Books on complex analysis (Ahlfors, Conway and Lang)

Question

To make my question slightly different from others, I would like to know how would you rate on the complex analysis books by Ahlfors, Conway and Lang?

I had a basic course on complex analysis during undergraduate (and you could imagine it's mostly about computing integrals and residues), and would like to learn more about the theory. There exist many good books, and the three books aforementioned are the ones I like the most. Of course I don't and won't have time to study all these three books in detail, so I have to pick one.

The coverage of these books seem to be similar (except Conway's second volume, which should not be compared to others' single volume book). These three books contain rigorous proofs, so it's kind of hard to choose.

Of course if you have read any two of them or all three of them you are very welcome to compare these books. If you ask me where am I headed to I would say I want to learn something about several complex variables.

Also, if you think there is some book better than these three, you are welcome to mention it.

Answer 1

I have a hard time avoiding blatant self-promotion here...

I don't know Lang. Ahlfors is of course a classic. I have a lot of issues with Conway. (My complaints are with the first volume, which it turns out he wrote as a student! The second volume is full of great stuff.) Conway was the standard text here for years - I hated it so much I started using my own notes instead, which eventually became Complex Made Simple (oops. Well, there are things in there that are not in any other elementary text that I know of.)

Two examples that spring to mind regarding Conway:

He spends almost a page using the power series for $\log(1+z)$ to show that $\lim_{z\to0}\frac{\log(1+z)}{z}=1,$ evidently not recalling the definition of the derivative.

There's a chapter or at least a section on the Perron solution to the Dirichlet problem. There's an exercise, like the first or second exercise in the chapter, which a few decades ago I was unable to do. I sent him a letter explaining why it was harder than he seemed to think.

In the next edition the words "This exercise is hard" were added. A year or so later I realized the exercise was not just hard, it was impossible. Asks us to prove something false.

Seems very unimpressive - I complain I don't know how to do the exercise and he doesn't even bother to make sure it's correct.

Answer 2

Ahlfors was the second book I saw about complex analysis, c. 1971. It was my source for some years. Picked up a few things (albeit in a strange world-view) from W. Rudin's R-and-C.

Lang's book surely (I did have some acquaintance with him in the late 1970s) had good intentions, and was completely competent.

I have only marginal acquaintance with Conway's book.

As quite a few people would say, by this year, "look at my course notes for what I think this course/idea should address". :)

(http://www.math.umn.edu/~garrett/m/complex/)

It may be worth emphasizing that, while a certain body of inquiry that got itself labelled (accurately or not) "complex analysis" was a hot topic some decades ago, as far as I can tell, work that would self-label "complex analysis" these days would be considered not-so-contemporarily-hot.

But this is absolutely not to claim in any way that "complex analysis of one variable" is obsolete or archaic. Quite to the contrary, it is universally relevant in mathematics (things depend holomorphically on a parameter), and, apparently, in physics and engineering...