I am looking for a conformal map from a "polygon" to eg the upper half plane, which consists of circle segments instead of lines. So for example, it could be a quadrilateral ABCD, but where AB is a circle segment. The closest I can find is the Schwarz-Christoffel mapping.

Anyone has any tips?

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    $\begingroup$ It seems like the book "Schwarz-Christoffel Mapping" by Driscoll and Trefethen considers circular-arc polygons in section 4.10. $\endgroup$ – Maxime Fortier Bourque Dec 16 '14 at 15:08
  • $\begingroup$ I see. It is only 3 pages though and considering the other answer is quite involved, I doubt that these three pages will be useful. Do you happen to know the content of that chapter? Cant find the (full) book online.. $\endgroup$ – user2133437 Dec 16 '14 at 17:24

The mapping function is a solution of the Schwarz differential equation $$\frac{f'''}{f'}-\frac{3}{2}\left(\frac{f''}{f'}\right)^2=R(z),$$ where $R$ is a rational function with poles at the preimages of the vertices. The poles of $f$ are of second order, and the coefficients at the second order terms are determined by the angles. Schwarz equation can be reduced to a linear differential equation of the form $y''+Ry/2=0$. In the case of a triangle, this is a hypergeometric equation. In the case of a quadrilateral, this is a Heun equation.

Literature: Courant, Geometrische Funktionentheorie, Caratheodory, Funktionentheorie, II, (There is an English translation), Golubev, Vorlesungen über Differentialgleichungen im Komplexen, (transled from Russian).

The case of a triangle is completely understood (see any book on hypergeometric function) The case of a quadrilateral is already quite complicated, and there are many unsolved questions about quadrilaterals and Heun's equation. See for example arXiv:1409.1529 for some recent results about this, and literature there.

EDIT. More concrete examples can be found here arXiv:1111.2296 and here arXiv:1110.2696 and in references to these papers.

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  • $\begingroup$ Thank you. The particular problem I have indeed includes a quadrilateral. So if I understand it correctly, when mapping the polygon to a riemann sphere the angles in the polygon becomes the conical singularities? Also, do you think there is any simplification if one deals with a quadrilateral where only one segment is a circle segment, and the rest straight lines? $\endgroup$ – user2133437 Dec 16 '14 at 17:17
  • $\begingroup$ Also, as I wrote to the other commenter, do you know the content of the chapter on "circular arc-polygons" in the book "Schwarz-Christoffel Mapping" by Driscoll? Is it related to the stuff you discuss? $\endgroup$ – user2133437 Dec 16 '14 at 17:33
  • $\begingroup$ @user213437: the literature on conformal maps of quadrilaterals is enormous. I am not familiar with that particular book you refer to, but I suppose that it contains some standard material on this. $\endgroup$ – Alexandre Eremenko Dec 16 '14 at 18:05
  • $\begingroup$ @user213437: if you state your problem precisely, perhaps I can give a more specific reference. $\endgroup$ – Alexandre Eremenko Dec 16 '14 at 18:06
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    $\begingroup$ :The problem is in the following link: s11.postimg.org/50fiu0e03/problem.png Basically I want to map the shaded region in the left image to the shaded region in the right image, conformally except at A. Initially I wanted an exact analytic expression for the mapping, but maybe that is too much to ask. $\endgroup$ – user2133437 Dec 16 '14 at 18:54

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