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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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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.

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I have already looked through Conway - it uses "analytic function" instead of "holomorphic" - see, for example, page 34. –  Andrei Sipoș Nov 4 '12 at 16:41
    
He does use different terms for different things. He does not use them interchangeably. For example, Conway uses "differentiable" to mean that a function has a complex derivative, he uses "analytic" to mean that it is continuously differentiable, and finally he just says "has a power series expansion" to mean precisely that. –  Adrián Barquero Nov 4 '12 at 16:53
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It seems to me a little weird to dismiss such a good textbook because the author didn't use "analytic" instead of saying "has a power series expansion". But in any case, to each his own =) –  Adrián Barquero Nov 4 '12 at 16:57
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