Continued Fractions and Conformal Mappings for Domains with Angel Points

20180927 https://doi.org/10.14419/ijet.v7i4.7.23039 
Conformal mapping, approximation, continued fraction, complex variables, rational function. 
Abstract
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 which boudary possesses acute internal angles. Then we construct the method of rational root approximation in the right complex halfplane. First we construct the square root approximation and consider 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 halfplane. These approximations converge to the algebraic root functions , , , . 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 is the ratio k/N the worse is the approximation. Also we give the examples that illustrate the conformal mapping construction.
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How to Cite
N. Ivanshin, P., & ., . (2018). Continued Fractions and Conformal Mappings for Domains with Angel Points. International Journal of Engineering & Technology, 7(4.7), 409419. https://doi.org/10.14419/ijet.v7i4.7.23039Received date: 20181203
Accepted date: 20181203
Published date: 20180927