Complex Analysis

Abstract

Complex analysis, the theory of functions of a complex variable, is one of the most powerful mathematical instruments of applied mathematicians, engineers, and physicists. We benefit greatly from large segments of this theory. In this chapter we discuss those areas of complex analysis that we need in order to make the presentation as self-contained as possible. We will keep the treatment brief and will not present any proofs of statements or theorems. Instead we demonstrate that the claim holds for specific examples and thereby making the statement at least believable. The interested reader is referred to Churchill (Complex variables and applications, 1960) [1] for further reading. We start by introducing basic concepts like analytic functions, singular points, poles, residues, and contour integration. Then we continue with response functions. A fundamental property that all physical systems have and that is needed for the interactions treated in this book to occur is that the system responds to an external perturbation. The response of the system to various perturbations are described by response functions, correlation functions. The differential equations all show time-reversal symmetry but the response functions are all retarded time-correlation functions which means that the response comes after the perturbation; they obey the causality principle. This means that they have some characteristic properties in the complex frequency plane. We show how this is handled starting from the simplest of systems, the vacuum.