Here's the answer: an exponential times a sine/cosine wave is what you already have in rectangular coordinates. The Bessel functions show up because they are the cylindrical coordinate equivalent of sine, cosine and exponential. "What does that mean," you say? It is quite simple, as sine, cosine, the exponential function and all Bessel functions are defined as the solutions to specific "fundamental" differential equations. The basic thread between solving Laplace's equations in any coordinate system is, you separate variables and break down the equations in other ways until you have factored them into a set of standard ("prime?" I don't know if that's a good word, I'm not a mathematician) DEs that somebody like Bessel has tabulated the numerical solution of.
