Complex Analysis

This new edition of a classic textbook develops complex analysis from the established theory of real analysis by emphasising the differences that arise as a result of the richer geometry of the complex plane. Key features of the authors' approach are to use simple topological ideas to translate visual intuition to rigorous proof, and, in this edition, to address the conceptual conflicts between pure and applied approaches head-on. Beyond the material of the clarified and corrected original edition, there are three new chapters: Chapter 15, on infinitesimals in real and complex analysis; Chapter 16, on homology versions of Cauchy's theorem and Cauchy's residue theorem, linking back to geometric intuition; and Chapter 17, outlines some more advanced directions in which complex analysis has developed, and continues to evolve into the future. With numerous worked examples and exercises, clear and direct proofs, and a view to the future of the subject, this is an invaluable companion for any modern complex analysis course.

• Using simple topological ideas of continuity and connectivity, this textbook explains the differences between real and complex analysis as a consequence of the richer geometry of the complex plane and teaches students to translate visual intuition into rigorous proof
• Introduces a simple formal definition of an extension field containing infinitesimal quantities to show the connection between pure and applied approaches - students grasp the continuing evolution of mathematical ideas
• Includes supplementary material showing in more detail the changes between the previous edition and this one