# Books on complex analysis (Ahlfors, Conway and Lang)

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.

• I like Ahlfors, especially because it covers multiply-connected versions of the Riemann mapping theorem. Conway is serviceable but a bit dry. – user98602 Aug 4 '15 at 17:23
• Let me introduce an excellent one from the French mathematician Henri Cartan - Elementary Theory of Analytic Functions of One or Several Complex Variables (Dover Books on Mathematics). amazon.com/Elementary-Analytic-Functions-Variables-Mathematics/… – mathcounterexamples.net Aug 4 '15 at 17:29
• Theory of Functions of a Complex Variable (2 or 3 Vol depending edition) by A. I. Markushevich. – Aram Aug 4 '15 at 22:01

## 1 Answer

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 $\lim_{z\to0}\log(1+z)/z=1,$ 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.

• @DavidC.Ulrich Is that really a true story? That sounds so hard to believe that the author was so out of touch with the contents of the text even after your alerting specific attention. Apparently Conway's priority is not tending to correcting that gross erratum. In fact, it makes me curious as to the author's depth. What was the pseudo theorem if I may ask? – Mark Viola Aug 4 '15 at 17:52
• True story. Regarding "gross erratum", he didn't know it was wrong - I didn't tell him, didn't know at the time. Don't recall exactly - the book's at the office and it's too hot to visit the office except for something more important than this. But like I said, it was regarding the Perron solution to the Dirichlet probem; the counterexample was a punctured disk... – David C. Ullrich Aug 4 '15 at 17:57
• Interesting. See something about you actually is. And when you get to the office, just stay "cool." – Mark Viola Aug 4 '15 at 18:02
• I don't see a problem like you describe in my copy of the book. (Of course, I didn't solve each problem to verify... but there doesn't seem to be an indicator that 'this problem is hard!'). Maybe it was changed between printings. – user98602 Aug 4 '15 at 18:40
• I imagine you have a recent edition and you're looking in the chapter towards the end on the Dirchlet problem? (The solution for "arbitrary" domains via Perron, not the stuff about Poisson integrals on a disk.) If so then yes maybe he changed it. – David C. Ullrich Aug 4 '15 at 18:43