Friday, July 7, 2023

Integral of the Bessel Function

Have you ever been reading a book, making good progress with everything making sense, and then you suddenly stop at say “wait… what?”. That happened to me recently as I was reading Homework Problem 31 in Chapter 12 of Intermediate Physics for Medicine and Biology. (Wait…what? I’m a coauthor of IPMB! How could there be any surprises for me?) The problem is about calculating the two-dimensional Fourier transform of 1/r, and it supplies the following Bessel function identity 

An equation for the integral of the Bessel function J0(kr).

The function J0 is a Bessel function of the first kind of order zero. What surprised me is that if you let x = kr, you get that the integral of the Bessel function is one,

An equation for the integral of the Bessel function J0(x), which equals one.

Really? Here’s a plot of J0(x).

A plot of the J0(x) Bessel function versus x.

It oscillates like crazy and the envelope of those oscillations falls off very slowly. In fact, an asymptotic expansion for J0 at large x is

An asymptotic expression for the J0 Bessel function at large argument.

The leading factor of 1/√x decays so slowly that its integral from zero to infinity does not converge. Yet, when you include the cosine so the function oscillates, the integral does converge. Here’s a plot of

An expression for the integral of the Bessel function J0(x') from 0 to x.

A plot of the integral of the J0 Bessel function.

The integral approaches one at large x, but very slowly. So, the expression given in the problem is correct, but I sure wouldn’t want to do any numerical calculations using it, where I had to truncate the endpoint of the integral to something less than infinity. That would be a mess!

Here’s another interesting fact. Bessel functions come in many orders—J0, J1, J2, etc.—and they all integrate to one.

Who’s responsible for these strangely-behaved functions? They’re named after the German astronomer Friedrich Bessel but they were first defined by the Swiss mathematician Daniel Bernoulli (1700–1782), a member of the brilliant Bernoulli family. The Bernoulli equation, mentioned in Chapter 1 of IPMB, is also named for Daniel Bernoulli. 

There was a time when I was in graduate school that I was obsessed with Bessel functions, especially modified Bessel functions that don’t oscillate. I’m not so preoccupied by them now, but they remain my favorite of the many special functions encountered in physics.

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