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Schwarz–Christoffel mappings to unbounded multiply connected polygonal regions

Published online by Cambridge University Press:  10 April 2007

DARREN CROWDY
DARREN CROWDY*
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen's Gate, London, SW7 2AZ.
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Abstract

A formula for the generalized Schwarz–Christoffel conformal mapping from a bounded multiply connected circular domain to an unbounded multiply connected polygonal domain is derived. The formula for the derivative of the mapping function is shown to contain a product of powers of Schottky–Klein prime functions associated with the circular preimage domain. Two analytical checks of the new formula are given. First, it is compared with a known formula in the doubly connected case. Second, a new slit mapping formula from a circular domain to the triply connected region exterior to three slits on the real axis is derived using separate arguments. The derivative of this independently-derived slit mapping formula is shown to correspond to a degenerate case of the new Schwarz–Christoffel mapping. The example of the mapping to the triply connected region exterior to three rectangles centred on the real axis is considered in detail.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 142 , Issue 2 , March 2007 , pp. 319 - 339
Copyright
Copyright © Cambridge Philosophical Society 2007

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References

REFERENCES

[1] Ablowitz, M. J. and Fokas, A. S.. Complex Variables (Cambridge University Press, 1997).Google Scholar
[2] Akhiezer, N. I.. Translations of Mathematical Monographs 79, Elements of the theory of elliptic functions (American Mathematical Society, 1970).Google Scholar
[3] Akhiezer, N. I.. Aerodynamical investigations. Ukrain. Akad. Nauk. Trudi. Fiz.-Mat. Viddilu 7 (1928).Google Scholar
[4] Baker, H.. Abelian Functions (Cambridge University Press, 1995).Google Scholar
[5] Beardon, A. F.. A primer on Riemann surfaces. London. Math. Soc. Lecture Note Ser. 78 (Cambridge University Press, 1984).Google Scholar
[6] Crowdy, D. G.. The Schwarz–Christoffel mapping to multiply connected polygonal domains. Proc. Roy. Soc. A 461 (2005), 2653–2678.Google Scholar
[7] Crowdy, D. G. and Marshall, J. S.. Conformal mappings between canonical multiply connected domains. Comput. Methods Funct. Theory 6 (1) (2006), 59–76.CrossRefGoogle Scholar
[8] DeLillo, T. K., Elcrat, A. R. and Pfaltzgraff, J. A.. Schwarz–Christoffel mapping of the annulus. SIAM Rev. 43 (2001), 469–477.Google Scholar
[9] DeLillo, T. K., Elcrat, A. R. and Pfaltzgraff, J. A.. Schwarz–Christoffel mapping of multiply connected domains. J. d'Analyse Math. 94 (2004), 17–48.Google Scholar
[10] Driscoll, T. A. and Trefethen, L. N.. Schwarz–Christoffel mapping. Cambridge Mathematical Monographs (Cambridge University Press, 2002).CrossRefGoogle Scholar
[11] Driscoll, T. A.. SC Toolbox, www.math.udel.edu/~driscoll/SC/.Google Scholar
[12] Hejhal, D. Theta functions, kernel functions and Abelian integrals. Mem. Amer. Math. Soc. No. 129 (1972).Google Scholar
[13] Henrici, P.. Applied and Computational Complex Analysis, Vol. 3 (Wiley Interscience, 1986).Google Scholar
[14] Hu, C. Algorithm 785: A software package for computing Schwarz–Christoffel conformal transformations for doubly connected polygonal regions. ACM Trans. Math. Software 24 (1998), 317–333.Google Scholar
[15] Koebe, P.. Abhandlungen zur theorie der konformen abbildung. Acta Math. 41 (1914), 305–344.Google Scholar
[16] Komatu, Y.. Darstellung der in einem Kreisringe analytischen Funktionem nebst den Anwendungen auf konforme Abbildung uber Polygonalringebiete. Japan J. Math. 19 (1945), 203–215.Google Scholar
[17] Mumford, D., Series, C. and Wright, D.. Indra's Pearls (Cambridge University Press, 2002).CrossRefGoogle Scholar
[18] Nehari, Z.. Conformal Mapping (McGraw-Hill, 1952).Google Scholar
[19] Whittaker, E. T. and Watson, G. N. A Course of Modern Analysis (Cambridge University Press, 1927).Google Scholar