Computation of the Riemann Map using Integral Equation Method and Cauchy’s Integral Formula Ali Hassan Mohamed Murid Nurul Akmal Mohamed Department of Mathematics, Faculty of Science Universiti Teknologi Malaysia 81310 Skudai, Johor, Malaysia ahmm@mel.fs.utm.my, akmalmohdy@yahoo.com Abstract The Riemann map is a conformal mapping that maps a simply connected region to a unit disk. Such a map has applications in fluid mechanics, electrostatics, and image processing. We present a numerical procedure for the computation of the Riemann map based on two stages. First we compute the boundary values of the Riemann map for the region we wish to map by solving an integral equation. Then we compute the Riemann map in the interior of the region using the well-known Cauchy’s integral formula. Due to periodicity, trapezoidal rule is the most appealing procedure for these computations. We also provide some results of our numerical experiments using epitrochoid (”apple”) as a test region. Keywords: Riemann map, Integral equation, Cauchy’s integral formula. 1. Introduction An important and familiar tool of science and engineering since the develop- ment of complex analysis is conformal mapping. Conformal mapping uses functions of complex variables to transform complicated boundaries to sim- pler, more manageable configurations. A conformal mapping has the special property that angles between curves are preserved in magnitude as well as in direction (see Figure 1). Thus any set of orthogonal curves in the z -plane would therefore appear as another set of orthogonal curves in the w-plane. An important fact about conformal mapping which accounts for much of its applications is that it the Laplace’s equation is invariant under conformal mapping (Henrici, 1974, Section 5.6). This forms the basis of a method of solving numerous two-dimensional boundary-value problems such as the Dirichlet problem and the Neumann problem. 1