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I'm currently self-studying some complex analysis. My background is limited: single- and multivariable calculus, linear algebra, introductory Fourier analysis and matrix theory. Each course, with multivariable calculus being a bit of an exception, has emphasized theory and proofs. I've also done the first four chapters of "Baby Rudin" in my spare time.

The complex analysis book by Stein and Shakarchi is interesting. At times, the book omits quite a few steps that are not so obvious to me. I'm not unfamiliar with this from Rudin who I found was very clear despite the rather terse style. This book will sometimes omit more than I would like. For example, on pages 42-45 the book gives two examples of how real integrals can be solved using Cauchy's theorem, and I had to do a significant amount of filling in the steps and this made me kind of lose track of the bigger idea. I don't like being spoon-fed things because I like to think but at times filling in the steps too much gets in the way of the learning. At least this is how I experience it.

My gripe with the book, however, is with the exercises. Many of the exercises, I find, are very difficult. With Rudin, I struggled because the material was abstract: working so rigorously in general metric spaces was extremely challenging for me but I still managed to do at least half the exercises in each chapter. I can at best a third of the exercises in S&S.

Any advice on how to better absorb the material in this book? It seems to have a pretty interesting approach with many interesting topics (Zeta function, prime number theorem, Riemann mapping theorem, elliptic functions) including some that appear early (Runge's theorem, Hadamard's factorization theorem). So there's plenty of wonderful mathematics here but the exercises are making studying this terribly difficult.

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    $\begingroup$ The issue is probably that you don't have enough experience with basic real analysis. I mean two things by this. First, you probably need more practice with making standard arguments that S&S take for granted you can make without effort. Second, there are things in the later chapters of Rudin that you will need (e.g. uniform convergence). A possible solution: read and do more exercises out of Rudin first. $\endgroup$ Commented Mar 21, 2014 at 21:45
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    $\begingroup$ For what it's worth, I think the exercises in S&S are on the easier side, especially compared to some of the ones in Ahlfors. This is probably another indication your experience is the problem. $\endgroup$ Commented Mar 21, 2014 at 21:47
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    $\begingroup$ I'll give you a gem that no one talks about... Paulka's Introduction to Complex Function Theory. This is an amazing first read. Then you can move on to something like Stein. Before Paulka I read Churchill. That was a gem as well for me as an undergrad. In my opinion, things like Conway and Stein are good for the 2nd or 3rd reads. $\endgroup$ Commented Nov 14, 2017 at 3:02
  • $\begingroup$ "How do I study..." - very carefully. $\endgroup$ Commented Nov 14, 2017 at 4:04

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Generally, one takes a "complex variables" course before taking "complex analysis", just as one takes calculus before taking real analysis. So it kind of depends on what your goal is. If you want to learn how to deal with complex functions, like taking derivatives, path integrals, infinite series etc. I would buy a book with "complex variables" or "theory of functions of a complex variable" in the title. A complex analysis book is going to be driven more towards theorem proving. Generally the proofs you get in a calculus class (like calc I, calc II, calc III) are not really proofs at all but more like proof-sketches. In general it takes an entire semester of real analysis (sometimes called advanced calculus) to prove all the results in a semester of calculus. For instance, in real analysis I you generally prove all the results from Cal I. I can't speak to your class in particular but there just really isn't enough time in a semester to do all that. Generally they teach you how to use calculus first, as that's all a lot of professions need, and then they leave the rigorous development of calculus to later courses. It's similar with complex functions. You've bought the real analysis I analog of complex functions and not the Cal I analog of complex functions. The author is going to assume you are very experienced in mathematical thinking/proofs and is going to make leaps in certain areas based on that assumption. If your goal is theorem proving, I would buy a book on real analysis first ( I probably wouldn't go with rudin), as certain results from it are going to be pre-requisites in a sense, and possibly a book on theorem proving. "Mathematical thinking: Logic and Proofs) is a good one. In general Math Majors will take a "proofs course" on how to write proofs and proof methods/strategies prior to taking theorem driven courses like analysis and abstract algebra.

I should add that as a math major I maintained a perfect gpa in my classes with virtually no attendance. I just read the book. So it is certainly doable. That is, you don't need to actually take the class in order to learn the material. I wouldn't get discouraged in that sense. The one nice thing about learning it at a university is the sequence/order in which you should study certain material is laid out for you. It's going to be extremely difficult to follow a complex analysis text without having learned real analysis first.

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  • $\begingroup$ I don't think a proofs course is strictly necessary; I just learned proof-writing on the fly in my abstract algebra class. Same with complex analysis; I did not need to take a complex variables class beforehand to understand the material. $\endgroup$ Commented May 7, 2021 at 22:20
  • $\begingroup$ @ExponentBOT I didn't say complex variables was a necessary pre-req...just real analysis. Did you not have a discrete math class that covered sets, relations, combinatorics, modular arithmetic, etc. before taking abstract algebra? $\endgroup$ Commented May 8, 2021 at 20:36
  • $\begingroup$ No, I never took a discrete math class. I went from calculus III to abstract algebra and managed quite well. $\endgroup$ Commented May 8, 2021 at 21:25
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    $\begingroup$ Nicely done. That's not an ideal progression imo though. At a minimum abstract algebra can seem completely pointless without having had some prior exposure to abstract structures like matrices, vector spaces over R or C, congruence classes, etc. Also many of those topics in discrete math are actually necessary in building the theories of groups,rings,fields, etc. and to learn them "on the fly" as you progress through the course would likely require moving through the curriculum at too fast a pace for many people to manage. $\endgroup$ Commented May 10, 2021 at 4:56

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