Complex Analysis

Princeton Lectures in Analysis #2

by Elias M Stein Rami Shakarchi, Rami Shakarchi

With this second volume we enter the intriguing world of complex analysis From the first theorems on the elegance and sweep of the results is evident The starting point is the simple idea of extending a function initially given for real values of the argument to one that is defined when the argument is complex From there one proceeds to the main properties of holomorphic functions whose proofs are generally short and quite illuminating the Cauchy theorems residues analytic continuation the argument principle With this background the reader is ready to learn a wealth of additional material connecting the subject with other areas of mathematics the Fourier transform treated by contour integration the zeta function and the prime number theorem and an introduction to elliptic functions culminating in their application to combinatorics and number theory Thoroughly developing a subject with many ramifications while striking a careful balance between conceptual insights and the technical underpinnings of rigorous analysis Complex Analysis will be welcomed by students of mathematics physics engineering and other sciences The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them Numerous examples and applications throughout its four planned volumes of which Complex Analysis is the second highlight the far reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in depth considerations of complex analysis measure and integration theory and Hilbert spaces and finally further topics such as functional analysis distributions and elements of probability theory

Genres: Mathematics, Textbooks, Reference, Nonfiction, Science, Technical

400 pages, Paperback

First published April 7, 2003

Reviews

[Geoffrey Lee · Rating 1 out of 5]

This book straight up sucks. Stein has a way of making beautiful results look ugly.

[Eric · Rating 4 out of 5]

the only higher math textbook i’ve ever read and thought was Well written (not saying much because only read gunning)

[Will · Rating 5 out of 5]

The story here is far more complicated than its real cousin, but also far prettier. An active imagination is recommended.

[Reader · Rating 2 out of 5]

I have to went through Ahlfors and Brown&Churchill's book to see how many steps Stein acrobatically skipped. This is a book for the students who are mathematically matured enough to fill in the gap as it goes, or just someone who has already learned it from other books, but then he would not have needed to read this from the beginning. The general style of this book reminds me of a particularly annoying professors who enjoys riddling with his students by giving all kinds of subtle suggestion that only the level of professionals of himself would have the prerequesite knowledge to appreciate. Let's be clear: a book is generally for the purpose of communication, and a textbook is generally to communicate the information from people who know it to people who don't. This book failed both. One could argue that leaving all these gaps for the students to fill could inspire the students to solve the problem themselves and 'enjoy the fun of exploration and finding the theorem like a real mathematician.'. Well, in that case, why don't just abandon the textbook at all? Why not just spend the next 300 years that you don't have on building the whole edifice of modern mathematics alone? I am pretty sure that would bring one with inquisitive mind no bound of fun, and if that is not enough, strip yourself and go to an uninhabitated island and grow a technology tree on your own.

I feel it important to mention it that this is the first book review I ever written because I simply do not care to do this kind of thing. I am only driven by the inproportionate bitterness reading this particular book, more precisely Chapter 3, 5 and 8. Elias' book on Fourier analysis and Real analysis are much more enjoyable and in fact suitable for a student to learn through this two books, but Functional analysis is as bad, if not ever worse than this book. Luckily I had the fortune to read this book beforehand to realise this problem could occur, and stop plowing through after I saw how he present Hahn-Banach's theorem.

[Hobart Mariner · Rating 4 out of 5]

Pretty close to a perfect [Edit: on thinking over the sections I disliked, I'm thinking more like "very good"] textbook on complex that favors a heavily ``hard-analytic'' approach with lots of Fourier transforms, in line with the previous volume in the series. The number of exercises (split into routine exercises and more challenging problems) is impressive. Lots of good practice with basic analytic techniques like putting a lower bound on a positive quantity, big-O estimates, etc.

Think the last chapter, about the use of theta functions to prove number theoretic statements, was fun but would have been better if replaced with something on several complex variables. Another area where the choice of content was iffy was the lengthy bit on Schwarz-Christoffel conformal mappings. Just flat didn't care that much, given how gnarly some of the work got. Also had a bone to pick with how they proved the uniform convergence in the elliptic functions section. It seems to me that it's better to prove an explicit lower bound for |m*tau+ n| in terms of c*(|m|+|n|) rather than to bust out this wavy approximation notation that they make very little use of. Also during the proof that holomorphic bijections have nonzero derivative they got a little handwavy with the shrinking of circles (where does w come from...I know it can be justified but the order of the shrinking is important). But these are pretty pedantic things to obsess over. It's a great book.

[Kent Sibilev · Rating 5 out of 5]

I evaluate books on complex analysis by checking if they cover Zeta and Elliptic functions. Otherwise, it is like preparing a tasty meal without ever eating it. This book has it all. As a second part of the series the book covers the theory and some important applications in the number theory and asymptotics.

[William Easton · Rating 5 out of 5]

9.1/10 Great textbook. At times it could be a little too keen to skip steps, but altogether quite clear. The second half of the book was less appealing to me personally, maybe it was just the style of proofs or the results themselves, but they were definitely less analytical. Also how hard is it to add a box at the end of your proofs? Overall very good though.

[NoName66 · Rating 5 out of 5]

first complex analysis course, very informative, finished first 8 chapters. The exercises are hard, my friends.

[Rachel Schmidt · Rating 3 out of 5]

Haley said I had to put my complex analysis textbook on my goodreads. Here it is. I read it. Peace out.

[annina · Rating 5 out of 5]

Read chapters 1-10.
It's great, the only flaws are that they don't end proofs with a ◻ and that definitions don't "stand out" as well from the rest of text.

[Tyler Han · Rating 4 out of 5]

Pretty good first course in complex analysis. Stuff on the prime number theorem is done well imo. Chapter 1-7ish are great, everything after I would recommend something else.

[dead letter office · Rating 4 out of 5]

this is the only book on complex analysis i've tried, so i don't have much basis for comparison, but it's pretty nice and relatively easy to learn from. part of that might be the subject matter, though, which is pretty inherently elegant and would be hard to muck up too badly.

[Evan · Rating 4 out of 5]

Applications as I recall are pretty much strictly to pure math (# theory, etc), but I thought this was a great presentation of this material.

[Saman · Rating 4 out of 5]

The approach is pure and deep. A good book to understand complex analysis and it's applications in some of difficult areas of math like number theory.

[Francesko Bici]

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