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Computational Methods and Function Theory

, Volume 17, Issue 4, pp 689–713 | Cite as

Fast and Accurate Computation of the Logarithmic Capacity of Compact Sets

  • Jörg LiesenEmail author
  • Olivier Sète
  • Mohamed M. S. Nasser
Article

Abstract

We present a numerical method for computing the logarithmic capacity of compact subsets of \(\mathbb {C}\), which are bounded by Jordan curves and have finitely connected complement. The subsets may have several components and need not have any special symmetry. The method relies on the conformal map onto lemniscatic domains and, computationally, on the solution of a boundary integral equation with the Neumann kernel. Our numerical examples indicate that the method is fast and accurate. We apply it to give an estimate of the logarithmic capacity of the Cantor middle third set and generalizations of it.

Keywords

Logarithmic capacity Transfinite diameter Chebyshev constant Conformal map Lemniscatic domain Boundary integral equation Cantor middle third set 

Mathematics Subject Classification

65E05 30C30 30C85 31A15 

Notes

Acknowledgements

We thank Thomas Ransford for bringing to our attention the analytic formula for the capacity of two unequal disks (Example 4.7). We also thank Nick Trefethen for sharing the numerical results on the capacity of the Cantor middle third set he obtained together with Banjai and Embree.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Jörg Liesen
    • 1
    Email author
  • Olivier Sète
    • 2
  • Mohamed M. S. Nasser
    • 3
  1. 1.Institute of MathematicsTU BerlinBerlinGermany
  2. 2.Mathematical InstituteUniversity of OxfordOxfordUK
  3. 3.Department of Mathematics, Statistics and PhysicsQatar UniversityDohaQatar

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