Complex Analysis: a Self-Study Guide

by Terrence P. Murphy

This is not a college textbook. With a college textbook and course, you are expected to develop the skills to independently prove a medium-difficult complex analysis theorem. Our goal here is more modest. After studying this book, you should have the skills to read, follow, and understand someone else's proof of a complex analysis theorem. My first published book is A Study of Bernhard Riemann’s 1859 Paper . In Riemann’s famous paper, he makes several important advances in mathematics and speculates (the "Riemann Hypothesis") about the location of the zeros of the Zeta function. My Riemann book is intended to bring an understanding of Riemann's paper to a wider audience by bridging the gap between John Derbyshire’s excellent but less technical book, Prime Obsession , and Harold Edward's excellent but highly technical book, Riemann’s Zeta Function . To advance your knowledge beyond Derbyshire’s book, you must have a good understanding of complex analysis (call it knowledge at the "hobbyist" level). Several people have said they want to get to the "hobbyist" level so they can better understand my Riemann book. However, when they look at Ahlfors and other well-known (400+ page) textbooks on complex analysis, it feels like a mountain just a little too high to climb. This book is intended to help you advance to the "hobbyist" level. Let’s assume you want to read my Riemann book. This book may be suitable Another possibility. You are currently a student and have just finished a course in real analysis. Your course in complex analysis is coming soon. You want to ace the course and are looking for a leg up. If you invest time studying this book (some skimming allowed), your efforts will be rewarded. We assume you have a good understanding of the following The above list essentially describes knowledge obtained from a course in real analysis or advanced calculus. More than anything, you must be familiar with epsilon-delta proofs. In this book, we develop much of the central theory of complex analysis, using standard/classic proofs. In our version of those proofs, we provide enough detail so the reader can follow the proofs (without outside help) from beginning to end. At the end of each chapter, we demonstrate what was learned, either with proofs of supplemental theorems or with exercises for the reader. We provide answers to all exercises in the book Appendix. Chapters 1 through 6 are the warm-up, with important definitions and with a review of familiar concepts (and an extension of those concepts to the complex plane). Chapters 7 through 13 are the heart of the book, where we study the central theorems of complex function theory, from Cauchy's Integral Theorem through the Residue Theorem. We end with a chapter on the Argument Principle, which Riemann used to determine the number of zeros of the Zeta function inside a region of height T in the critical strip.

Genres: Mathematics

175 pages, Paperback

Published March 16, 2022

Reviews

[Katie Meyers · Rating 5 out of 5]

Complex Analysis: A Self-Study Guide is a concise, exceptionally clear, and refreshingly accessible introduction to one of mathematics’ most intricate subjects. Terrence P. Murphy succeeds masterfully in his stated goal: guiding readers to a “hobbyist-level” understanding of complex analysis deep enough to confidently follow proofs, yet approachable for independent learners who may feel overwhelmed by traditional 400-page texts.

The book is structured with admirable clarity. Chapters 1 through 6 provide a smooth warm-up, revisiting essential real analysis concepts and extending them naturally into the complex plane. From Chapters 7 to 13, the guide enters the heart of complex function theory Cauchy’s Integral Theorem, the Residue Theorem, and the Argument Principle presented with enough detail and explanation to make each proof genuinely understandable. The writing is thoughtful and precise, and Murphy’s ability to demystify classically challenging concepts is one of the book’s greatest strengths.

What sets this guide apart is its philosophy: it does not try to turn the reader into a researcher but into someone mathematically fluent enough to engage with advanced literature such as the author’s companion work on Riemann’s 1859 paper. Each chapter provides meaningful demonstrations, supplemental theorems, and exercises, all of which reinforce the reader’s growth. With solutions included, self study becomes not only possible but deeply rewarding.

For students preparing for a course, hobbyists aspiring to read more advanced works, or curious minds seeking an accessible path into complex analysis, this guide is a valuable, confidence-building resource. Clear, practical, and surprisingly engaging it accomplishes exactly what it promises.