Textbooks and Instruction

[Yasiru]

I think it's better to have a separate thread for texts and leave the 'Mathematical Literature' one for books about maths and popular maths.

What books have you found to be the best/most instructive in whatever field you've taken a course in or have expertise in?

[Yasiru]

Doesn't seem to be much activity going on here. I suppose I'll go first-

From my experience, for Complex Analysis it doesn't get better than Ahlfors.
An intuitive (not rigorous), geometric treatment is from Needham's Visual Complex Analysis, which I used to complement Ahlfors and my course notes. (It was very surprising that I liked this latter, because geometric intuition has often eluded me.)

I wasn't able to take the second course in complex analysis offered in my final year unfortunately, so I'm not certain what good texts there are beyond the scope of these two.

[Adam]

My proffesors claim Real and Complex Analysis by Rudin is one of the ultimate books for that.

[Yasiru]

That's a very, very difficult path (if achievable at all when you're first coming to the subject).
Real and Complex Analysis is a brilliant supplement to advanced analysis courses (I used it as a reference/supplementary text along with Royden's Real Analysis for a course in Measure Theory and Fourier Analysis in my final year), but the treatment of complex analysis is, well, even 'terse' doesn't seem a good enough description to how sparing it is (though elegant, no doubt).
Best savoured after a course (or two) in more hands on complex analysis I think.

We had really good notes and the lectures were superb, so it occurs to me now that just Ahlfors by itself might be quite difficult as well. Another reference I remember using a couple of times is Introduction to Complex Analysis by H A Priestley.

Is your background in algebra, Adam?

[Yasiru]

Yes, I wouldn't recommend big Rudin for most people without a gentler introduction to measure theory first (and ideally, an introduction to complex analysis). Even his Principles of Mathematical Analysis might be a little too terse for a beginner, especially if you're learning on your own without a lecturer or other references to help (in this regard, while I haven't looked at them, apparently Amann and Escher, and Courant are worthy alternatives). But once you're matured enough and think it worth the effort, going through the sequel gives you a very impressive, unified picture of analysis.

Needham is all about geometric insights (which comes with the caution that rigour is not often given its due). I typically tend to favour the more abstract, formalism-based approach, but it had surprising mileage with the geometric flavour.

Have you taken topology yet (or plan to)?

[Adam]

Yasiru wrote: "That's a very, very difficult path (if achievable at all when you're first coming to the subject).
Real and Complex Analysis is a brilliant supplement to advanced analysis courses (I..."

Yeah, I do mostly Algebra. Though I am trying, painfully to learn how to think geometrically more. Since I think it is a decent skill to have. I only mentioned Rudin because you said you had already done work in Complex Analysis. Rudin is indeed terse and I wouldn't recommend Little Rudin for a beginner. I think a more gentle introduction is good for a first encounter with such a deep subject.

My professor used Introduction to Analysis, which I thought was a decent introductory text. While being a much gentler text, I still think you need a professors help to supplement what is actually going on. Like most analysis texts then tend to provide completed proofs, and do not show the secrets behind what is going on. That is mostly my opinion anyway. Maybe someday I will write an introductory Analysis text that is not necessarily done in the canonical Analysis way, but is written for a student to understand. This has been my problem with most Analysis texts I've read, there hasn't been an illuminating one for a beginner.

Fatema, if you are interested in going on in Topology on your own I highly recommend Topology.

[Yasiru]

@Fatema- I suppose analysis seems harder because of the careful considerations of rigour at each step.
You might like topology though, exactly for the visual aspect. I prefer Topology by K Jänich to the Munkres text Adam recommends (which is standard at many places, and very clear for beginners, but isn't terribly motivated in my view).
On the other hand, some popular accounts to encourage you are The Shape of Space and Euler's Gem: The Polyhedron Formula and the Birth of Topology (superb reads even if you don't go on to do any topology).


@Adam- I'm not very good at that either. Even though I've always liked analysis a little better than algebra (though after being introduced to Galois theory, I'm now torn between them), geometric intuitions remain a point of difficulty for me. On that note, you would probably like Needham's Visual Complex Analysis too.
Big Rudin is definitely good, but only with some appreciation of measure theory and a more technique-oriented course of complex analysis.
I've not seen that introductory analysis text before, but almost any such intro is gentler than Rudin. That said, I think the heavy rigour is essential for getting anywhere with analysis. This is fine for a university course built around a textbook like Rudin, where the lecturer can elaborate on things and give different proofs where the text is obscure, etc. (this is being hopeful you have a good lecturer who sees these needs of course), but if you're not so fortunate, or trying to teach yourself, it can feel like testing yourself against a brick wall. On that note, G H Hardy's A Course of Pure Mathematics is, even more than a century after publication, still one of the best ways you can go. The rigour is there, but enthusiastic motivation and discussion too.

[Lisa]

Everyone here seems familiar with big and little Rudin, but how about Real Analysis by Royden . It is thorough and I feel much gentler than Rudin.

For complex analysis, I prefer the two volume set The Theory of Functions of a Complex Variable by A.I. Markushevich. In my advisor's words "he leaves no stone unturned".

[Zero]

Wow, can't believe my profs never used any of these when I was an undergrad

[Yasiru]

I did like Royden better. The problems were very helpful.

Markushevich looks formidable.

Robby wrote: "Wow, can't believe my profs never used any of these when I was an undergrad"

I think the Rudin pair and perhaps Ahlfors are fairly popular choices. The others may depend on the instructor or where the institution is. Most I mentioned were referred to in my course syllabi and helped me from time to time, but books like Hardy's older analysis text or Needham's less formal complex analysis one don't seem likely to find themselves as recommended texts.

[Miguel Ángel]

All the books in complex variables cited (Ahlfors, Rudin, Needham, ...) they're OK. But, if you're interested in analytic number theory, then the book of Freitag & Busam: Complex Analysis it's a good training. And also for people with a geometric flavour preferences, then the book of Jones & Singerman: Complex Functions: An Algebraic and Geometric Viewpoint it's a superb election. And finaly, no doubt, the work of Markushevich: The Theory of Functions of a Complex Variable (Vol. 1 & 2) it's THE BOOK.

[NumberLord]

I'm glad someone found a good book for analytic number theory. There are many good ones out there. For algebraic number theory, Number Fields is the best I've seen.