Reference request for undergraduate complex analysis.

Question

I am a second year student studying electrical engineering. I self-study pure mathematics and want to pursue a career as a mathematician.

What are some prerequisites for studying complex analysis? And what are some good books from which i can start studying?

I have studied abstract algebra from I.N.Hernstein and Artin, and have some knowledge about real analysis.

Answer 1

• You should know calculus.
• You should know how mathematical reasoning is done: what theorems and proofs are.
• It might help to know some things that are often not taught in first-year calculus: epsilons and deltas, things like the difference between pointwise convergence and uniform convergence, suprema and infima, etc. Knowing some things about power series can help. What is a radius of convergence? Why do power series converge uniformly on subsets of their region of convergence that are bounded away from the boundaries of the region, if if the convergence is at most pointwise rather than uniform on the region as a whole? PS added later: Why can power series be differentiated term by term in the interior of the region of convergence?
• You should understand some algebra and geometry of complex numbers. Why is multiplication by a nonzero complex number the same as rotating and dilating? (Probably you've seen a lot of that if you've studied electrical engineering.) Why is $e^{i\theta}$ equal to $\cos\theta+i\sin\theta$? That last can be looked at in several different ways, including but definitely not limited to power series. The fact that multiplying by a nonzero complex number is rotating and dilating explains immediately why differentiable functions of a complex variable are conformal mappings at all points where the derivative is not zero. The concept of conformal mapping is sometimes encountered at a very elementary level in cartography courses. In mathematics courses, it is often first encountered when you learn complex variables.

Answer 2

I think Micheal Hardy gives you a definitive answer below as far as prerequisites for complex analysis goes-I don't think I can improve on his response. As far as I'm concerned, there's no better introduction to the beautiful subject of complex analysis then Theodore Gamelin's Complex Analysis. It begins at a very basic level, requiring only simple calculus and builds slowly to PHD exam level topics such as Julia sets and meromorphic functions. It also has one of the most complete and basic presentations of Riemann surfaces you'll ever find in a complex analysis text. It's beautifully written with a host of examples, pictures and a ton of excellent problems. It also has many applications to geometry and physics. There are a legion of great books on complex analysis at various levels, but if I had to learn complex analysis by a fixed deadline and had to make sure I learned it right, that's the one I'd pick.

Answer 3

Michael Hardy's answer is excellent. For a specific text I would recommend Flanigan's "Complex Variables." It is extremely clear with many pictures - aiding the geometric aspects M.H. mentioned.

This is a Dover publication, so in contrast to the exorbitant cost of most math texts, this is a bargain.

http://www.amazon.com/Complex-Variables-Dover-Books-Mathematics/dp/0486613887/ref=sr_1_1?s=books&ie=UTF8&qid=1409247924&sr=1-1&keywords=flanigan+complex

Answer 4

I recommend you a book that I thoroughly enjoyed when I studied complex analysis:

"An Introduction to Complex Analysis", by Agarwal, Perera and Pinelas.

This book is divided into 50 lectures that will take you from the very basics of complex analysis to advanced topics; it starts with the usual introductory material and then goes through the Cauchy integral formula, power series, the residue theorem and evaluation of real integrals by contour integration, conformal mappings, and harmonic functions. It ends with brief sections on the Riemann zeta function, Riemann surfaces, and the Bieberbach conjecture. Almost every theorem/lemma/proposition is proved in a rigorous way.

At the end of each lecture there are numerous exercises to test your knowledge; hints are provided for most of them so, if you're studying on your own, you won't be kept in the dark forever wondering how to approach a specific problem.

If you're lucky enough, perhaps you'll even be able to access the digital version of the book for free through your university's library platform, at SpringerLink.

Answer 5

“Complex Variables and Applications” by Brown and Churchill is pretty nice. It's compact, and the later chapters show applications of the theory to some physics problems.

Answer 6

My undergrad course used Complex Analysis by Bak & Newman. One great benefit is that solutions to all of the exercises are given in the back.