# Complex Analysis textbook - specific criteria

This is my first question on this site and I hope I won't screw it up.

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.)

1. 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.

2. 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 $\int_a^b f(\gamma(t)) \gamma^\prime(t)\;\mathrm{d}t$).

3. 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.

4. 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.)

5. 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?

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