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.