Continued Fractions and Conformal Mappings for Domains with Angle Points
DOI:
https://doi.org/10.61841/w9p2wy33Keywords:
Conformal mapping, approximation, continued fraction, complex variables, rational function. MSC. 30C30, 30C20Abstract
Here we construct the conformal mappings with the help of the continued fraction approximations. We first show that the method of [19] works for conformal mappings of the unit disk onto domains with acute external angles at the boundary. We give certain illustrative examples of these constructions. Next we outline the problem with domains in which boudary possesses acute internal angles. Then we construct the method of rational root approximation in the right complex half-plane. First we construct the square root approximation and consider the approximative properties of the mapping sequence in Theorem 1. Then we turn to the general case, namely the continued fraction approximation of the rational root function in the complex right half-plane. These approximations converge to the algebraic root functions N z, N ∈N, z ∈C, and Rez > 0. This is proved in Theorem 2 of the aricle. Thus we prove convergence of this method and construct conformal approximate mappings of the unit disk onto domains with angles and thin domains. We estimate the convergence rate of the approximation sequences. Note that the closer the point is to zero or infinity and the lower the ratio k/N, the worse the approximation. Also, we give examples that illustrate the conformal mapping construction.
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