Is there a simple construction of a confomral mapping of the half-plane onto a "circular trianagle", i.e. a domain whose sides are the arcs of three circles.

## 2 Answers

Yes, it is (how simple, is a matter of opinion). You may always assume that $0,1,\infty$ on the boundary of upper half-plane are preimages of the vertices. Now suppose that the inner angles of your triangle are $\pi\alpha_j,$ and let us assume that $\sum\alpha_j$ is not an odd integer (Euclidean triangles must be considered separately and this case is in fact simpler).

Then your mapping function is a solution of the Schwarz differential equation $$\frac{f'''}{f'}-\frac{3}{2}\left(\frac{f''}{f'}\right)^2 =\frac{1-\alpha_1^2}{2z^2}+\frac{1-\alpha_2^2}{2(z-1)^2}+\frac{\alpha_1^2+\alpha_2^2-\alpha_3^2-1}{2z(z-1)}.$$ This is written in Hurwitz-Courant (with a misprint which I corrected), and in Caratheodory, vol. II.

This is not as scary as it may look, because in fact if $F(z)$ is the right hand-side, then $f=y_1/y_2$ where $y_1$ and $y_2$ are linearly independent solutions of the linear differential equation $$y''+(F/2)w=0,$$ which in our case is a hypergeometric equation. Its solutions (hypergeometric functions with real parameters) are special functions, and "everything is known" about them. I mean explicit power series, location of zeros, asymptotics, integral representations, explicit analytic continuation, tables, and so on. Function $f$ contains 3 arbitrary constants which can be determined from positions of vertices of your triangle.

For the case of Euclidean triangle (bounded by straight lines, and with $\sum\alpha_j=1$) the function is more explicit: it is the Schwarz--Christoffel integral $$f(z)=C\int_a^z\zeta^{\alpha_1-1}(\zeta-1)^{\alpha_2-1}d\zeta$$ where $a$ and $C$ are constants, and can be determined from positions of two vertices of your triangle.

Refs. A. Hurwitz and R. Courant, Vorlesungen uber allgemeine Funktionentheorie, (available in German and Russian)

C. Caratheodory, Theory of functions of a complex variable, vol. II (available in German and English). Most of the vol. II is devoted to this subject.

F. Klein, Vorlesungen über die Hypergeometrische Funktion, Berlin 1933 (the whole book is devoted to the subject).

W. Koppenfels and F. Stahlman, The Practice of conformal Mappings (available in German and Russian).

The next case in complexity, circular quadrilaterals, is much more complicated and still remains a research subject.

Edit: A good modern source in English is S. Donaldson, Riemann surfaces, Oxford, 2011, Thm. 29.

The Christoffel-Schwarz formula for polygons bounded by arcs of circles is considered in the PhD thesis Computation of Conformal Maps by Modified Schwarz-Christoffel Transformations (1990) by Louis H. Howell. This thesis is accessible in the Internet.