Determining parameters of conformal mappings from the upper halfplane onto straight-line periodic polygons with double symmetry and onto circular periodic polygons

The paper solves the problem of constructing conformal mappings from the half-plane onto a periodic polygon. A periodic polygon $\Delta$ is a simply connected domain with symmetry of translation, i.e. it has the property $L(\Delta)=\Delta$, where $L(w)=w+2\pi$. We consider a polygon with a boundary consisting of a countable number of circular arcs. Moreover, it has a unique prime end at infinity, fixed under the shift $L(w)$. We use a Schwarz-type differential equation for the representation of the mapping. There is a classical problem of determining parameters for equations of this type. They are the preimages of polygon's vertices under the mapping and additional accessory parameters. To determine these parameters, we generalize Kufarev's method. It was proposed for solving the problem of finding parameters in the Schwarz–Christoffel integral. The method, based on Loewner's differential equation, reduces the problem to the Cauchy problem for a system of ordinary differential equations. There is a differential equation of the Loewner type for periodic polygon. Separately, we consider periodic polygons that have mirror symmetry with respect to a couple of vertical lines; their boundaries consist of straight line segments. We give an example of mapping of the half-plane onto a specified periodic polygon with a boundary consisting of circular arcs and determine its parameters using Kufarev's method.