I have the problem: $$u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}(u_{\theta\theta}),0<\theta<2\pi, 0<R_1<r<R_2$$
$$u(R_1,\theta)=f(\theta), u(R_2,\theta)=g(\theta)$$ $$u(r,0)=u(r,2\pi),u_\theta(r,0)=u_\theta(r,2\pi)$$
And this seems like a pretty standard Laplace equation on an annulus problem. I'm able to perform separation of variables and carry on until I get to the point
$$u(r,\theta)=\sum_{0}^{\infty}[(A_ncos(n\theta)+B_nsin(n\theta))(C_nr^n+D_nr^{-n})]$$
Here is where I am stuck, as I do not know how to apply the boundary condition: $$u(R_1,\theta)=f(\theta), u(R_2,\theta)=g(\theta)$$ to the question. In most cases that I have seen, the application of one of these two equations allows a simplification of some kind, eliminating or allowing the determination of C or D. From there a fourier series is used. I do not see how this is the case here, due to the equations both being equal to a function of theta.
Any help is appreciated.