Complex Analysis textbook - specific criteria
I'm looking for a text (textbook, lecture notes etc.) on Complex Analysis that meets some very specific desiderata. I've already searched through books recommended here and on MathOverflow, but so far I haven't found anything to suit my needs. (I should mention that I took a one-semester course in C.A. two years ago which was presented in this way, so I may be a little bit partial here.)
Firstly, the terms "holomorphic" and "analytic" should not be used interchangeably. Although there is no mathematical mistake as long as the power series expansion theorem is not implicitly assumed, it's good to have some distinction of meaning.
Complex integrals should be done in their general form, i.e. with Riemann sums over arbitrary (rectifiable) curves, not just over $\mathcal{C}^1$ curves (with the integral defined as $\int_a^b f(\gamma(t)) \gamma^\prime(t)\;\mathrm{d}t$).
Definitions (like the integral one above) should emphasise the conceptual side of a notion, not the computational side. E.g. in the course I took the winding number was defined like this, not like this.
The Cauchy integral theorems should be presented using homotopy/homology theories. (As a counterexample, the Stein/Shakarchi book proves them only on particular cases of contours.)
It would be nice to have short introductions to topics which stem from complex function theory - like sheaf theory, Riemann surfaces or analytic number theory - but I think that I already narrowed the answer space too much.
What can you recommend?
Solution 1:
I think that John Conway's Functions of One Complex Variable is precisely what you want. For example it defines the complex integrals using rectifiable curves as you want, also it does the homotopy version of Cauchy's theorem, it defines the winding number as you want, and it even has a chapter on analytic continuation and Riemann surfaces (with a discussion of sheaves of germs).
Now, maybe it does not have a lot of analytic number theory in it, but it has some relevant sections on the factorization of the Sine function, and on the Gamma and Zeta functions.
Finally, another book worth looking at is Lars Ahlfors' Complex Analysis. Although I personally prefer Conway's book.
Solution 2:
Henri Cartan's classic "Elementary Theory of Analytic Functions of One or Several Complex Variables" ticks 1, 3, 5 and partially 4. It is my favorite besides the book by B. V. Shabat "Introduction to complex analysis", Parts I and II (who says in the introduction that he had been inspired by Cartan's book); sadly only Part II of that book is translated from Russian.
A new book by Serge Lvovski "Principles of Complex Analysis" might tick a lot of points (judging by the table of contents) and overall looks promising (given a superb real analysis book by Lvovski).
For number 5 the books by Freitag and Busam "Complex Analysis" and "Complex Analysis II" might be good choices (some analytic number theory, Riemann surfaces, rudiments of several variables, uniformization).