This is the first primary introductory textbook on complex variables and analytic functions to make extensive use of functional illustrations. Aiming to reach undergraduate students entering the world of complex variables and analytic functions, the authors utilize graphics to visually build on familiar cases and illustrate how these same functions extend beyond the real axis. The book covers several important topics that are omitted in nearly all recent texts, including techniques for analytic continuation and discussions of elliptic functions and of Wiener Hopf methods. It presents current advances in research, highlighting the subject s active and fascinating frontier.