Random walk – the stochastic process formed by successive summation of independent, identically distributed random variables – is one of the most basic and well-studied topics in probability theory. For random walks on the integer lattice ℤd, the main reference is the classic book by Spitzer (1976). This text considers only a subset of such walks, namely those corresponding to increment distributions with zero mean and finite variance. In this case, one can summarize the main result very quickly: the central limit theorem implies that under appropriate rescaling the limiting distribution is normal, and the functional central limit theorem implies that the distribution of the corresponding path-valued process (after standard rescaling of time and space) approaches that of Brownian motion.

Researchers who work with perturbations of random walks, or with particle systems and other models that use random walks as a basic ingredient, often need more precise information on random walk behavior than that provided by the central limit theorems. In particular, it is important to understand the size of the error resulting from the approximation of random walk by Brownian motion. For this reason, there is need for more detailed analysis. This book is an introduction to the random walk theory with an emphasis on the error estimates. Although “mean zero, finite variance” assumption is both necessary and sufficient for normal convergence, one typically needs to make stronger assumptions on the increments of the walk in order to obtain good bounds on the error terms.

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.