Conformal mapping of half-plane onto circual numerable polygon with double symmetry

Recently conformal mapping of the upper half-plane onto simply connected domains of the half-plane type with the symmetry of transfer along the real axis by $2\pi$, with a boundary consisting of circular arcs, straight line segments and rays have been used in mathematical physics. In the paper it is proved that the conformal mapping of the upper half-plane onto such domain that has the additional property of symmetry with respect to the vertical straight $\omega=\pi+i\upsilon, \upsilon\in \mathbb{R}$ is a solution of a differential equation of the third order of Christoffel-Schwarz equation type for circular polygons. The received equation depends on the values of the angles at the finite number of vertices, their counter images, the accessory parameters. The proof is based on the Riemann-Schwarz principle of symmetry and the Christoffel-Schwarz formula for circular polygons. The system of two linear algebraic equations for the accessory parameters has been written. For mapping onto the specific circular numerable-polygon with double symmetry, the diffenerential equation, equivalent to the Fuchs class equation with three singular points, has been reduced to the Gauss equation. The map is represented in terms of hypergeometric integrals.