The 3 Best Books on Complex Analysis

I describe my three favorite books for an introduction to complex analysis, and conclude with some remarks about a few other books. Hope this is helpful for both students and instructors!

0:00 Book 1: Greene and Krantz
6:08 Book 2: Stein and Shakarchi
10:14 Book 3: Ablowitz and Fokas
13:44 Other books

Books mentioned:

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[ryancreedon]

Used the Ablowitz + Fokas book in my grad applied complex class. It's an excellent text---keeps the theorems grounded with explicit computations. Honestly, it could work for an intro undergrad class, too. The asymptotics and Riemann-Hilbert sections are fascinating reads and incredibly useful to anyone interested in Nonlinear PDEs.

[hanznoa]

Great Video, absolutely love hearing you talk about math books. I find the perspective you take on books really distinct from other book reviewer what makes it so informative. Thx for the Video and keep doing that sort of Videos. Would really love if you could go deeper into the separate books and talk a little more about them, like how did it feel working through them, what prerequisites should you have, what chapters should you focus on and what are "nice to have". Pretty much like you did for the First book, just also for the others please haha

[gilloon]

Greene and Krantz is not a great place to start on the subject!
The best books to start are: Gamelin, “ Complex Analysis “ ; Palka, “An introduction to Complex Function Theory” ; Marsden and Hoffman, ” Basic Complex Analysis “

[qsfrankfurt]

Ablowitz and Fokas is great. It's very well presented and great for visualization as well as concepts.

[billmorrigan]

Excellent books. Thank you for the excellent advice.
On top of that I can add a few more textbooks on complex variables that I found to be very good:
_Complex Variables and Applications_ by Churchill
_A First Course in Complex Analysis_ by Zill (simple, easy book).
There's also a stunning book called _Visual Complex Analysis_ by Tristan Needham. I'm gonna quote the author here: "The present book openly challenges the current dominance of purely symbolic logical reasoning by using new, visually accessible arguments to explain the truths of elementary complex analysis". In my opinion it is a must-have-addition to books on complex variables.

[pdeva]

What do you think of ‘Complex Analysis with Applications’ by Asmar and Grafakos?

[Censorededs]

I'd love to see your take on a new by Donald Marshall "Complex Analysis"; when I learned complex analysis we used Ahlfors, Rudin, Stein and Shakarchi, and Marshall. Also could you look at Terry Tao's 246ABC lecture notes on his blog. I'd love to see an in-depth take on different approaches to different major theorems in each book, and possibly some interpretation on what students might be more comfortable with what types of approaches.

I first learned from Stein's Complex Analysis, and I hated it. I then signed up for the grad complex sequence and fell in love with Complex Analysis. I'm currently going back through Complex Analysis as I forgot everything over the years rediscovering my love for the subject.

[whig]

If I could only pick one, if would be "Theory of Functions of a Complex Variable" (3 volumes in one), by A. I. Markushevich, translated from Russian by Silverman. It's the book I found in the math library that always seemed to be the most helpful when I was stuck on something. I've since bought my own copy. Dover publishes a subset of the first volume which is "priced right" and sufficient for a "first look" at the subject. Walter Rudin's "Real & Complex Analysis" is also a very popular textbook--but not suitable for a first course.

[qnvk]

Great video. I used the Shakarchi book as an introduction but it was the only one. Therefore I didn’t have a lot to study from and there were parts where I was stuck and left without a good understanding of what was going on (the classic “and it is easy to see that…”). I’ll be checking these books for sure.

[Roshawn-cs]

Hi ! I want to know does springer textbooks have solutions and answer in COMPLEX ANALYSIS

AND WHAT THE BEST TEXTBOOKS FOR MORE PRACTICAL APPLIED COMPLEX ANALYSIS

THANK YOU

[anthonyymm]

Do you have any book recommendations for complex (differential) geometry from a more analytic perspective ? Most texts I come across I find to be too algebraic for my tastes.

[GalileanInvariance]

Complex Variables and Applications (Brown and Churchill) ... Not an 'analysis' textbook per se, but nonetheless a serious treatment of the subject (with proofs), and I would venture to say that more engineering and applied sciences students have learned the subject from this reference than any of the others you recommended.

[Roshawn-cs]

I want to known does springer textbooks have answers in them
or its better to get a functional analysis textbooks

Thank you

[vladimirlucic]

Great video. I would also point out thee two volume all-time classic by Einar Hille, "Analytic Function Theory".

[mimmyjau]

any thoughts on Lars Ahlfors' book? seems to be recommended quite often.

[buxeessingh]

I think the books by Flanigan and Fisher, both cheap from Dover, are great for intuition.

[julianwilson]

In addition to the famous Visual Complex Analysis by Needham (of which a 25th anniversary edition was recently published), check out Wegert's Visual Complex Functions: An Introduction with Phase Portraits as a companion to more traditional texts.

[aczajka]

Fun fact: Robert E. Greene taught Russel Crowe to play the violin for the movie Master and Commander.

[MrPhasespace]

I liked the old classics by Konrad Knopp.

[dbgsdc]

How about Thomas calculus ? Or James Stewart books which one is better, it will be so helpful, reply me