Abstract
AbstractA boundary integral equation method is presented for fast computation of the analytic capacities of compact sets in the complex plane. The method is based on using the Kerzman–Stein integral equation to compute the Szegő kernel and then the value of the derivative of the Ahlfors map at the point at infinity. The proposed method can be used for domains with smooth and piecewise smooth boundaries. When combined with conformal mappings, the method can be used for compact slit sets. Several numerical examples are presented to demonstrate the efficiency of the proposed method. We recover some known exact results and corroborate the conjectural subadditivity property of analytic capacity.
Publisher
Springer Science and Business Media LLC
Reference37 articles.
1. Anderson, G., Vamanamurthy, M., Vuorinen, M.: Conformal Invariants, Inequalities, and Quasiconformal Maps. Wiley, New York (1997)
2. Bell, S.: Numerical computation of the Ahlfors map of a multiply connected planar domain. J. Math. Anal. Appl. 120(1), 211–217 (1986)
3. Bell, S.: The Cauchy Transform, Potential Theory and Conformal Mapping, 2nd edn. CRC Press, Boca Raton (2016)
4. Bolt, M., Snoeyink, S., Van Andel, E.: Visual representation of the Riemann and Ahlfors maps via the Kerzman–Stein equation. Involve 3, 405–420 (2010)
5. Crowdy, D.G.: Finite gap Jacobi matrices and the Schottky–Klein prime function. Comput. Methods Funct. Theory 17, 319–341 (2017)