Recent decades have seen profound changes in the way we understand complex analysis. This new work presents a much-needed modern treatment of the subject, incorporating the latest developments and providing a rigorous yet accessible introduction to the concepts and proofs of this fundamental branch of mathematics. With its thorough review of the prerequisites and well-balanced mix of theory and practice, this book will appeal both to readers interested in pursuing advanced topics as well as those wishing to explore the many applications of complex analysis to engineering and the physical sciences.

* Reviews the necessary calculus, bringing readers quickly up to speed on the material

* Illustrates the theory, techniques, and reasoning through the use of short proofs and many examples

* Demystifies complex versus real differentiability for functions from the plane to the plane

* Develops Cauchy's Theorem, presenting the powerful and easy-to-use winding-number version

* Contains over 100 sophisticated graphics to provide helpful examples and reinforce important concepts