# Rudin for Complex Analysis?

• Analysis

## Main Question or Discussion Point

Hello, I was wondering how well is Rudin's Real and Complex Analysis for learning complex analysis, assuming that difficulty won't be an issue. Does it cover the standard material? Is it deep enough? Should I just read from elsewhere instead?

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QuantumQuest
Gold Member
Rudin's books are really good, but it would be helpful to tell what is your background and your goal, in order to give some further opinion / advice.

Hello, I was wondering how well is Rudin's Real and Complex Analysis for learning complex analysis, assuming that difficulty won't be an issue. Does it cover the standard material? Is it deep enough? Should I just read from elsewhere instead?
I would never recommend Rudin for anything. The book is horrible. It makes me physically sick that it's so popular. But ok, to see which book fits you, you need to tell more about yourself.

Anyway, I always liked Freitag (and Busam) two complex analysis books. They're very good. Barry Simon also treats complex analysis very comprehensively (with use of topology) in his comprehensive course on analysis.

I would never recommend Rudin for anything. The book is horrible. It makes me physically sick that it's so popular. But ok, to see which book fits you, you need to tell more about yourself.

Anyway, I always liked Freitag (and Busam) two complex analysis books. They're very good. Barry Simon also treats complex analysis very comprehensively (with use of topology) in his comprehensive course on analysis.
Thanks for the warning micromass. I am wondering now, what sequence of books would you recommend or are your favorite for an expansive coverage of everything an analysis student should learn, from real to complex to functional, from undergraduate to graduate, and beyond. Thanks in advance!

Thanks for the warning micromass. I am wondering now, what sequence of books would you recommend or are your favorite for an expansive coverage of everything an analysis student should learn, from real to complex to functional, from undergraduate to graduate, and beyond. Thanks in advance!
That really depends on you. What is your background, what is your goal? What is the style of math you like (geometric,rigorous, foundational, algebraic, historical, blabla)? If you can give me a brief explanation of these things, I'll list you a road map.

My background is that I like rigor and foundational stuff. I have experience in Algebra, some previous analysis (from Rudin POMA up to ch. 4), topology, etc. I feel that difficulty in a book isn't a problem for me, I usually figure it out, or just do a quick look online if I am stuck on a concept. I hope that helps.

OK, so let's start with what I consider to be the beginning. As first analysis book, I recommend

Bloch's "real numbers and real analysis" https://www.amazon.com/dp/0387721762/?tag=pfamazon01-20
Perhaps since you went through Rudin already this book won't be necessary anymore, but this book really is a gem. First, it proves everything. It takes nothing for granted, except some basic set theory. It starts of with Peano axioms and actually constructs the integers, rationals and reals. It's the only book I know that does this in such detail. There is a lot of very cool stuff in the other parts of the book too. For example, he proves several equivalents to the completeness axioms, which cannot be found in standard texts. Also, he goes into what we mean with "area" and proves that an integral is an area. It's neat little stuff like this that make Bloch a real beautiful book. There are many historical notes too. Also fun are notes from the author in which he tells us why he approached a topic a certain way.
Let's end it with this: I'm a professional mathematician and I still learn new things from this book. I doubt many people can say that of Rudin...

A good alternative to Bloch/Rudin would be Apostol's "mathematical analysis". https://www.amazon.com/dp/0201002884/?tag=pfamazon01-20 I describe Apostol as a pretty dry book. It doesn't motivate things well. But it contains a lot of important material. For example, he goes into detail in how you can treat series (associativity, commutativity, etc.). I don't find these kind of things in many other texts. Perhaps because other texts find it too boring. But I think it's still important. Apostol is really nice if you don't mind dry books. Also, the problems in Apostol can be hard.

Then you might want to do a bit of multivariable analysis. A standard reference is Spivak "calculus on manifolds". https://www.amazon.com/dp/0805390219/?tag=pfamazon01-20 This book isn't especially good, but it's pretty short and does the job. It goes on to prove Stokes' theorem in pretty nice generality. The problems are very insightful. But perhaps you will want a less dense text.

Then it is time for Carother's "real analysis". https://www.amazon.com/dp/0521497566/?tag=pfamazon01-20 This is without any doubt the best analysis text I've seen so far. Every concept is motivated in much much detail. There are many many problems, some of which are pretty easy (but important), others which can be hard. I still love browsing through this book every now and then. I think it's an absolute gem. It covers metric spaces, function spaces and Lebesgue integration. One possible flaw is that it doesn't cover Lebesgue integration outside $\mathbb{R}$, but I don't think it's much of a problem.

So now you might be ready for some topology. Munkres seems to be the standard reference here, but I don't really enjoy the book all that much, especially from the point of view of an analyst. If you can handle it, Willard is much better, but also a lot harder. Perhaps I recommend going through Lee's excellent book first https://www.amazon.com/dp/1441979395/?tag=pfamazon01-20 and then Willard https://www.amazon.com/dp/0486434796/?tag=pfamazon01-20

Then you might want to do some measure theory. Bartle's book is very good here: https://www.amazon.com/dp/0471042226/?tag=pfamazon01-20 It does everything in a suitable generality. It covers the standard topics very neatly. The problems are perhaps easy, but very very insightful. A lot of nice things are hidden in the problems.

Functional analysis is of course very important too. The book by Kreyszig is the perfect introduction to it: https://www.amazon.com/dp/0471504599/?tag=pfamazon01-20 You can perfectly read the book after a book like Bloch, but the longer you wait, the more you'll get out of it. The book might be too easy for you, but I adored it start to finish. You won't find a better introduction to functional analysis like this. But it is just an introduction after all, don't expect a deep knowledge of functional analysis after it.

Complex analysis then. The two books by Freitag (and Busam) are extremely well written. https://www.amazon.com/dp/3540939822/?tag=pfamazon01-20 and https://www.amazon.com/dp/3642205534/?tag=pfamazon01-20 They have very cool problems and very good exposition. They go very deep inside of complex analysis too. You can read these books after something of the level of Spivak's multivariable book.

Conway's abstract analysis book might fit in after these https://www.amazon.com/dp/0821890832/?tag=pfamazon01-20 Very nice book which goes towards operator theory. It starts of by building entire measure theory (which you should probably already know) in a very nonstandard way. Goes onto do C*-algebras.

Lang's real analysis and functional analysis is a true gem too. https://www.amazon.com/dp/0387940014/?tag=pfamazon01-20 Does things in a very great generality. For example, treat entire multivariable calculus (like inverse function theorem), but on Banach spaces. Treats measures on Banach spaces, etc. Really good and insightful read.

Then you might be ready for the true masterpiece of analysis. This is Simon's Comprehensive analysis course. Contains about everything you want to know about analysis and more. It starts of at the very beginning, and starts by doing topology and measure theory. But it does it at a very high level. https://www.amazon.com/dp/1470410982/?tag=pfamazon01-20

There must be a lot I've missed of course. For example, ODE and PDE theory is obviously important to analysis too, and is perhaps not treated very well in all the books I've linked. But I've got to stop somewhere. Tell me if there's anything special you're interested in.

OK, so let's start with what I consider to be the beginning. As first analysis book, I recommend

Bloch's "real numbers and real analysis" https://www.amazon.com/dp/0387721762/?tag=pfamazon01-20
Perhaps since you went through Rudin already this book won't be necessary anymore, but this book really is a gem. First, it proves everything. It takes nothing for granted, except some basic set theory. It starts of with Peano axioms and actually constructs the integers, rationals and reals. It's the only book I know that does this in such detail. There is a lot of very cool stuff in the other parts of the book too. For example, he proves several equivalents to the completeness axioms, which cannot be found in standard texts. Also, he goes into what we mean with "area" and proves that an integral is an area. It's neat little stuff like this that make Bloch a real beautiful book. There are many historical notes too. Also fun are notes from the author in which he tells us why he approached a topic a certain way.
Let's end it with this: I'm a professional mathematician and I still learn new things from this book. I doubt many people can say that of Rudin...

A good alternative to Bloch/Rudin would be Apostol's "mathematical analysis". https://www.amazon.com/dp/0201002884/?tag=pfamazon01-20 I describe Apostol as a pretty dry book. It doesn't motivate things well. But it contains a lot of important material. For example, he goes into detail in how you can treat series (associativity, commutativity, etc.). I don't find these kind of things in many other texts. Perhaps because other texts find it too boring. But I think it's still important. Apostol is really nice if you don't mind dry books. Also, the problems in Apostol can be hard.

Then you might want to do a bit of multivariable analysis. A standard reference is Spivak "calculus on manifolds". https://www.amazon.com/dp/0805390219/?tag=pfamazon01-20 This book isn't especially good, but it's pretty short and does the job. It goes on to prove Stokes' theorem in pretty nice generality. The problems are very insightful. But perhaps you will want a less dense text.

Then it is time for Carother's "real analysis". https://www.amazon.com/dp/0521497566/?tag=pfamazon01-20 This is without any doubt the best analysis text I've seen so far. Every concept is motivated in much much detail. There are many many problems, some of which are pretty easy (but important), others which can be hard. I still love browsing through this book every now and then. I think it's an absolute gem. It covers metric spaces, function spaces and Lebesgue integration. One possible flaw is that it doesn't cover Lebesgue integration outside $\mathbb{R}$, but I don't think it's much of a problem.

So now you might be ready for some topology. Munkres seems to be the standard reference here, but I don't really enjoy the book all that much, especially from the point of view of an analyst. If you can handle it, Willard is much better, but also a lot harder. Perhaps I recommend going through Lee's excellent book first https://www.amazon.com/dp/1441979395/?tag=pfamazon01-20 and then Willard https://www.amazon.com/dp/0486434796/?tag=pfamazon01-20

Then you might want to do some measure theory. Bartle's book is very good here: https://www.amazon.com/dp/0471042226/?tag=pfamazon01-20 It does everything in a suitable generality. It covers the standard topics very neatly. The problems are perhaps easy, but very very insightful. A lot of nice things are hidden in the problems.

Functional analysis is of course very important too. The book by Kreyszig is the perfect introduction to it: https://www.amazon.com/dp/0471504599/?tag=pfamazon01-20 You can perfectly read the book after a book like Bloch, but the longer you wait, the more you'll get out of it. The book might be too easy for you, but I adored it start to finish. You won't find a better introduction to functional analysis like this. But it is just an introduction after all, don't expect a deep knowledge of functional analysis after it.

Complex analysis then. The two books by Freitag (and Busam) are extremely well written. https://www.amazon.com/dp/3540939822/?tag=pfamazon01-20 and https://www.amazon.com/dp/3642205534/?tag=pfamazon01-20 They have very cool problems and very good exposition. They go very deep inside of complex analysis too. You can read these books after something of the level of Spivak's multivariable book.

Conway's abstract analysis book might fit in after these https://www.amazon.com/dp/0821890832/?tag=pfamazon01-20 Very nice book which goes towards operator theory. It starts of by building entire measure theory (which you should probably already know) in a very nonstandard way. Goes onto do C*-algebras.

Lang's real analysis and functional analysis is a true gem too. https://www.amazon.com/dp/0387940014/?tag=pfamazon01-20 Does things in a very great generality. For example, treat entire multivariable calculus (like inverse function theorem), but on Banach spaces. Treats measures on Banach spaces, etc. Really good and insightful read.

Then you might be ready for the true masterpiece of analysis. This is Simon's Comprehensive analysis course. Contains about everything you want to know about analysis and more. It starts of at the very beginning, and starts by doing topology and measure theory. But it does it at a very high level. https://www.amazon.com/dp/1470410982/?tag=pfamazon01-20

There must be a lot I've missed of course. For example, ODE and PDE theory is obviously important to analysis too, and is perhaps not treated very well in all the books I've linked. But I've got to stop somewhere. Tell me if there's anything special you're interested in.

OK, so let's start with what I consider to be the beginning. As first analysis book, I recommend

Bloch's "real numbers and real analysis" https://www.amazon.com/dp/0387721762/?tag=pfamazon01-20
Perhaps since you went through Rudin already this book won't be necessary anymore, but this book really is a gem. First, it proves everything. It takes nothing for granted, except some basic set theory. It starts of with Peano axioms and actually constructs the integers, rationals and reals. It's the only book I know that does this in such detail. There is a lot of very cool stuff in the other parts of the book too. For example, he proves several equivalents to the completeness axioms, which cannot be found in standard texts. Also, he goes into what we mean with "area" and proves that an integral is an area. It's neat little stuff like this that make Bloch a real beautiful book. There are many historical notes too. Also fun are notes from the author in which he tells us why he approached a topic a certain way.
Let's end it with this: I'm a professional mathematician and I still learn new things from this book. I doubt many people can say that of Rudin...

A good alternative to Bloch/Rudin would be Apostol's "mathematical analysis". https://www.amazon.com/dp/0201002884/?tag=pfamazon01-20 I describe Apostol as a pretty dry book. It doesn't motivate things well. But it contains a lot of important material. For example, he goes into detail in how you can treat series (associativity, commutativity, etc.). I don't find these kind of things in many other texts. Perhaps because other texts find it too boring. But I think it's still important. Apostol is really nice if you don't mind dry books. Also, the problems in Apostol can be hard.

Then you might want to do a bit of multivariable analysis. A standard reference is Spivak "calculus on manifolds". https://www.amazon.com/dp/0805390219/?tag=pfamazon01-20 This book isn't especially good, but it's pretty short and does the job. It goes on to prove Stokes' theorem in pretty nice generality. The problems are very insightful. But perhaps you will want a less dense text.

Then it is time for Carother's "real analysis". https://www.amazon.com/dp/0521497566/?tag=pfamazon01-20 This is without any doubt the best analysis text I've seen so far. Every concept is motivated in much much detail. There are many many problems, some of which are pretty easy (but important), others which can be hard. I still love browsing through this book every now and then. I think it's an absolute gem. It covers metric spaces, function spaces and Lebesgue integration. One possible flaw is that it doesn't cover Lebesgue integration outside $\mathbb{R}$, but I don't think it's much of a problem.

So now you might be ready for some topology. Munkres seems to be the standard reference here, but I don't really enjoy the book all that much, especially from the point of view of an analyst. If you can handle it, Willard is much better, but also a lot harder. Perhaps I recommend going through Lee's excellent book first https://www.amazon.com/dp/1441979395/?tag=pfamazon01-20 and then Willard https://www.amazon.com/dp/0486434796/?tag=pfamazon01-20

Then you might want to do some measure theory. Bartle's book is very good here: https://www.amazon.com/dp/0471042226/?tag=pfamazon01-20 It does everything in a suitable generality. It covers the standard topics very neatly. The problems are perhaps easy, but very very insightful. A lot of nice things are hidden in the problems.

Functional analysis is of course very important too. The book by Kreyszig is the perfect introduction to it: https://www.amazon.com/dp/0471504599/?tag=pfamazon01-20 You can perfectly read the book after a book like Bloch, but the longer you wait, the more you'll get out of it. The book might be too easy for you, but I adored it start to finish. You won't find a better introduction to functional analysis like this. But it is just an introduction after all, don't expect a deep knowledge of functional analysis after it.

Complex analysis then. The two books by Freitag (and Busam) are extremely well written. https://www.amazon.com/dp/3540939822/?tag=pfamazon01-20 and https://www.amazon.com/dp/3642205534/?tag=pfamazon01-20 They have very cool problems and very good exposition. They go very deep inside of complex analysis too. You can read these books after something of the level of Spivak's multivariable book.

Conway's abstract analysis book might fit in after these https://www.amazon.com/dp/0821890832/?tag=pfamazon01-20 Very nice book which goes towards operator theory. It starts of by building entire measure theory (which you should probably already know) in a very nonstandard way. Goes onto do C*-algebras.

Lang's real analysis and functional analysis is a true gem too. https://www.amazon.com/dp/0387940014/?tag=pfamazon01-20 Does things in a very great generality. For example, treat entire multivariable calculus (like inverse function theorem), but on Banach spaces. Treats measures on Banach spaces, etc. Really good and insightful read.

Then you might be ready for the true masterpiece of analysis. This is Simon's Comprehensive analysis course. Contains about everything you want to know about analysis and more. It starts of at the very beginning, and starts by doing topology and measure theory. But it does it at a very high level. https://www.amazon.com/dp/1470410982/?tag=pfamazon01-20

There must be a lot I've missed of course. For example, ODE and PDE theory is obviously important to analysis too, and is perhaps not treated very well in all the books I've linked. But I've got to stop somewhere. Tell me if there's anything special you're interested in.
Bloch book is really good, but one alternative is Terrance Tao's Analysis I and II. He also constructs the number system clearly and rigorously, and going in-depth into the foundational mathematics before taking a journey to the introductory analysis. I also really liked Landau's Foundations of Analysis and Shilov's Elementary Real and Complex Analysis (same level as Rudin but Shilov's motivates the concepts very well and goes in-depth to the topology).

I do not know if you read the following book: Introduction to Topology and Modern Analysis by George Simmons. He treats the topology from the analytic view.

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Bloch book is really good, but one alternative is Terrance Tao's Analysis I and II. He also constructs the number system clearly and rigorously, and going in-depth into the foundational mathematics before taking a journey to the introductory analysis. I also really liked Landau's Foundations of Analysis and Shilov's Elementary Real and Complex Analysis (same level as Rudin but Shilov's motivates the concepts very well and goes in-depth to the topology).
With respect to foundations, Bloch goes more indepth than any of these books. There are many very cool results in Bloch that you can never see anywhere else. I know Landau and Tao construct everything from scratch too, but Bloch simply goes deeper. Some examples of results in Bloch that you won't find in many other books:
- Categoricalness of $\mathbb{R}$
- Existence and uniqueness of decimal expansions is treated in full detail. Repeating decimal expansions are also treated in full detail.
- Induction and recursion theorems are treated in full detail.
- Many equivalence of completeness are obtained, including a full proof that the intermediate value theorem is equivalent to completeness
- Content is treated in full detail and justifies the claim that integrals compute areas.
- A full and rigorous computation of areas/circumferences of the circle (which is surprisingly difficult to do, since the usual approach using integrals does NOT work).
- A proof that $\pi$ is irrational.
- A full discussion of when a function is Riemann integrable (theorem of Lebesgue)
- A detailed discussion of a continuous nowhere differentiable function.

I do not know if you read the following book: Introduction to Topology and Modern Analysis by George Simmons. He treats the topology from the analytic view.
I know the book, but I don't find it suitable for a self-study of topology.

Bloch book is really good, but one alternative is Terrance Tao's Analysis I and II. He also constructs the number system clearly and rigorously, and going in-depth into the foundational mathematics before taking a journey to the introductory analysis. I also really liked Landau's Foundations of Analysis and Shilov's Elementary Real and Complex Analysis (same level as Rudin but Shilov's motivates the concepts very well and goes in-depth to the topology).
With respect to foundations, Bloch goes more indepth than any of these books. There are many very cool results in Bloch that you can never see anywhere else. I know Landau and Tao construct everything from scratch too, but Bloch simply goes deeper. Some examples of results in Bloch that you won't find in many other books:
- Categoricalness of $\mathbb{R}$
- Existence and uniqueness of decimal expansions is treated in full detail. Repeating decimal expansions are also treated in full detail.
- Induction and recursion theorems are treated in full detail.
- Many equivalence of completeness are obtained, including a full proof that the intermediate value theorem is equivalent to completeness
- Content is treated in full detail and justifies the claim that integrals compute areas.
- A full and rigorous computation of areas/circumferences of the circle (which is surprisingly difficult to do, since the usual approach using integrals does NOT work).
- A proof that $\pi$ is irrational.
- A full discussion of when a function is Riemann integrable (theorem of Lebesgue)
- A detailed discussion of a continuous nowhere differentiable function.

I know the book, but I don't find it suitable for a self-study of topology.
Ah! Thank you for the detailed comparison! Do you still recommend a student to read Bloch book from the first page to the last page? I read Rudin and Shilov (and little bits of Apostol), but I did not read Bloch in detail. Is there any interesting topic Bloch has that other books I mentioned do not have (except for the ones you listed)?

I thought Simmons make a great preparation for the books like Rudin-RCA and Lang-RFA.

Ah! Thank you for the detailed comparison! Do you still recommend a student to read Bloch book from the first page to the last page? I read Rudin and Shilov (and little bits of Apostol), but I did not read Bloch in detail. Is there any interesting topic Bloch has that other books I mentioned do not have (except for the ones you listed)?
I you already read Rudin/Shilov/Apostol, then reading Bloch is not necessary anymore. You might want to skim it to see if anything interesting pops out. But since you're obviously well acquainted with some real analysis already, I'd think that a more advanced approach will suit you more.

I thought Simmons make a great preparation for the books like Rudin-RCA and Lang-RFA.
Don't get me wrong, Simmons is a cool book. It would make a very neat preparation for the books you mentioned. But here are some things I find lacking:

1) No discussion of nets (or filters) at all. For a book that pretends to do topology from an analytic point of view, I find that an unforgivable omission.
2) No mention is simply connectedness/homotopy/homology/winding numbers. I find this omission sad because these topological notions are central in the study of complex analysis. Few books really do justice to the interplay between complex analysis and topology.
3) No mention of initial and final topologies (although admittedly there is a very small section on weak topology), which is sad because these notions of weak and weak* topologies in Banach setting is illuminated a lot by these things.
4) Exercises which I think do not go deep enough.

I you already read Rudin/Shilov/Apostol, then reading Bloch is not necessary anymore. You might want to skim it to see if anything interesting pops out. But since you're obviously well acquainted with some real analysis already, I'd think that a more advanced approach will suit you more.

Don't get me wrong, Simmons is a cool book. It would make a very neat preparation for the books you mentioned. But here are some things I find lacking:

1) No discussion of nets (or filters) at all. For a book that pretends to do topology from an analytic point of view, I find that an unforgivable omission.
2) No mention is simply connectedness/homotopy/homology/winding numbers. I find this omission sad because these topological notions are central in the study of complex analysis. Few books really do justice to the interplay between complex analysis and topology.
3) No mention of initial and final topologies (although admittedly there is a very small section on weak topology), which is sad because these notions of weak and weak* topologies in Banach setting is illuminated a lot by these things.
4) Exercises which I think do not go deep enough.
Thank you very much for the advice. I am taking the elementary topology course that uses the Munkres, and I think Simmons make a good companion to Munkres since Simmons illuminate the set theory very well. I do like his discussions of basic terms like open and closed sets.

It is just my opinion, but I think Simmons wanted to present just enough topology so he can discuss the algebra and analysis later on.

Oh by the way, do you know which is a better book to start learning the complex analysis on my own? Anlfors or Lang?

I would never recommend Rudin for anything. The book is horrible. It makes me physically sick that it's so popular. But ok, to see which book fits you, you need to tell more about yourself.

Anyway, I always liked Freitag (and Busam) two complex analysis books. They're very good. Barry Simon also treats complex analysis very comprehensively (with use of topology) in his comprehensive course on analysis.
Could you please briefly expand on what is wrong with Rudin in your opinion? Do you mean both of them?