width.
#4 andy.s
July 5, 2009

aaaahhh crap. caught by the less than sign bug again.

ahem…

The reason this is weird is that the hypothetical circuit starts reacting when t is less than 0, meaning it reacts before it receives a signal. It’s a-causal.
#5 travc
July 5, 2009

Small world… mathematically speaking at least.

I just ran across this function again for the first time in a long time yesterday. I’m doing acoustic processing, and numerical approximations of sinc are quite useful (and common) designing real world filters.

Sox even has a sinc filter effect. (If you ever work with acoustic stuff, you really should know and love sox.)

BTW: Out of the order infinite^2 possible equations, it really is amazing how a small number are so common when describing physical processes. I don’t buy the metaphysical crap… but humans did pick a really good basis for our math.
#6 CCPhysicist
July 5, 2009

Learn something every day. I say ” zero” when I see that guy, not “sinc”.

Until now, I only knew that function as a special case (the zeroth spherical Bessel function j_0(x) and thus related to J_{1/2} by factors I forget). Many of the features you mention are a result of it being part of the leading term in an expansion of a spherical wave in 3-space, thus making it commonplace in scattering theory.
#7 Uncle Al
July 5, 2009

avoid boring y’all Shouldn’t that be “all y’all” to be inclusive?
#8 complex field
July 5, 2009

I feel l’Hopital’s Rule coming on…
#9 Paul Murray
July 6, 2009

Yeah – intuitively, f(x)=sin(x) becomes very close to f(x)=x when x is near zero (its obvious if you think about in terms of rotational motion), so sin(x)/x becomes very close to x/x = 1 .
#10 Chris
July 6, 2009

The function is well-behaved at zero if you replace sin() with its power series expansion.
#11 Anonymous
July 7, 2009

Wait why would there be
‘an infinitely tiny single point missing at the apex of the central peak.’?
It looks like there’s only one place where division by zero occurs.
#12 Anonymous
July 7, 2009

woops, i completely misread that. That’s what I get for trying to read this before I’ve had my coffee proper.