Applying simple boundary conditions to a multi-variable function. If I have the following general differential equation solution (to Laplace equation):
$$
\phi(r,\theta)= A+B\ln(r)+\sum_{n=1}^{\infty}\left(C_nr^n+\frac{D_n}{r^n}\right)\left( E_n \cos(n\theta) + F_n \sin(n\theta) \right) \quad \ \quad  (*)
$$
subject to $\phi(r=0,\theta=0)=0$, why is the particular solution
$$
\phi(r,\theta)= \sum_{n=1}^{\infty}C_nF_n r^n \sin(n\theta) \quad ?
$$
A priori, why $A=B=D_n=0 \ \forall \ n$ is clear. With these conditions $(*)$ becomes
$$
\phi(r,\theta)=\sum_{n=1}^{\infty}C_nr^n\left( E_n \cos(n\theta) + F_n \sin(n\theta) \right)
$$
However, I dont see why $E_n =0$, the above already satisfies the boundary conditions. Moreover couldn’t one keep $D_n\neq0$ provided $E_n=0$?
To me it seems as if there is not enough information to uniquely determine $\phi(r,\theta)$.
 A: I agree with you. There is not enough information to get your particular solution from your general solution given the only boundary condition provided.
From your general solution, setting $\phi(r = 0, \theta) =0$, leads to $A = B = 0$ and $D_n = 0$. You are then left with
$$ \phi(r,\theta)=\sum_{n=1}^{\infty}C_nr^n\left( E_n \cos(n\theta) + F_n \sin(n\theta) \right),$$
or, with a redefinition of coefficients,
$$ \phi(r,\theta)=\sum_{n=1}^{\infty}r^n\left( \alpha_n \cos(n\theta) + \beta_n \sin(n\theta) \right).$$
The values of $\alpha_n$ and $\beta_n$ cannot be uniquely determined from the only condition you provide $\phi(r=0, \theta) =0$. Instead, we would need some additional condition like $F(\theta) = \phi(r = R, \theta)$, which would allow us to compute $\alpha_n$ and $\beta_n$ according to
$$\alpha_n = \frac{1}{\pi R^n}\int^{2\pi}_{0} F(\theta) \cos(n \theta), \quad \beta_n = \frac{1}{\pi R ^n}\int^{2\pi}_{0} F(\theta) \sin(n \theta).$$
Sometimes the integration domain is $[-\pi, \pi]$ in which case we could use symmetry to eliminate one of the integrals. In either case, I think you're missing a boundary condition.
