A construction method for conformal mapping of the unit disk onto a Riemann surface

P. N. Ivanshin

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 160 (2018) no. 4, pp. 738-749

Résumé

A construction method for approximate conformal mapping of the unit disk onto a Riemann surface (a map with a self-overlapping image) has been described. An example has been provided to illustrate the applicability of the method to conformal mapping of the unit disk onto a two-sheeted covering of the domain by a Riemann surface. The function construction is based on the approximate solution of the second kind Fredholm integral equation by reducing it to the finite linear equation system, so the construction is easily programmable. The necessary and sufficient condition for the function given on the closed curve to be the boundary value of some function analytic in the region on the Riemann surface bounded by the given curve is naturally somewhat different from that for one-connected and one-sheeted domains. We have applied this condition for a multiply-sheeted region. Let $z(\zeta)$ be the function that maps the unit disk onto a multiply-sheeted region conformally. For the function $\displaystyle \phi (z) = \ln({\zeta (z)}/{z})$, we write the equations similar to that for one-connected and one-sheeted domains. Note that for our example with two-sheeted domain it is necessary to divide the right-hand side of our relations by $3$ for the points on the contour sections bounding the nonunivalent region. The solution then repeats the steps of the one-sheeted domain situation for our case.

Détail

Keywords: conformal mapping, Riemann surface, analytic function, Fredholm equation.