What do you think about Lang Textbooks?

[dreamtheater]

What do people think about Lang's textbooks?

I am learning Complex Analysis for the first time from Lang. (literally the 1st time I'm seeing ANY sort of complex analysis/calculus, although I have extensive experience with real analysis).

I have gone through quite a love-hate relationship with this book. At first, I hated it with a PASSION, namely because it seemed a little too "wink-wink" look-how-clever-this-mathematics-is feel without being too systematic, formal, or elegant.

But then I fell in love with Lang's decision to systematically cover formal power series from an algebraic point-of-view before ever considering convergence.

But then I also still hate the book because it seems to bury really important techniques/results in "examples". (Even a simple remark, like "the above is important." would help.)

For those with more experience with Lang's numerous textbooks, what do you think? Lang's textbooks seem to have a specific flavor that is antithetical to the flavor of, say, Royden/Fitzpatrick or Munkres. Some have criticized Lang for churning out too many textbooks, without much care.

Just interested in what people think about Lang.

P.S. Also, any good companion textbooks for Lang's Complex Analysis?

 

[snipez90]

I read the first two chapters back when I was taking real analysis. I remember disliking the experience because Lang didn't provide much motivation for me to study the subject. In the first chapter, I think he does a review of basic topology before talking about complex differentiability. It felt out of place since he only needed the notion of an open set to introduce complex differentiability. Moreover I don't think he discussed connectedness, which seems like one of the more important topological notions in basic complex analysis. He probably does this later on, but still the topology in the first chapter seems out of place.

When he does talk about complex differentiability, he starts by proving the basic differentiation rules from calculus. Again I don't hold it against him because it's review. When he gets to the Cauchy-Riemann equations, I think he did a clever derivation that related complex differentiability to real differentiability, and that's all. He basically made it very hard to distinguish between complex differentiability and real differentiability. He does not hint at all at why complex differentiability is stronger or why some basic functions are not complex differentiable. He does not introduce formal differential operators to demonstrate the difference between complex functions of two real variables and complex functions of a complex variable. Finally he spends like 2 pages discussing conformal mapping before turning to power series...

As for the formal power series, I didn't particularly care. This is after all analysis, so I would have preferred discussing convergence right away. I think some of the stuff he did at the end like the local maximum modulus principle was neat but overall I still had the impression that I had no clue what complex analysis was about.

I think the first two chapters were a total of about 90 pages, and chapter 3 is on Cauchy's theorem. Having studied complex analysis this past few weeks from other texts, I gather that it shouldn't take that much text to capture the flavor of complex analysis. I'm reading Stein and Shakarchi's Complex Analysis and in the first 100 pages they discuss complex differentiability, Cauchy's theorem, contour integration, Schwarz reflection principle, Runge's theorem, residue theory, maximum modulus principle, etc. More importantly for me, the authors actually discuss the results they aim to prove and why those results distinguish complex analysis in its own right.

Incidentally, the text I'm using cites Lang as a reference, so it's probably a good companion text if you ever get tired of Lang. I'll actually be using Lang in the fall but I think I'll mostly be using it for the problems or as a reference. As for previous exposure to Lang, I liked his basic calculus text :P. His undergraduate analysis text was pretty good for multivariable topics, but I didn't read much else of it. My friend was using his text for differentiable manifolds I think and apparently it started with category theory, so go figure :P.