A Numerical Method for Analytic Continuation of Conformal Mappings

摘要

A numerical analytic continuation of holomorphic functions is obtained by solving an initial value problem for the Cauchy-Riemann equations. The method is applied to computing conformal mappings and determining their analytic continuations. Let Omega denote a bounded, open, simply connected domain in the complex plane, C. Assume the boundary (partial differentiation Omega) is an analytic curve. Let D be the unit disk. By Z = F (w), a conformal mapping from D onto Omega is denoted. The analyticity of (partial differentiation Omega) assures that F(w) can be continued analytically across every point of the unit circle. Two problems, assuming F(w) is known on modulus w = 1, are considered: (1) determine the analytic continuation of F(w) into C for D, and (2) determine F(w) in D. For both problems, a solution is computed by a finite difference method.