# Conformal map onto a circle, from an identification space composed of five squares

I am looking to derive a conformal map for the problem illustrated in this image. I've read a bit about how to map a square onto a circle, but I'm struggling to extend the concepts for the domain at hand. I don't have a rigorous mathematical background (mech. engineer in computational fluid dynamics), so I would appreciate if someone here could advise me on the route that I should take in order to derive such a conformal mapping.

The end application is to generate a smooth computational mesh that looks like this. I have generated a mesh like this using other means, but the smoothness of the mesh vertices is not sufficient for extremely fine meshes. This results in spurious oscillations in the numerical problem I am trying to solve.

• This is a question more appropriate for Mathematics StackExchange. This site is intended for research-level mathematics. Nov 22, 2019 at 11:06
• @Carl-Fredrik Broda: you think this is not research level mathematics? Perhaps you know the answer? Nov 22, 2019 at 13:25
• @AlexandreEremenko I do not know the answer, no. The problem seemed to me as one which was more appropriate for SE, as it seemed as one that might be readily solved by someone with a more "rigorous mathematical background'' in conformal mappings, as OP puts it, and there might be a higher chance of an answer to this question on SE. But my judgement on this might well be wrong! Nov 22, 2019 at 13:47
• @Carl-Fredrik Nyberg Brodda, thanks for the suggestion! I have posted it on there as well. Nov 23, 2019 at 22:45
• @niran90 It is an interesting question no doubt, so I hope you find the answer you're looking for! Nov 23, 2019 at 23:06

Edited. Using symmetry lines, break the original $$D$$ (5 squares) into 8 trapezoids with angles $$\pi/4,\pi/2,\pi/2,3\pi/4$$. This trapezoid must be mapped conformally onto a sector which makes $$1/8$$-th of the disk. Under this map all angles at the corners are preserved, except $$3\pi/4$$ which becomes $$\pi$$. Then Christoffel-Schwarz leads to the following integral $$C\int_0^z \zeta^{-3/4}(\zeta-a)^{-1/2}(\zeta-1)^{-1/2}d\zeta,$$ where $$a$$ is the accessory parameter, and it is determined from the equation $$\int_0^a=-3i\int_a^1.$$ This equation can be easily solved numerically since the absolute values of right and left sides are monotone in the opposite directions (one increasing another decreasiing as functions of $$a$$). The definite integral involved in the equation for the accessory parameter can be expressed in terms of hypergeometric function, Gradshtein, Ryzhik, p. 317, item 3.197, but I am not sure how much this helps. I would just find $$a$$ numerically.