While studying for a comprehensive exam, I have come across this old problem:
Consider the Helmholtz equation in the $\mathbb{R}^2$ plane $$u_{xx} + u_{yy} + \omega^2u = 0$$ Derive an integralrepresentation for the axisymmetric solution, i.e. a solution $u(x,y) = U(r)$ where $r = \sqrt{x^2 + y^2}$, by superposition of simple functions.
It later asks me to study the asymptotics for $r \to 0$ and $r \to \infty$ but this isn't difficult once you have the integral form. So the first step was to change to polar coordinates and use the $\theta$ independent form of the laplacian to get:
$$\frac{1}{r} u' + u'' + \omega^2u = 0.$$
However, I'm not finding any "simple solutions". I've tried various substitutions like $u = e^{k(r)}$ but only get non-linear ODEs that I can't solve. I know that this is related to the be Bessel's equation for $n = 0$, but even then, the only references I could find in solving an integral representation is to first solve the ODE with an series solution, then start with the integral and show that the integral has the same series. Certainly this is not what the problem is looking for. Any ideas??