Conformal maps between simply connected domains with piecewise real algebraic boundary

Question

Between polygons in $\mathbb C\cup\{\infty\}$ (including the "single side polygons", hemispheres, disks) the Schwartz-Christoffel mappings give arguably explicit conformal maps. For polygons with few angles those are well known special functions -hypergeometric for triangles to halfplanes and elliptic integrals for rectangles. We also have domains with piecewise quadratic boundaries to which we can write explicit transformations. For instance sectors of disks map to disks via powers -removing the origin to make this conformal- and to halfplanes via Möbius transformations.

My question is whether we can write explicit transformations between domains piecewise bounded by higher degree real algebraic curves, like cubic or quartic, and a disk. There are numerical methods but can we use special functions?

This is related to calculating periods in the sense of Kontsevich-Zagier, though my motivation is rather low-level.

EDIT: I may consider as "explicit" integrals of algebraic functions. If you can give formulas in the form of "generalized Schwarz-Christoffel integrals" but for arbitrary polynomials defining piecewise the boundary that would already be satisfying.

Answer 1

I disagree that the Schwarz-Christoffel formula is "reasonably explicit", except in the case of triangle and rectangle, and very few other cases. The reason is that Schwarz-Christoffel formula for $n\geq 3$ contains unknown "accessory" parameters. Determination of these parameters requires inversion of some rather complicated integrals.

Even less explicit it is for polygons whose sides are arcs of circles. For this case, only circular triangles are reasonably well understood. (See Klein's book Forlesungen uber hypergeometrische Funktion. In English, Caratheodory, Function theory, vol. II).

The case of circular quadrilaterals has been intensively studied since the second half of 19th century. In no way you can call the conformal map on a generic circular quadrilateral "explicit". The mapping is a ratio of two solutions of the Heun equation, and about solutions of this equation not so much is known. Some people would call them "special functions", but they are not included in Whittaker Watson, except the special case of Lame equation. So one can say that there is no explicit answer in any sense already for a generic circular quadrilateral. Some special quadrilaterals were subject of much research.

The literature about them is enormous, it goes under the names "Heun equation", (a special case is the Lame equation), "accessory parameters", "Painleve VI", and there is a lot of "physics" literature, old and modern, with keywords like "conformal blocks", etc.

Of course there are some very special cases when regions are bounded by other algebraic curves, like ellipse, parabola, some cycloids or lemniscates. But these are very special cases.

A reasonable account of what one can do explicitly is: Werner von Koppenfels, and Friedmann Stallmann, Praxis der konformen Abbildung, Berlin, Springer-Verlag. (1959). It is somewhat out of date but not much.

You can glance at my own recent papers (all available on the arxiv) dedicated to some special cases of circular quadrilaterals, and even one on pentagons, arXiv:1611.01356.