I'm working on a 2D fluid simulation based on Stokes flow. A point disturbance in the fluid at $(r, \phi)$ (polar coordinates) of $F$ can be cancelled at the surface of a circle of radius $R > r$ by smearing point disturbances of various strengths around the perimeter of the circle. I'm trying to determine what these point disturbances are at different points on the circle. Mathematically this looks like solving for $f(t)$ in this integral:
$$2 \int_0^{2\pi} f(\theta + t) (\frac{1}{2} -\ln(2 R \sin\frac{t}{2})) + \overline{f(\theta + t)} e^{i (2 \theta + t)} dt \\= F (1-\ln(r^2 + R^2 - 2 r R \cos(\theta - \phi))) + \overline{F}\frac{r e^{i \phi} - R e^{i \theta}}{r e^{-i \phi} - R e^{-i \theta}} $$
for $\phi, \theta, t, r, R \in \mathbb{R}, F \in \mathbb{C}, 0 \le r < R, f: \mathbb{R} \to \mathbb{C}$, and $\overline{F}$ is the complex conjugate of $F$.
I'm trying to determine roughly what $f(t)$ looks like, though I don't necessarily need the exact function. For instance, empirically I know when $r = 0$, $f(t) = a e^{i (2t + \psi)} + z$ for some unknown $a, \psi \in \mathbb{R}, z \in \mathbb{C}$.
However I have no idea how to even begin generalizing that when $r \ne 0$. Any help getting pointed in the right direction would be appreciated.