# What is a good, hi-tech textbook on complex analysis?

I am looking for an introductory textbook for Complex Analysis that is hi-tech.

All the books I have looked at suffer from the same problem; they're only assuming that the reader is familiar with is basic real analysis, and thus, are by design, low-tech.

I'm looking for a textbook that:

• Doesn't shy away from treating the Riemann sphere as a manifold, and clearly distinguishes it from $\mathbb{C}$, so it's easy to keep track of where my functions live.
• Gives the statement of Cauchy's theorem in a modern, algebraic topological language of (co)homology
• Actually compares the theorems, where applicable, to the $2$-dimensional real case with more than passing remarks
• Doesn't give whacky definitions of topological properties (eg. simple connectedness)

This isn't a complete list, but this should give you a good idea about what I mean by hi-tech.

Additional extras:

• Has a sane statement of Liouville's theorem. Why say that "bounded entire functions are constant" when you could be saying "the image of a function $\mathbb{C} \to \mathbb{C}$ is dense or a single point"?
• Covers basic multivariable complex analysis
• Treats the logarithm, etc. as functions from a Riemann surface, rather than the clumsy "multifunction" treatment
• Treats power series formally and then passes to convergent ones

I basically want someone like John M. Lee to write a complex analysis book. (His book on Smooth Manifolds is about as good as textbooks get, in my opinion.)

The closest I have found was Cartan's text, but I'm hoping that someone on this site might know something even better.

Many thanks!

• Not sure "hi-tech" is the word you are looking for. More like "advanced." – Thomas Andrews Nov 26 '14 at 23:51
• @ThomasAndrews, well, it depends on what you mean by advanced. I don't want the theorems to be (much) more advanced than the ones you can find in say Ahlfors. But I do want the presentation to be advanced, yes. – user40167 Nov 27 '14 at 0:02
• "Has a sane statement of Liouville's theorem." You're being silly here. The proof of your claim follows from the classical statement of Liouville, and since Liouville stated in such a way, and it is this variant one uses mostly, it is useless to go against historical inertia and tradition. – Pedro Tamaroff Nov 27 '14 at 1:02
• @key: I'm not sure whether the book "Complex Analysis in One Variable" by R. Narsasimhan & Y. Nievergelt will fall into your $"{\bf hi}$-${\bf tech}"$ book list or not, but you can take a look at it. books.google.co.in/… – Krish Nov 27 '14 at 1:13
• I would echo Narasimhan. Also, look at Forster's Riemann Surfaces book, along with Hörmander's SCV (which covers one variable in 20 pages or so, but then uses serious graduate real analysis). – Ted Shifrin Nov 27 '14 at 1:42

## 1 Answer

Berenstein/Gay: Complex variables. Imo it satisfies the first four points. Concerning a sane statement of Liouville's theorem: the usual statement is the sane one; if you don't understand this, think harder about it. It also does not treat complex analysis in several variables, for this you should take a look at Hörmander; I am also not sure as far as power series are concerned; knowledge of formal power series is quite "low tech" (as you would put it), and so including it would be odd.

[I would describe Lee's smooth manifold as rather "low tech", in particular his rather unsophisticated treatment of $\otimes$ and algebra in general (the same is true for most algebraic topology textbooks apart from perhaps Spanier), and of de Rham cohomology. In any case it does a good job as far as geometry is concerned, and this is what it is about. In the same way you can't expect a textbook on analysis to give the most elegant/sophisticated treatment of algebra related topics, but as you are looking for something like Lee, I guess you are not looking for something that has Bourbaki level.]

• Thank you for your answer. Berenstein/Gay looks very nice! A few points: formal power series are definitely a layer of abstraction over convergent power series. I only added it to the list, because I enjoyed Cartan's treatment of them. I'm unfortunately not familiar with the works of Bourbaki, but in any case, if you have other good books in mind, please share. – user40167 Nov 29 '14 at 1:07
• @krey certainly they are more abstract, but elementary in the same sense as the intermediate value theorem is, so they are not something sophisticated or anything. – Mister Benjamin Dover Nov 29 '14 at 15:41