A Review of Complex Variable Theory

Abstract

Before developing the mathematical foundations of the Complex Variable Boundary Element Method (CVBEM), the basic tools needed for the development of the method need to be reviewed. In this chapter, a brief surrmary of the prerequisite complex variable theory is presented. The basic definitions of analytic function theory are reviewed first, and the necessary line integral theory including the Cauchy theorems is then addressed. Because the use of computers in engineering and applied mathematics problems is now a conmon practice, computer programming for a complex polynomial approximation on a problem boundary is included in the review. Additionally, a few topics of applied complex variables are reviewed at the end of the chapter. These are not required for the development of the basic CVBEM, but will be useful for study of CVBEM applications to engineering problems. For a more extensive review of the subject, the complex variable theory, the reader is referred to the standard text or reference books; e.g. Churchill, 1974; Kaplan, 1952; Butkov, 1968.