It is well known that the conformal mapping of a simply [or doubly] connected domain

$D$ onto the unit disk [or a circular ring] can be reduced to the solution of a Dirichlet problem. The author solves the latter by the ‘charge simulation method’ which has been studied extensively by several Japanese authors: Murashima, Katsurada, Okamoto and the present author. In this method the solution of the Dirichlet problem is approximated by

$G\left(z\right)={\sum}_{i=1}^{N}{Q}_{i}log|z-{\zeta}_{i}|$ with unknown coefficients

${Q}_{i}$, where the charge points

${\zeta}_{i}$ lie outside

$\overline{D}$. To determine the

${Q}_{i}$ it is required that

$G\left({z}_{j}\right)=b\left({z}_{j}\right)$ for

$N$ collocation points

${z}_{j}\in \partial D$. The error is defined by

${E}_{G}=max|G\left({z}_{j+\frac{1}{2}}\right)-b\left({z}_{j+\frac{1}{2}}\right)|$, where

${z}_{j+\frac{1}{2}}\in \partial D$ are intermediate points. The determination of

$G$ thus leads to the solution of a

$N\times N$ linear system for the

${Q}_{i}$. – The paper fully covers earlier work and gives a unified treatment of three types of mapping problems. The choice of the charge points is discussed, and the condition of the

$N\times N$ matrix is studied. Results of 11 numerical examples are given and compared with previous experiments. Good results are reported in all cases provided that no concave corners on

$\partial D$ occur.