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!
Best 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.]