I'm trying to describe diffusion through a solid cylinder by following Crank's "The Mathematics of Diffusion". I more or less follow the method of adapting the diffusion equation for a cylinder, and using separation of variables to get the general equation. But when it comes time to introduce the boundary conditions we see Bessel's functions appear and this is where I get lost. I have never seen these before (I am not a mathematician!) and I was wondering if someone could give me some tips for how to use an equation like the one attached (sorry, I don't know how to write it in the post). Specifically, what are the alpha terms? Crank's book says they are the roots of the bessel function, but I feel like this means you go around in circles? Any help would be greatly appreciated.