ABSTRACT

We will present Theodorsen’s integral equation and establish the convergence of the related iterative method for the standard case of mapping the unit circle onto the interior (or exterior) of almost circular and starlike regions, both containing the origin. A trigonometric interpolation scheme is presented, and Wegmann’s iterative and Newton’s method for numerically solving this equation are discussed. The last two methods are based on a certain Riemann-Hilbert problem, which turns out to be a linearized form of a singular integral equation of the second kind. Unlike the classical iterative method, the solution of the linearized problem in Wegmann’s method for the conformal map of the unit circle can be represented explicitly in terms of integral transforms, which leads to a quadratic convergent Newton-like method that avoids the numerical solution of a system of linear equations and thus becomes more economical. Theodorsen’s integral equation has specific significance in the theory of airfoils.