Determination of accessory parameters for mapping onto a numerable polygon

We consider a simply connected region of the half-plane type with symmetry of translation along the real axis by $2\pi$ and such that a part of the boundary from a point $w_0$ to a point $w_0+2\pi$ consists of a finite number of straight line segments and rays. The region is called a numerable polygon with symmetry of translation along the real axis by $2\pi$. Conformal mappings of the upper half-plane onto a numerable polygon find applications in some problems of hydrodynamics, heat conduction problems, microwave theory, etc. The representation of conformal mappings of the half-plane onto a numerable polygon with symmetry of translation along the real axis by $2\pi$ is known in a form of the Christoffel–Schwarz type integral. Different efficient numerical methods of finding the accessory parameters included in the classical Christoffel–Schwarz integral have been developed; one of them was proposed by P. P. Kufarev. In this paper, the problem of finding the accessory parameters in the Christoffel–Schwarz integral for mapping onto a numerable polygon with symmetry of translation by $2\pi$ along the real axis is reduced to the problem of integrating a system of ordinary differential equations with Cauchy initial conditions by use of an idea of P. P. Kufarev. The system of differential equations is derived using the Christoffel–Schwarz formula for mapping onto a numerable polygon and the differential equation of the Loewner type for mapping the half-plane onto the plane with cuts along pairwise disjoint simple curves $\gamma_m$ tending to infinity, $\gamma_m=\gamma_0+2\pi m$, $m\in\mathbb Z$.