(And a related question: Where should an analytic number theorist learn about Bessel functions?)
Bessel functions occur quite frequently in analytic number theory. One example, Corollary 4.7 of Iwaniec and Kowalski, says the following. Let $r(m)$ be the number of representations of $m$ as two squares, and suppose that $g$ is smooth and compactly supported in $(0, \infty)$. Then,
$$\sum_{m = 1}^{\infty} r(m) g(m) = \pi \int_0^{\infty} g(x) dx + \sum_{m = 1}^{\infty} r(m) h(m),$$
where
$$h(y) = \pi \int_0^{\infty} g(x) J_0(2 \pi \sqrt{xy}) dx.$$
$J_0(x)$ is a Bessel function, and I+K follow with four equivalent integral expressions — the equivalence of which is not at all obvious by looking at them. Looking at Lemma 4.17, the relevance appears to be that Bessel functions arise when you take Fourier transforms of radially symmetric functions.
Another example comes from (3.8) of this paper of Miller and Schmid, and the relevance comes from the identity
$$\int_0^{\infty} J_0(\sqrt{x}) x^{s – 1} dx = 4^s \frac{\Gamma(s)}{\Gamma(1 – s)},$$
where the gamma factors come from functional equations of $L$-functions. Okay, if this is true, then I understand why we care, but it seemed a bit deus ex machina to me.
There are many other examples too, for example the Petersson formula in the theory of modular forms, etc. There are $I$-Bessel functions, $K$-Bessel functions, $Y$-Bessel functions, etc., all of which seem to satisfy a dizzying number of highly nontrivial identities, and reading Iwaniec and Kowalski one gets the sense that an expert should have the ability to recognize and manipulate them on sight. They also provide references to, e.g., (23.451.1) of a book by Gradhsteyn and Rizhik, and although I confess I have not looked at it, I can infer from the formula number that it is not the sort of thing I might read on an airport layover.
Meanwhile, Wikipedia tells me that they naturally arise as solutions of certain partial differential equations. Looks extremely interesting, although I'm afraid I am not an expert in PDE.
As an analytic number theorist, how might I make friends with these objects? How should I look at them, and what conceptual frameworks do they fit in? Thank you!
(ed. Thanks to everyone for informative answers! I could only accept one answer but +1 all around)
Best Answer
Radial Fourier transforms provide a good, consistent perspective on most of the theory. The Fourier transform $\widehat{f}(t)$ of a function $f \colon \mathbb{R}^n \to \mathbb{R}$ is given by the integral of $f(x) e^{2\pi i \langle x,t \rangle} \, dx$ over $x \in \mathbb{R}^n$. If $f$ is a radial function (i.e., $f(x)$ depends only on $|x|$), then we can radial symmetrize everything and the exponential function averages out to a radial function. Specifically, we get $$ \widehat{f}(t) = 2\pi |t|^{-(n/2-1)} \int_0^\infty f(r) J_{n/2-1} (2 \pi r |t|) r^{n/2} \, dr. $$ The precise factors are a little annoying, but basically this just means $J_{n/2-1}$ is what you get when you radially symmetrize an exponential function in $n$ dimensions. It's easy to see that if you symmetrize $e^{2\pi i \langle x,t \rangle}$ by averaging over all $x$ on a sphere, then you get a radial function of $t$, and furthermore as you vary the radius of the sphere you just rescale the function. So the one function $J_{n/2-1}$ captures all of this, modulo scaling.
One consequence is that Bessel functions inherit the orthogonality of the exponential functions (i.e., the different scalings are orthogonal), so they also inherit all the consequences of orthogonality. For example, this is really where the differential equation comes from. There's a strong analogy between Bessel functions and orthogonal polynomials, where rescaling the Bessel function corresponds to varying the degree of the polynomial.
You also get certain qualitative results for free: for example, the product of two Bessel functions should be an integral of Bessel functions with positive coefficients, since this corresponds to saying the product of two radial, positive-definite functions remains positive definite. You can write down the coefficients explicitly, but sometimes all you need is nonnegativity, and in any case this point of view makes it easy to believe that there should be an explicit formula.
This is basically a low-brow version of the representation theory approach. Basically, ordinary Fourier analysis studies $L^2(\mathbb{R}^n)$ under the action of the translation group $\mathbb{R}^n$. If you look at the full group of isometries of $\mathbb{R}^n$ (including the orthogonal group), then it's just a little more elaborate, and the Bessel functions arise as zonal spherical functions. It's worthwhile working through this perspective, but in practice just thinking about radial Fourier analysis gives you most of the benefits with less machinery.