Effective Conformal Transformation of Smooth, Simply Connected Domains.

1. Statement of Problem.-Many problems in plane potential theory can be solved easily, if one knows how to map the interior (or exterior) of a given closed curve C in the z-plane, conformally and one-one onto the interior (or exterior) of the unit circle t = e"' in the t-plane. This can be done by tabulated functions in some cases, but even in such cases, the solution often involves one or more symbolic parameters, whose numerical determination is awkward. Therefore, there is great interest in obtaining direct iterative or other numerical procedures, applicable to all sufficiently regular domains. Many such schemes have been proposed, of which the most widely used is due to Theodorsen and Garrick.' The analytical formulae involved in these schemes have all been known, or could have been derived easily, since the time of Riemann, Schwarz and Neumann. However, there is a tremendous variation in the accuracy which can be achieved in a limited number of steps, and in the amount of data which must be stored, depending on the method used. We have tried to develop schemes practicable for high-speed computing machines with limited "memories," giving good accuracy in a limited (105-106) number of steps. We outline below two new schemes, one for computing t = f(z) from z, and the other for computing z = g(t) from t, which seem to us to offer various advantages. In addition, an analogous scheme for computing steady plane flows with free boundaries is described. 2. First Solution.-To compute f(z), it seems most convenient to consider