Using Separation of variables for $\nabla^2 u$ in polar coordinates I got to the general solution:
$$u(r,\theta)=A_0 + B_0 + \sum_{n=1}^{\infty} \left[ \left( A_nr^n+C_nr^{-n} \right)cos(n\theta) + \left( B_nr^n+D_nr^{-n} \right)sin(n\theta) \right] $$
and I have the initial condition $u(a,\theta)=0$
My question is that, if given the general solution can just assume that, due to the divergencies at $r=0$ and $r\rightarrow \infty$ , $ B_0=C_n=D_n=0 $ ? Then: $$ u(r,\theta)=A_0+\sum_{n=1}^\infty \left[A_nr^ncos(n\theta)+B_nsin(n\theta)\right]$$
and use Fourier integrals to find the coefficients.
If not, I evaluate in r=a, then:
$$u(a,\theta)=0=A_0 + B_0 + \sum_{n=1}^{\infty} \left[ \left( A_na^n+C_na^{-n} \right)cos(n\theta) + \left( B_na^n+D_na^{-n} \right)sin(n\theta) \right] $$ but how do I proceed from there?