I have Laplace's equation in polar coordinates,
$\frac{1}{r} \frac{\delta}{\delta r}(r \frac{\delta u}{\delta r}) + \frac{1}{r^{2}} \frac{\delta ^{2} u}{\delta \theta ^{2}} =0$
with initial data, $u = f(\theta )$, $\frac{\delta u}{\delta r} = g(\theta )$ for $ r=1$. Here f,g are real analytic and of period $2 \pi$ for all real $\theta\:$. I have the general form of the solution using the separation of variables but I have no clue how to find the coefficients. Thanks.
Edit: Here is the work I have done.
Using a solution of the form $u(r,\theta ) = R(r) \Theta (\theta)$ I arrived at the two equations $\Theta (\theta)''+ \lambda \Theta (\theta) =0$ and $r^{2}R''(r)+rR'(r)-r^{2}R(r)=0$. Solving these and combining the solutions I came up with the general solution of $u(r, \theta) = \frac{a_{0}}{2} +d_{0}\ln(r) +\sum_{n \geq 1} \frac{r^{2}}{a^{2}}[a_n \cos(n\theta) +b_n \sin(n \theta)] + \sum_{n \geq 1} \frac{a^{2}}{r^{2}}[c_{n} \cos(n \theta) + d_{n} \sin(n\theta)]$
From there I don't know how to find the coefficients from the boundary conditions.