Numerical methods for elliptic partial differential equations have been the subject of many books in recent years, but few have treated the subject of complex equations. In this important new book, the author introduces the theory of, and approximate methods for, nonlinear elliptic complex equations in multiple connected domains. Constructive methods are systematically applied to proper boundary value problems which include very general boundary conditions. Approximate and numerical methods, such as the Newton imbedding method, the continuity method, the finite element method, the difference method and the boundary integral method, as well as their applications, are discussed in detail. The book will be of interest to all scientists studying the theory or applications of complex analysis.