The Newton method for solving the Theodorsen integral equation (Q1068975)

Theodorsen's integral equation for a starlike region given by the polar coordinates (\(\phi\),\(\rho\) (\(\phi)\)) of its boundary is solved by applying Newton's method (in a Banach space). It is shown that the linear integral equation for the Newton corrections can be reduced to a Riemann- Hilbert problem on a disk. This approach, which a posteriori was also found in concealed form in an old Russian paper by \textit{B. A. Vertgejm} [Dokl. Akad. Nauk SSSR 119, 12-14 (1958; Zbl 0080.288)], is likely to become important in numerical conformal mapping since in practice these Riemann-Hilbert problems can be solved very efficiently using FFTs. Under the assumption that \(\rho\) '' is Lipschitz continuous the local quadratic convergence is proved, and under the assumptions \(\rho ''\in L^{\infty}\) and \(\| \rho '/\rho \|_{\infty}<1/3\) the global convergence of the (undiscretized) iteration is established.