This problem is due to Mathews and Walker's Mathematical Methods of Physics, exercise 5.1.
On the 2D plane, suppose we have a series of coplanar charged strips of line charge density $\lambda$ and length $2a$, separated by width $2b$. That is, there are strips of charged strips at intervals $(2n(a+b) - a, 2n(a + b) + a)$ for each integer $n$. The problem asks me to use the method of conformal transformations to find the electrostatic potential arising from this system, that is, a function $V(x,y)$ such that $\nabla^2 V = 0$ everywhere and $V$ is constant on each strip.
I have tried to come up with conformal maps such that every strip is created by the same interval via some multivalued map, but I found dealing with branch cuts quite confusing. I also tried to solve for just one charged strip, on the region $-(a+b) < x < a+b$ with periodic boundary conditions on the edges. However, enforcing such conditions while having the function be holomorphic was challenging.
I'd like to get some recommendations on the conformal mapping to use for this problem, and possibly some techniques for coming up with them on the first place. I personally dislike using conformal maps since they feel more like a gut feeling-guided guesses than algorithmic methods for solutions. If there are any general rules of thumb that could help with finding appropriate conformal maps, I'd love to hear them