Complex Analysis Books - Best Textbooks in Complex Analysis

Complex analysis is a subject that sounds more difficult than it is. Many students have expressed to me that complex analysis shocked them by being easy and fun (relative to other advanced math courses), and the comparatively accessible-sounding "Real analysis" similarly surprised them by being painfully abstract and brutally difficult.

Advice for students: Don't be intimidated. This is a fun subject! If you have a tough time understanding the book your professor has chosen, get a supplemental book at a more accessible level.

Advice for teachers and professors: More so than any other subject, complex analysis has the ability to really wow students with deep mathematical truths that are bizarre and profoundly beautiful. It is also a subject that can be approached in many different ways. Rather than strictly taking one approach to the subject, try to give students a feel for the different facets of the subject. You can alternatively emphasize geometrical reasoning, the topological aspects of the subject, clever manipulations of infinite series, connections to number theory, or more direct practical applications.

Whether you're a student or a teacher, but especially for teachers, I think it is worthwhile to expose yourself to as many different perspectives on this subject as possible.

It's hard for me to pick a favorite book on the subject of Complex Analysis, but if I had to pick one single book, it would be this one. I think it is best for advanced undergraduates and beginning graduate students, an ideal text for learning Complex Analysis from for the first time.

Strengths: Clear writing, rich motivation and discussion. Proofs are clear and accessible, and the book makes good use of geometry to explain concepts and develop intuition. Covers a good amount of ground. Gives equal weight both to practical (applied mathematics) applications as well as applications in pure math. Lastly, the earlier sections are presented in ways that it is easy for more advanced students to skip over them, making the book useful for a wide range of ability levels.

Downsides: This book does not provide rigorous proofs of all theorems it covers. It is oriented primarily towards undergraduates and beginning graduate students, and people going deeper in the subject will probably require a more rigorous book.

This book is a newer edition (blue cover) of an older book by Churchhill and Brown (Red Cover). This is the clearest, most accessible introduction to the subject of complex analysis that I have ever read and worked with. I recommend it especially at the undergraduate level, for engineers and physicists, and for self-study. It is the most accessible and easiest to work through book of all the texts I review here.

Strengths: Unparalleled clarity and detail. I find I can read the proofs in this book more easily than any other, without ever getting stuck. This makes the book outstanding for self-study. I think this would be a great first introduction to the subject.

Weaknesses: This book will probably move too slow for some students, and will be less useful for graduate students studying pure mathematics. The emphasis is on clarity and fullness of explanation. It is not comprehensive as an advanced reference, however, and I don't think it would be as useful for a graduate course or at an advanced level.

This is an outstanding book which I think is not best for all uses, but serves a very important purpose. I recommend it as a study text, as it is poorly suited as a concise reference.

This is a good second book on complex analysis to purchase, and it would also be outstanding for teachers and professors to purchase and work through before teaching an intro complex analysis course. This is a book for someone who wants to dedicate a lot of time to really mastering the subject and exploring all the nuances of it. It is advanced, but in a completely different way from the dry "theorem-proof" approaches of other advanced texts.

Pros: Exceptionally thorough. Numerous exercises, some of which are very tough and go quite deep into the subject. I think the exercises are my favorite part of the book. I especially like the exercises which explore counterexamples to converses of theorems.

Cons: Not the easiest book to read; requires extensive thought to get through. I often find the diagrams less-than-fully helpful. Probably not best suited to someone studying the subject for the first time.

This is a classic book on complex analysis. I find it clear and accessible, but it has a faster pace and more minimal style than most undergrads would be used to. This is a text for a serious mathematician, with a high standard of rigor. It covers all the key results in the subject, but it stays relatively compact by avoiding some of the more specialized topics. I recommend it for students and teachers who prefer a more dry, theorem-proof style that gets to the point and expects the reader to do more of the work to understand the material.

Pros: I found this book very clear to read once I got to the appropriate level. I found the order of topics natural, and the exercises to be at just the right difficulty level.

Cons: Requires experience with in mathematical analysis/advanced calculus, as well as a certain level of mathematical sophistication (at least 3 proof-oriented math classes). I think this book is most accessible to students who have already studied some complex analysis. This book is also a bit pricey.

Serge Lang is best known for his book Algebra, which I gave outstanding reviews, but which is admittedly tough to read. This book is much more accessible and is completely different from that one--less radical and much more elementary. People seem to have mixed feelings about this book, and I can see why, as I explain below.

Pros: This book provides a more topological approach to complex analysis. Excellent motivation is provided for usually abstract theorems, especially in later chapters. As someone who thinks topologically and has better intuition for topological reasoning than pure symbolic logic, I found this book easier to understand than most.

Cons: Inconsistent pacing: I found the beginning portions of the book to move painfully slow (especially for people who have prior background in rigorous mathematical analysis/advanced calculus), but the end of the book seems too fast, especially given the pace of the beginning. I also did not particularly like the order or choice of topics.

Greene and Krantz' Function Theory of One Complex Variable is yet another unorthodox and I think under-appreciated book on complex analysis, that is one of my favorite books on the subject.

This book's primary goal is to present complex analysis more in terms of its relationships with multi-variable calculus (Calc 3), and de-emphasize its relationship to topology. I want to preface this review by saying that I am someone who loves topology, and tends to think topologically, especially when it comes to complex analysis. So I was initially skeptical of the goals and philosophy behind this text.

But when I began working with it, my mind was opened. The approach works, and quite beautifully. The authors have come up with something very novel.

Pros: This book will likely seem much more accessible and natural to people who are coming straight out of subjects like Calc 3 (Multivariable Calculus) and differential equations. I think this book is actually so accessible that the authors have over-stated the requirements to work with it and understand it. I think this book would be accessible to most undergraduates who have taken Calc 3. My favorite part of this book is the rich, explanatory prose. The writing is captivating and ample motivation is provided for the various theorems.

Cons: At a few points in the book, proofs in the book have a rather tedious and ugly plug-and-chug approach, which I find a little tiresome and not particularly good for developing intuition. Exercises are a bit inconsistent in difficulty level and I found some of them to be too easy, especially for a text marketed as a graduate-level text.

This is a intermediate-level textbook. I found it very clear, but it picks up the pace much faster than some of the other textbooks I review on this page. Students with a strong background in abstract mathematics and the guidance of a good teacher may be able to delve right into this book without prior background in complex analysis, but for self-study or undergraduates, I'd recommend starting with an easier book.

Pros: Gets into deep results quickly. Very clear, especially to people who are used to working through smaller details on their own.

Cons: A few major theorems are left to exercises, which can make this book a bit less useful as a reference. Moves pretty fast and requires sufficient mathematical maturity. Probably a bit too dense for most students encountering complex analysis for the first time.

So far I've only found one free online textbook on this subject that I would recommend; thankfully, it is a pretty good one:

• Complex Analysis by George Cain - At Georgia Tech, by chapter in PDF format.

Students are often tempted to think that if a book is easier to understand, it is a better choice. This is not always the case. Some easier-to-read books may be sloppy in their use of logic, making them a poor choice for students who are going on in mathematics, who will need to understand the material in a deeper, mathematically rigorous way later.

Challenging problem sets and even more concise proofs can also provide useful exercise, which can help develop your mathematical skill and prepare you for doing higher-level research later on.

But many teachers, especially professors at research-oriented universities, make the opposite mistake, assigning concise, math-oriented texts to classes of engineers and scientists who may never need the higher-level mathematics. In complex analysis, there are a broad range of levels of rigor that you can find in textbooks. I recommend finding the level that is right for you. Teachers, don't try to force students to study at a higher or lower level than they want.

But students, don't go straight for the easiest book if you really want to understand the subject inside and out. Sometimes having a more elementary book on hand can help you work your way through a denser, more rigorous text, so that you get the best of both worlds.