Abstract
The problem of constructing a conformal mapping from the upper half-plane to a circular polygon is considered. The preimages of the vertices of the polygon and accessory parameters are determined by the generalized method of P. P. Kufarev for determining the parameters in the Christoffel–Schwarz integral. The method is based on the Loewner equation with boundary normalization. The problem of constructing a mapping from a half-plane onto the exterior of a polygon with a boundary consisting of straight-line segments is solved separately. Examples of mappings whose parameters are found by the Kufarev method are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
I. A. Aleksandrov, Theory of Functions of a Complex Variable [in Russian], Tomsk (2002).
I. A. Aleksandrov, Parametric Continuations in the Theory of Univalent Functions, Nauka, Moscow (1976).
B. G. Baybarin, “On a numerical method of defining the parameters of the Schwarz derivative for a function, which conformally maps the half-plane onto a circular domain,” Tr. Tomsk. Univ. Mat. Mekh., 189, 123–136 (1966).
E. N. Bereslavskii, “Modeling the motion of groundwater from pits fenced with Zhukovsky sheet piles,” Vestn. Sankt-Peterburg. Univ. Ser. 10. Prikl. Mat. Inform. Prots. Upravl., 13, No. 2, 124-137 (2017).
E. N. Bereslavskii, “On integrating in the closed form of one class of Fuchsian equations and its applications,” Izv. Vyssh. Ucheb. Zaved. Mat., No. 9, 3–5 (1989).
L. I. Chibrikova, “On piecewise holomorphic solutions of Fuchsian equations,” in: Boundary-Value Problems and Their Applications [in Russian], Cheboksary (1986), pp. 136–148.
Yu. V. Chistyakov, Numerical method of defining functions that conformally map the circle onto polygons [in Russian], Ph.D. thesis, Tomsk (1953).
V. Ya. Gutlyanskii and A. O. Zaidan, “On conformal mapping of polygonal regions,” Ukr. Mat. Zh., 45, No. 11, 1669–1680 (1993).
T. R. Hopkins and D. E. Roberts, “Kufarev’s method for determination the Schwarz-Christoffel parameters,” Numer. Math., 33, No. 4, 353–365 (1979).
S. D. Howison and J. R. King, “Explicit solutions to six free-boundary problems in fluid flow and diffusion,” IMA J. Appl. Math., 42, No. 2, 155–175 (1989).
I. A. Kolesnikov, “Defining accessory parameters for a mapping onto a polygon with a countable number of vertices,” Tr. Tomsk. Univ. Mat. Mekh., No. 2 (28), 18–28 (2014).
I. A. Kolesnikov, “On the problem of determining parameters in the Schwarz equation,” Probl. Anal. Issues Anal., 7 (25), No. 2, 50–62 (2018).
W. von Koppenfels and F. Stallmann, Praxis der konformen Abbildung, Springer-Verlag, Berling–Göttingen–Heidelberg (1963).
P. P. Kufarev, “On one method of numerical defining the parameters in the Christoffel–Schwarz integral,” Doll. Akad. Nauk SSSR., 57, No. 6, 535–537 (1947).
P. P. Kufarev, Collected Works [in Russian], Tomsk (2009).
N. N. Nakipov and S. R. Nasyrov, “Parametric method of finding accessory parameters in generalized Christoffel–Schwarz integral,” Uch. Zap. Kazan. Univ. Ser. Fiz.-Mat. Nauki., 158, No. 2, 202–220 (2016).
S. R. Nasyrov, Geometric Problems of the Theory of Branched Coverings of Riemannian Surfaces [[inRussian], Magarif, Kazan (2008).
S. R. Nasyrov and L. Yu. Nizamieva, “Finding of accessory parameters for mixed inverse boundary-value problem with a known polygonal part of boundary,” Izv. Saratov. Univ. Nov. Ser. Mat. Mekh. Inform., 11, No. 4, 34–40 (2011).
Nepritvorennaia L. M., “Finding of unknown parameters of the conformal mapping of the upper half-plane onto an arbitrary circular quadrangle,” Ukr. Mat. Zh., 23, 261–268 (1971).
M. V. Pomeranets and N. L. Yakimov, Solution of the problem on the motion of depression curves in the stationary formulation [in Russian], Preprint VINITI No. 1366–B94 (1994).
A. R. Tsitskishvili, “On constructing analytic functions that conformally map the half-plane onto circular polygons,” Differ. Uravn., 21, No. 4, 646–656 (1985).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 175, Proceedings of the XVII All-Russian Youth School-Conference “Lobachevsky Readings-2018,” November 23-28, 2018, Kazan. Part 1, 2020.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Kolesnikov, I.A. On the Search for Parameters of a Conformal Mapping from a Half-Plane to a Circular Polygon. J Math Sci 272, 803–815 (2023). https://doi.org/10.1007/s10958-023-06474-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-023-06474-2