Abstract: The purpose of this article is to systematically explore the geometric properties of finite and infinite Blaschke products, as well as of the Dirichlet functions generated by them. Computer-generated graphics are intended to illustrate symmetries, to portray their fundamental domains, and to explore their global mapping properties. They are also instrumental in the extrapolation of the properties of finite Blaschke products to the case of infinite ones, where non-isolated singular points appear. The behavior of an analytic function in the neighborhood of an isolated essential singular point is described by Picard’s theorem, yet this theorem is not applicable to essential non-isolated singular points. The geometric approach we are using allows us to deal with such situations in both the cases of infinite Blaschke products and the Dirichlet functions generated by these products.

Keywords: fundamental domains, Blaschke product, Dirichlet functions, analytic continuation, conformal mapping, pre-images of lines and circles

Pages: 116-136