Continued Fractions and Conformal Mappings for Domains with Angel Points
Keywords:Conformal mapping, approximation, continued fraction, complex variables, rational function.
Here we construct the conformal mappings with the help of the continued fraction approximations. We first show that the method of  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 half-plane. 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 half-plane. 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.
 Abzalilov, D.F., Shirokova, E.A (2016). The approximate conformal mapping onto simply and doubly connected domains. Complex Variables and Elliptic Equations. pp. 1-12.
 Ahlfors, L (1978). Complex Analysis: An introduction to the theory of analytic functions of one complex variable. Third edition, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co.
 Aptekarev, A.I., Yattselev, M.L (2016). Approximations of algebraic functions by rational ones â€” functional analogues of diophantine approximants. Preprint of Keldysh Institute of Applied Mathematics RAS, Moscow.
 Aptekarev, A.I., Yattselev, M.L (2015). Pade approximants for functions with branch points â€“ strong asymptotics of Nuttallâ€“Stahl polynomials. Acta Math. 215. 217Ð²Ð‚â€œ280
 Cuyt, A., Petersen, V.B (2008). Brigitte Verdonk, Haakon Waadeland. William B. Jones, Handbook of Continued Fractions for Special Functions, Springer.
 DeLillo, Th.K., Elcrat, A.R (1993). A Fornberg-like conformal mapping method for slender regions. Journal of Computational and Applied Mathematics, 46. pp. 49-64.
 Driscoll, T.A., Trefethen, L.N (2002). Schwarz-Christoffel mapping. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge.
 Driscoll, T.A., Vavasis, S.A (1998). Numerical conformal mapping using cross-ratios and Delaunay triangulation. SIAM J. Sci. Comput. 19. pp. 1783â€“1803.
 Fornberg, B (1980). A numerical method for conformal mappings// SIAM J. Sci. Comput. Vol.1, No. 3. pp. 386-400.
 Fornberg, B.A (1984). A numerical method for conformal mapping of doubly connected regions. SIAM J. Sci. Statist. Comput. 5. 771Ð²Ð‚â€œ783.
 Heinhold, J., Albrecht, R (1954). Zur Praxis der konformen Abbildung. Rend. Circ. Mat. Palermo 3. 130Ð²Ð‚â€œ148.
 Khovanskii, A.N (1963). The application of continued fractions and their generalizations to problems in approximation theory. P. Noordhoff N. V., Groningen.
 Luteberget, B.S (2010). Numerical approximation of conformal mappings. Norwegian University of Science and Technology Department of Mathematical Sciences.
 Porter, R.M (2005). History and Recent Developments in Techniques for Numerical Conformal Mapping in Proceedings of the International Workshop on Quasiconformal Mappings and their Applications (IWQCMA05).
 Rodin, B., Sullivan, D (1987). The convergence of circle packings to the Riemann mapping. Journal of Differential Geometry, 26 (2). pp. 349â€“360.
 Shirokova, E.A (2014). On approximate conformal mapping of the unit disk on an simply connected domain. Russia Mathematics (Iz VUZ). Vol. 58. No. 3. pp. 47â€“56.
 Shirokova, E.A., Abzalilov, D.F (2016). Methods of approximate conformal mapping of the canonical domain on simply connected and doubly connected domains, Materials Intern. Conf. in algebra, analysis and geometry. - Kazan: Kazan. Univ; AN RT Press. pp. 77-78. (in Russian).
 Shirokova, E.A., El-Shenawy, A (2018). A Cauchy integral method of the solution of the 2D Dirichlet problem for simply or doubly connected domains. Numerical Methods for Partial Differential Equations. https://doi.org/10.1002/num.22290
 Shirokova, E.A., Ivanshin, P.N (2016). Approximate Conformal Mappings and Elasticity Theory. Journal of Complex Analysis, Vol. 2016, Article ID 4367205. p. 8.
 Shirokova, E.A., Ivanshin, P.N (2011). Spline-interpolation solution of one Elasticity Theory Problem. Bentham Science Publishers.
 Wegmann, R (2005). Methods for numerical conformal mapping. In Handbook of complex analysis: geometric function theory. 2. pp. 351-477.
 Wegman, R., Murid, A.H.M., Nasser, M.M.S (2005). The Riemann-Hilbert problem and the generalized Neumann kernel. J. Comput. Appl. Math. 182. pp. 388-415.
 Wegman, R (1986). An iterative method for conformal mapping. J. Comput. Appl. Math. 14. pp. 7-18.
 Zeng, W., Gu, X.D (2013). Ricci Flow for Shape Analysis and Surface Registration: Theories. Algorithms and Applications, Springer.
 Zygmund, A (2002). Trigonometric series. Vols. I, II, Cambridge University Press.
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