Conformal mapping of unbounded multiply connected regions onto canonical slit regions

We present a boundary integral equation method for conformal mapping of unbounded multiply connected regions onto five types of canonical slit regions. For each canonical region, three linear boundary integral equations are constructed from a boundary relationship satisfied by an analytic function on an unboundedmultiply connected region. The integral equations are uniquely solvable. The kernels involved in these integral equations are the modified Neumann kernels and the adjoint generalized Neumann kernels

Numerical computation of the conformal map onto lemniscatic domains

We present a numerical method for the computation of the conformal map from unbounded multiply-connected domains onto lemniscatic domains. For $\ell$-times connected domains the method requires solving $\ell$ boundary integral equations with the Neumann kernel. This can be done in $O(\ell^2 n \log n)$ operations, where $n$ is the number of nodes in the discretization of each boundary component of the multiply connected domain. As demonstrated by numerical examples, the method works for domains with close-to-touching boundaries, non-convex boundaries, piecewise smooth boundaries, and for domains of high connectivity.Comment: Minor revision; simplified Example 6.1, and changed Example 6.2 to a set without symmetr

Fast and accurate computation of the logarithmic capacity of compact sets

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