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CONFORMAL SLIT MAPS IN APPLIED MATHEMATICS

Part of: Geometric function theory

Published online by Cambridge University Press:  17 September 2012

DARREN CROWDY
DARREN CROWDY*
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK (email: d.crowdy@imperial.ac.uk)
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Abstract

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Conformal slit maps play a fundamental theoretical role in analytic function theory and potential theory. A lesser-known fact is that they also have a key role to play in applied mathematics. This review article discusses several canonical conformal slit maps for multiply connected domains and gives explicit formulae for them in terms of a classical special function known as the Schottky–Klein prime function associated with a circular preimage domain. It is shown, by a series of examples, that these slit mapping functions can be used as basic building blocks to construct more complicated functions relevant to a variety of applied mathematical problems.

Keywords

conformal slit mapsSchottky–Klein prime functionmultiply connected

MSC classification

Secondary: 30C20: Conformal mappings of special domains

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 53 , Issue 3 , January 2012 , pp. 171 - 189
Copyright
Copyright © Australian Mathematical Society 2012

References

[1]Baker, H. F., Abelian functions: Abel’s theorem and the allied theory of theta functions (Cambridge University Press, Cambridge, 1897).Google Scholar
[2]Bergman, S., The kernel function and conformal mapping, Volume 5 of Mathematical Surveys (American Mathematical Society, Providence, RI, 1950).CrossRefGoogle Scholar
[3]Burnside, W., “On functions determined from their discontinuities and a certain form of boundary condition”, Proc. Lond. Math. Soc. 22 (1890) 346–358; doi:10.1112/plms/s1-22.1.346.CrossRefGoogle Scholar
[4]Crowdy, D. G., “Analytical formulae for source and sink flows in multiply connected domains”, Theor. Comput. Fluid Dyn., to appear; doi:10.1007/s00162-012-0258-x.Google Scholar
[5]Crowdy, D. G., “The Schwarz–Christoffel mapping to bounded multiply connected polygonal domains”, Proc. R. Soc. A 461 (2005) 2653–2678; doi:10.1098/rspa.2005.1480.CrossRefGoogle Scholar
[6]Crowdy, D. G., “Schwarz–Christoffel mappings to unbounded multiply connected polygonal regions”, Math. Proc. Cambridge Philos. Soc. 142 (2007) 319–339; doi:10.1017/S0305004106009832.Google Scholar
[7]Crowdy, D. G., “Multiple steady bubbles in a Hele-Shaw cell”, Proc. R. Soc. A. 465 (2009) 421–435; doi:10.1098/rspa.2008.0252.Google Scholar
[8]Crowdy, D. G., “A new calculus for two-dimensional vortex dynamics”, Theor. Comput. Fluid Dyn. 24 (2010) 9–24; doi:10.1007/s00162-009-0098-5.Google Scholar
[9]Crowdy, D. G., “The Schottky–Klein prime function on the Schottky double of planar domains”, Comput. Methods Funct. Theory 10 (2010) 501–517.Google Scholar
[10]Crowdy, D. G., “Frictional slip lengths for unidirectional superhydrophobic grooved surfaces”, Phys. Fluids 23 (2011) 072001; doi:10.1063/1.3605575.Google Scholar
[11]Crowdy, D. G., Fokas, A. S. and Green, C. C., “Conformal mappings to multiply connected polycircular arc domains”, Comput. Methods Funct. Theory 11 (2011) 685–706.Google Scholar
[12]Crowdy, D. G. and Green, C. C., “The Schottky–Klein prime function”,http://www2.imperial.ac.uk/∼dgcrowdy/SKPrime.Google Scholar
[13]Crowdy, D. G. and Green, C. C., “Analytical solutions for von Kármán streets of hollow vortices”, Phys. Fluids 23 (2011) 126602; doi:10.1063/1.3665102.Google Scholar
[14]Crowdy, D. G. and Marshall, J. S., “Analytical formulae for the Kirchhoff–Routh path function in multiply connected domains”, Proc. R. Soc. A 461 (2005) 2477–2501; doi:10.1098/rspa.2005.1492.Google Scholar
[15]Crowdy, D. G. and Marshall, J. S., “Conformal mappings between canonical multiply connected domains”, Comput. Methods Funct. Theory 6 (2006) 59–76.CrossRefGoogle Scholar
[16]Crowdy, D. G. and Marshall, J. S., “Computing the Schottky–Klein prime function on the Schottky double of planar domains”, Comput. Methods Funct. Theory 7 (2007) 293–308.CrossRefGoogle Scholar
[17]DeLillo, T. K., “Schwarz–Christoffel mapping of bounded, multiply connected domains”, Comput. Methods Funct. Theory 6 (2006) 275–300.CrossRefGoogle Scholar
[18]DeLillo, T. K., Driscoll, T. A., Elcrat, A. R. and Pfaltzgraff, J. A., “Computation of multiply connected Schwarz–Christoffel maps for exterior domains”, Comput. Methods Funct. Theory 6 (2006) 301–315.CrossRefGoogle Scholar
[19]DeLillo, T. K., Elcrat, A. R. and Pfaltzgraff, J. A., “Schwarz–Christoffel mapping of multiply connected domains”, J. Anal. Math. 94 (2004) 17–47; doi:10.1007/BF02789040.Google Scholar
[20]Driscoll, T. A. and Trefethen, L. N., Schwarz–Christoffel mapping (Cambridge University Press, Cambridge, 2002).CrossRefGoogle Scholar
[21]Fay, J. D., Theta functions on Riemann surfaces, Volume 352 of Lecture Notes in Mathematics (Springer, New York, 1973).Google Scholar
[22]Goluzin, G. M., Geometric theory of functions of a complex variable, Volume 26 of Translations of Mathematical Monographs (American Mathematical Society, Providence, RI, 1969).Google Scholar
[23]Gustafsson, B., “Quadrature identities and the Schottky double”, Acta Appl. Math. 1 (1983) 209–240; doi:10.1007/BF00046600.Google Scholar
[24]Hejhal, D. A., Theta functions, kernel functions, and Abelian integrals, Volume 129 of Memoirs of the American Mathematical Society (American Mathematical Society, Providence, RI, 1972).Google Scholar
[25]Klein, F., “Zur Theorie der Abel’schen Functionen”, Math. Ann. 36 (1890) 1–83; doi:10.1007/BF01199432.Google Scholar
[26]Milne-Thomson, L. M., Theoretical hydrodynamics, 5th edn (Dover, New York, 2011).Google Scholar
[27]Nehari, Z., Conformal mapping (Dover, New York, 1952).Google Scholar
[28]Philip, J. R., “Flows satisfying mixed no-slip and no-shear conditions”, Z. Angew. Math. Phys. (ZAMP) 23 (1972) 353–372; doi:10.1007/BF01595477.CrossRefGoogle Scholar
[29]Quéré, D., “Wetting and roughness”, Ann. Rev. Mat. Res. 38 (2008) 71–99; doi:10.1146/annurev.matsci.38.060407.132434.CrossRefGoogle Scholar
[30]Saffman, P. G., Vortex dynamics (Cambridge University Press, Cambridge, 1992).Google Scholar
[31]Schiffer, M., “Recent advances in the theory of conformal mapping”, appendix to: R. Courant, Dirichlet’s principle, conformal mapping, and minimal surfaces (Interscience Publishers, New York, 1950).Google Scholar
[32]Schottky, F., “Über eine specielle Function welche bei einer bestimmten linearen Transformation ihres Arguments unverändert bleibt”, J. reine angew. Math. 101 (1887) 227–272.CrossRefGoogle Scholar
[33]Sedov, L. I., Two-dimensional problems in hydrodynamics and aerodynamics (Wiley, London, 1965).Google Scholar
[34]Taylor, G. I. and Saffman, P. G., “A note on the motion of bubbles in a Hele-Shaw cell and porous medium”, Q. J. Mech. Appl. Math. 12 (1959) 265–279; doi:10.1093/qjmam/12.3.265.CrossRefGoogle Scholar
[35]Valiron, G., Cours d’analyse mathématique: Théorie des fonctions, 3rd edn (Masson et Cie, Paris, 1966).Google Scholar
[36]Whittaker, E. T. and Watson, G. N., A course of modern analysis (Cambridge University Press, Cambridge, 1927).Google Scholar