The methods of integration of complex functions and their underlying theories are discussed in this chapter. The cornerstones in complex integration are the Cauchy–Goursat theorem and the Cauchy integral formula. A fascinating result deduced from the Cauchy integral formula is that if a complex function is analytic at a point, then its derivatives of all orders exist and these derivatives are analytic at that point. Other important theorems include Gauss' mean value theorem, Liouville's theorem, and the maximum modulus theorem.

Many properties of the complex integrals are very similar to those of the real line integrals. For example, when the integrand satisfies certain conditions, the integral can be computed by finding the primitive function of the integrand and evaluating the primitive function at the two end points of the integration path. However, there are other properties that are unique to integration in the complex plane.

In the last section, we link the study of conservative fields in physics with the mathematical theory of analytic functions and complex integration. The prototype conservative fields considered include the gravitational potential fields, electrostatic fields and potential fluid flow fields. The potential functions in these physical models are governed by the Laplace equation, and so their solutions are harmonic functions. Complex variables techniques are seen to be effective analytical tools for solving these physical models.

This textbook is intended to be an introduction to complex variables for mathematics, science and engineering undergraduate students. The prerequisites are some knowledge of calculus (up to line integrals and Green's Theorem), though basic familiarity with differential equations would also be useful.

Complex function theory is an elegant mathematical structure on its own. On the other hand, many of its theoretical results provide powerful and versatile tools for solving problems in physical sciences and other branches of mathematics. The book presents the important analytical concepts and techniques in deriving most of the standard theoretical results in introductory complex function theory. I have included the proofs of most of the important theorems, except for a few that are highly technical. This book distinguishes itself from other texts in complex variables by emphasizing how to use complex variable methods. Throughout the text, many of the important theoretical results in complex function theory are followed by relevant and vivid examples in physical sciences. These examples serve to illustrate the uses and implications of complex function theory. They are drawn from a wide range of physical and engineering applications, like potential theory, steady state temperature problems, hydrodynamics, seepage flows, electrostatics and gravitation. For example, after discussing the mathematical foundations of the Laplace transform and Fourier transform, I show how to use the transform methods to solve initial-boundary problems arising from heat conduction and wave propagation problems.

A power series with non-negative power terms is called a Taylor series. In complex variable theory, it is common to work with power series with both positive and negative power terms. This type of power series is called a Laurent series. The primary goal of this chapter is to establish the relation between convergent power series and analytic functions. More precisely, we try to understand how the region of convergence of a Taylor series or a Laurent series is related to the domain of analyticity of an analytic function. The knowledge of Taylor and Laurent series expansion is linked with more advanced topics, like the classification of singularities of complex functions, residue calculus, analytic continuation, etc.

This chapter starts with the definitions of convergence of complex sequences and series. Many of the definitions and theorems for complex sequences and series are inferred from their counterparts in real variable calculus.

Complex sequences and series

An infinite sequence of complex numbers, denoted by {zn}, can be considered as a function defined on a set of positive integers into the unextended complex plane. In other words, the sequence of complex numbers z1, z2, z3, … is arranged sequentially and defined by some specific rule.

In the earlier chapters, we have analyzed several prototype potential field problems, including potential fluid flows, steady state temperature distribution, electrostatics problems and gravitational potential problems. All of these potential field problems are governed by the Laplace equation. There is no time variable in these problems, and the characterization of individual physical problems is exhibited by the corresponding prescribed boundary conditions. The mathematical problem of finding the solution of a partial differential equation that satisfies the prescribed boundary conditions is called a boundary value problem, of which there are two main types: Dirichlet problems where the boundary values of the solution function are prescribed, and Neumann problems where the values of the normal derivative of the solution function along the boundary are prescribed. In other physical problems, like the heat conduction and wave propagation models, the time variable is also involved in the model. To describe fully the partial differential equations modeling these problems, one needs to prescribe both the associated boundary conditions and the initial conditions. The latter class is called an initial-boundary value problem. This chapter discusses some of the solution methodologies for solving boundary value problems and initial-boundary value problems using complex variables methods.

The link between analytic functions and harmonic functions is exhibited by the fact that both the real and imaginary parts of a complex function that is analytic inside a domain satisfy the Laplace equation in the same domain.

A complex function w = f(z) can be regarded as a mapping from its domain in the z-plane to its range in the w-plane. In this chapter, we go beyond the previous chapters by analyzing in greater depth the geometric properties associated with mappings represented by complex functions. First, we examine the linkage between the analyticity of a complex function and the conformality of a mapping. A mapping is said to be conformal at a point if it preserves the angle of intersection between a pair of smooth arcs through that point. The invariance of the Laplace equation under a conformal mapping is also established. This invariance property allows us to use conformal mappings to solve various types of physical problem, like steady state temperature distribution, electrostatics and fluid flows, where problems with complicated configurations can be transformed into those with simple geometries.

First, we introduce various techniques for effecting the mappings of regions. Two special classes of transformation, the bilinear transformations and the Schwarz–Christoffel transformations, are discussed fully. A bilinear transformation maps the class of circles and lines to the same class, and it is conformal at every point except at its pole. The Schwarz–Christoffel transformations take half-planes onto polygonal regions. These polygonal regions can be unbounded with one or more of their vertices at infinity. We also consider the class of hodograph transformations, where the roles of the dependent and independent variables are reversed.

This chapter begins with a discussion of the classification of isolated singularities of complex functions. The classification can be done effectively by examining the Laurent series expansion of a complex function in a deleted neighborhood around an isolated singularity. An isolated singularity can be either a pole, a removable singularity or an essential singularity. The various forms of behavior of a complex function near an isolated singularity are examined. Next, we introduce the definition of the residue of a complex function at an isolated singularity. We show how to apply residue calculus to the evaluation of different types of integral. The Fourier transform and Fourier integrals are considered, and the effective use of residue calculus for the analytic evaluation of these integrals is illustrated. The concept of the Cauchy principal value of an improper integral is introduced. We also consider the application of residue calculus to solving fluid flow problems.

Classification of singular points

By definition, a singularity or a singular point of a function f(z) is a point at which f(z) is not analytic. A point at which f(z) is analytic is called a regular point of f(z). A point z0 is called an isolated singularity of f(z) if there exists a neighborhood of z0 inside which z0 is the only singular point of f(z).