Abstract
We study numerical computation of conformal invariants of domains in the complex plane. In particular, we provide an algorithm for computing the conformal capacity of a condenser. The algorithm applies to a wide variety of geometries: domains are assumed to have smooth or piecewise smooth boundaries. The method we use is based on the boundary integral equation method developed and implemented in [1]. A characteristic feature of this method is that, with small changes in the code, a wide spectrum of problems can be treated. We compare the performance and accuracy to previous results in the cases when numerical data is available and also in the case of several model problems where exact results are available.
Highlights
During the past fifty years conformal invariants have become crucial tools for complex analysis
We review the application of the integral equation to compute the conformal mapping from doubly connected domains onto an annulus {z : q < |z| < 1}, q ∈ (0, 1), and present the MATLAB implementation of the method
If ρ is the unique solution of the boundary integral equation (9) and the piecewise constant function h is given by (10), the function f with the boundary values (11) is analytic in the domain G with f (∞) = 0, the conformal mapping Φ is given by z − z2 z − z1 ef (z), z ∈ G ∪ Γ, (16)
Summary
During the past fifty years conformal invariants have become crucial tools for complex analysis. In cases for which the analytic formulas are unknown, the computational performance may be analysed by observing convergence features of the results under successive refinements of the numerical model, and error estimates maybe based on general theory. In those relatively few cases we have found in the literature where the analytic formula is known, the true error estimate may be given.
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