Conformal Mapping via a Density Correspondence for the Double-Layer   Potential
  We derive a representation formula for harmonic polynomials and Laurent polynomials in terms of densities of the double-layer potential on bounded piecewise smooth and simply connected domains. From this result, we obtain a method for the numerical computation of conformal maps that applies to both exterior and interior regions. We present analysis and numerical experiments supporting the accuracy and broad applicability of the method. 
Matt Wala and Andreas Kl\"ockner
math.NA