Abstract
In 1969 Symm introduced a new method for the conformal mapping of a ring domain G onto a concentric circular ring domain. The method involved the solution of an integral equation of the first kind with logarithmic kernel, and although it was numerically successful, some theoretical questions remained open. In this paper we deal with these questions and prove the existence and uniqueness of the solution of the integral equation under mild conditions on ∂G. Most of our results are proved for regions of connectivity n ≧ 2. We report on experiments which give approximations to the module M of various ring domains.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Literatur
Bowman, F.: Notes on two-dimensional electric field problems. Proc. London Math. Soc. 39 (1935), 205–215.
Gaier, D.: Integralgleichungen erster Art und konforme Abbildung. Math. Z. 147 (1976), 113–129.
Hayes, J.K., Kahaner, D.K., und Kellner, R.G.: An improved method for numerical conformal mapping. Math. Comp. 26 (1972), 327–334.
Kühnau, R.: Gauss-Thomsonsches Prinzip minimaler Energie, verallgemeinerte transfinite Durchmesser und quasikonforme Abbildungen. Proceedings III. Romanian-Finnish Seminar on Complex Analysis, Bukarest 1976, Springer Lecture Notes 743 (1979), 140–164.
Priwalow, I.I.: Randeigenschaften analytischer Funktionen. 2.Auflage. Deutscher Verlag der Wissenschaften, Berlin 1956.
Sollbach, H.: Konforme Abbildung von zweifach zusammenhängenden Gebieten mit Integralgleichungen erster Art. Diplomarbeit Giessen 1979.
Symm, G.T.: An integral equation method in conformal mapping. Numer. Math. 9 (1966), 250–258.
Symm, G.T.: Numerical mapping of exterior domains. Numer. Math. 10 (1967), 437–445.
Symm, G.T.: Conformal mapping of doubly-connected domains. Numer. Math. 13 (1969), 448–457.
Weisel, J.: Lösung singulärer Variationsprobleme durch die Verfahren von Ritz und Galerkin mit finiten Elementen–Anwendungen in der konformen Abbildung. Mitt. Math. Sem. Giessen 138 (1979), 1–150.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1981 Springer Basel AG
About this chapter
Cite this chapter
Gaier, D. (1981). Das logarithmische Potential und die konforme Abbildung mehrfach zusammenhängender Gebiete. In: Butzer, P.L., Fehér, F. (eds) E. B. Christoffel. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5452-8_19
Download citation
DOI: https://doi.org/10.1007/978-3-0348-5452-8_19
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-5453-5
Online ISBN: 978-3-0348-5452-8
eBook Packages: Springer Book Archive