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Conformal mappings between canonical multiply connected domains. (English) Zbl 1101.30010
The authors show that the modified Green’s function of a multiply connected circular domain can be represented by the Schottky-Klein prime function associated with this domain. Using this result, explicit analytic formulae for the conformal mappings from circular domains to domains with parallel, radial or circular slits are constructed.

##### MSC:
 30C20 Conformal mappings of special domains 31A15 Potentials and capacity, harmonic measure, extremal length (two-dimensional)
##### References:
 [1] M. J. Ablowitz and A. S. Fokas, Complex Variables, Cambridge University Press, 1997. [2] H. Baker, Abelian Functions, Cambridge University Press, Cambridge, 1995. [3] A. F. Beardon, A Primer on Riemann Surfaces, London. Math. Soc. Lecture Note Ser. 78, Cambridge University Press, Cambridge, 1984. [4] E. D. Belokolos, A. I. Bobenko, V. Z. Enol’skii, A. R. Its and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer Verlag, 1994. [5] D. G. Crowdy and J. S. Marshall, Analytical formulae for the Kirchhoff-Routh path function in multiply connected domains, Proc. Roy. Soc. A. 461 (2005), 2477–2501. · Zbl 1186.76630 · doi:10.1098/rspa.2005.1492 [6] D. G. Crowdy and J. S. Marshall, The motion of a point vortex through gaps in walls, to appear in J. Fluid Mech. [7] D. G. Crowdy, Schwarz-Christoffel mappings to multiply connected polygonal domains, Proc. Roy. Soc. A 461 (2005), 2653–2678. · Zbl 1186.30005 · doi:10.1098/rspa.2005.1480 [8] D. G. Crowdy, Genus-N algebraic reductions of the Benney hierarchy within a Schottky model, J. Phys. A: Math. Gen. 38 (2005), 10917–10934. · Zbl 1092.37046 · doi:10.1088/0305-4470/38/50/004 [9] J. Gibbons and S. Tsarev, Conformal mappings and reductions of the Benney equations, Phys Lett. A 258 (1999), 263–271. · Zbl 0936.35184 · doi:10.1016/S0375-9601(99)00389-8 [10] P. Henrici, Applied and Computational Complex Analysis, Wiley Interscience, New York, 1986. [11] G. Julia, Lecons sur la representation conforme des aires multiplement connexes, Gaulthiers-Villars, Paris, 1934. [12] H. Kober, A Dictionary of Conformal Representation, Dover, New York, 1957. [13] P. Koebe, Abhandlungen zur Theorie der konformen Abbildung, Acta Mathematica 41 (1914), 305–344. · doi:10.1007/BF02422949 [14] V. V. Mityushev and S. V. Rogosin, Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Function Theory, Chapman & Hall/CRC, London, 1999. [15] D. Mumford, C. Series and D. Wright, Indra’s Pearls, Cambridge University Press, 2002. [16] Z. Nehari, Conformal Mapping, au]McGraw-Hill,_ New York, 1952. [17] M. Schiffer, Recent advances in the theory of conformal mapping, appendix to: R. Courant, Dirichlet’s Principle, Conformal Mapping and Minimal Surfaces, 1950. [18] M. Schmies, Computational methods for Riemann surfaces and helicoids with handles, Ph.D. thesis, University of Berlin, 2005.