Dorin Ghisa
York University, Toronto, Canada

Conformal self mappings of a given domain of the complex plane onto itself can be obtained by using Riemann mapping theorem in the following way. Two different conformal mappings f and g of that domain onto one of the standard domains: the unit disc, the complex plane, or the Riemann sphere are taken and the f composed with the inverse of g is what we are looking for. Yet, this is just a theoretical construction, since the Riemann mapping theorem does not offer any concrete expression of those functions. The Möbius transformations are concrete, but they can be used only for particular circular domains.

We are proving in this paper that conformal self mappings of fundamental domains of any analytic function can be obtained via Möbius transformations as long as we allow those domains to have slits. Moreover, those mappings enjoy group properties. Graphic illustrations are offered for the most familiar classes of functions. This is a joint work with Andrei-Florin Albişoru.