Customer Reviews

An Introduction to Complex Analysis in Several Variables, Third Edition (North-Holland Mathematical Library)

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I started to learn several complex variables a few weeks ago, and I noticed the absolute lack of textbooks on the subject. Probably the book that comes more naturally as an extension of undergraduate complex analysis is Gunning and Rossi, but this title is out-of-print (even finding a used copy is nearly impossible. Believe me, I've tried hard).

So, we have Hormander's book. Lars Hormander is known for writing high-level math texts (both in quality and difficulty), as seen in his famous 4-volume series about PDE's, and this book is no exception. His point of view is more related to his area of research (PDE's, again), and his demands for prerequisites are higher than GR (basics from Lebesgue integration, differential forms, algebra and point-set topology are more than welcome), but this book is a masterpiece of mathematical craftsmanship. The methods here developed are often unique, and the author presents the subject in a fully rigorous way. Along with the fact that it is one of the very few books on several complex variables still in print, this is a very valuable text, set in a high standard of excellence. My only complaint is the obscenely high price for a book so important. Several complex variables are an indispensable background for complex manifolds and algebraic geometry, and several important topics in theoretical physics (string theory, twistor theory, conformal field theory), and it's a shame that books like GR go out-of-print without any others for substituting them. Hormander's book doesn't go much deep in these directions, but you won't find any other book in print on the subject with such a high quality.

This book is pretty hard to read.
It's extremely terse, for one thing. In one place he said something followed from the Hahn-Banach theorem. It didn't _look_ related to the HB theorem. So I find a corollary to the HB theorem which does look related. But it requires the Riesz representation theorem to work. So I read the proof of the RR theorem for positive measures. That also doesn't quite do it. I puzzle a bit, then find there's a RR theorem for complex measures. And, Finally this one sly little sentence makes sense!
If you care about understanding the math, making sense of it, it is sure going to be slow reading!
But, a lot of it is also fairly straightforward.
It has a few typos, but for the most part it's carefully proofread. Typos are especially frustrating when his writing is so compressed, because there isn't much of a context to make sense of them.
He uses new symbols without defining them. You have to guess what they mean. For example, if V is an open set, C^k(V) means complex valued functions on V that are k times continuously differentiable. He does define that. Then he starts talking about C^k(K), where K is the closure of an open set, without saying what he means by being differentiable on the boundary. Maybe he only talks about C^k(K) if the boundary of K is nice enough that you can define a derivative on it? Then he mentions C^k_(0,1)(K). Leaving you to figure out from the context that he means differential forms of type (0,1) with coefficients in C^k(K). Later, he talks about a "schlicht domain" without defining schlicht.
He uses idiosyncratic notation, so it would be hard to use the book as a reference. You'd have to figure out what all the symbols mean, and he doesn't list them all in his symbol table at the start of the book.
But, I also felt I was hearing from a truly expert mathematician. I wasn't surprised when I found out he got a Fields medal.
It has been awfully frustrating reading about multivariable complex analysis, because a lot of the authors do make mistakes. Gunning and Rossi's book had a lot of mistakes. Robert Gunning's later book, "Introduction to holomorphic functions of several variables" also has many mistakes in proofs. The mistakes in Gunning and Rossi had been fixed up, but Bochner's tube theorem, which is new in the later book, had a wrong proof. An article by Sin Hitotumatu purported to prove the theorem, but there was a big gap in the proof! But Hormander, though it took me ages to puzzle through his writing, actually did prove the tube theorem. There were no mistakes in Hormander's book as far as I read it. Being right, not using up your time with long proofs that have mistakes in them, makes up for a great deal of obscurity. And reading his proofs can be a delight like watching the clever conjurations of a wizard.
I've only read the first 42 pages of Hormander's book. But I imagine these attributes I describe only get more intense once he's past the introductory things.
Later books on multivariable complex analysis have been written for people who have trouble with compressed mathematical gems like Hormander's book.
His book is referred to a lot by other books, although probably hardly anybody actually tries to follow his reasoning.
Laura

zooom.....Here's a clue; As I recall, he does all of one variable complex analysis in an introductory chapter less than 20 pages long, including a version of Cauchy's theorem more general than I wager most have ever seen.