Conformal mapping of the domain bounded by a circular polygon with zero angles onto the unit disc

Vladimir Mityushev

Annales Polonici Mathematici (1998)

  • Volume: 68, Issue: 3, page 227-236
  • ISSN: 0066-2216

Abstract

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The conformal mapping ω(z) of a domain D onto the unit disc must satisfy the condition |ω(t)| = 1 on ∂D, the boundary of D. The last condition can be considered as a Dirichlet problem for the domain D. In the present paper this problem is reduced to a system of functional equations when ∂D is a circular polygon with zero angles. The mapping is given in terms of a Poincaré series.

How to cite

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Vladimir Mityushev. "Conformal mapping of the domain bounded by a circular polygon with zero angles onto the unit disc." Annales Polonici Mathematici 68.3 (1998): 227-236. <http://eudml.org/doc/270219>.

@article{VladimirMityushev1998,
abstract = {The conformal mapping ω(z) of a domain D onto the unit disc must satisfy the condition |ω(t)| = 1 on ∂D, the boundary of D. The last condition can be considered as a Dirichlet problem for the domain D. In the present paper this problem is reduced to a system of functional equations when ∂D is a circular polygon with zero angles. The mapping is given in terms of a Poincaré series.},
author = {Vladimir Mityushev},
journal = {Annales Polonici Mathematici},
keywords = {conformal mapping; boundary value problem; functional equation; Dirichlet problem; Poincaré series},
language = {eng},
number = {3},
pages = {227-236},
title = {Conformal mapping of the domain bounded by a circular polygon with zero angles onto the unit disc},
url = {http://eudml.org/doc/270219},
volume = {68},
year = {1998},
}

TY - JOUR
AU - Vladimir Mityushev
TI - Conformal mapping of the domain bounded by a circular polygon with zero angles onto the unit disc
JO - Annales Polonici Mathematici
PY - 1998
VL - 68
IS - 3
SP - 227
EP - 236
AB - The conformal mapping ω(z) of a domain D onto the unit disc must satisfy the condition |ω(t)| = 1 on ∂D, the boundary of D. The last condition can be considered as a Dirichlet problem for the domain D. In the present paper this problem is reduced to a system of functional equations when ∂D is a circular polygon with zero angles. The mapping is given in terms of a Poincaré series.
LA - eng
KW - conformal mapping; boundary value problem; functional equation; Dirichlet problem; Poincaré series
UR - http://eudml.org/doc/270219
ER -

References

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  2. [2] T. Akaza and K. Inoue, Limit sets of geometrically finite free Kleinian groups, Tôhoku Math. J. 36 (1984), 1-16. Zbl0537.30035
  3. [3] S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, Springer, New York, 1992. Zbl0765.31001
  4. [4] È. N. Bereslavskiĭ, On integrating in closed form of a class of Fuchsian equations and its applications, Differentsial'nye Uravneniya 25 (1989), 1048-1050 (in Russian). 
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  7. [7] G. M. Golusin, Geometrische Funktionentheorie, Deutscher Verlag Wiss., Berlin 1957. 
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  9. [9] S. G. Michlin, Integral Equations, Pergamon Press, New York, 1964. 
  10. [10] V. V. Mityushev, Plane problem for the steady heat conduction of material with circular inclusions, Arch. Mech. 45 (1993), 211-215. Zbl0785.73055
  11. [11] V. V. Mityushev, Solution of the Hilbert problem for a multiply connected domain, Słupskie Prace Mat. Przyr. 9a (1994), 37-69. Zbl0818.30026
  12. [12] P. Ya. Polubarinova-Kochina, On additional parameters on the examples of circular 4-polygons, Prikl. Mat. i Mekh. 55 (1991), 222-227 (in Russian). 
  13. [13] A. R. Tsitskishvili, On construction of analytic functions which map conformally the half-plane onto circular polygons, Differentsial'nye Uravneniya 21 (1985), 646-656 (in Russian). 
  14. [14] V. I. Vlasov and S. L. Skorokhod, Analytical solution of the Dirichlet problem for the Poisson equation for a class of polygonal domains, Vychisl. Tsentr Akad. Nauk SSSR, Moscow, 1988 (in Russian). 
  15. [15] V. I. Vlasov and D. B. Volkov, The Dirichlet problem in a disk with a corner cut, Vychisl. Tsentr Akad. Nauk SSSR, Moscow, 1986 (in Russian). 
  16. [16] V. I. Vlasov and D. B. Volkov, Solution of the Dirichlet problem for the Poisson equation in some domains with a complex boundary structure, Vychisl. Tsentr Akad. Nauk SSSR, Moscow, 1989 (in Russian). Zbl0731.35029

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