# Learning Complex Analysis: Integrals vs. Power Series - ordering the development of results.

Over the last few months, I have been visiting elementary complex analysis.

My exposure to complex analysis is pretty much limited to the material in three books: Ahlfors, Bak/Newman, and Churchill/Brown. I don't care much for Churchill/Brown, but really enjoy the other two.

Both Ahlfors and Bak/Newman open up the rabbit hole by proving the Cauchy Integral Theorem for functions analytic on a rectangle, and then for functions analytic on a rectangle except for (ostensibly) a finite set of points.

They both then show the Cauchy formula representing the function as a contour integral on a circle.

After that, things begin to diverge.

Bak/Newman develops the power series representation for analytic functions, and uses this to show analytic functions are infinitely differentiable. From this he gets Morera's theorem, the Uniqueness theorem, etc.

Ahlfors instead proceeds to differentiate the Cauchy formula under the integral to show an analytic function is infinitely differentiable, to get Morera's theorem, the Uniqueness theorem, etc., without really mentioning power series. He leaves power series to (slightly) later development.

(One thing I like about Ahlfors' approach is that, as far as I can tell, you can derive the differentiability of power series inside their radius of convergence without using the same kind of $\delta - \epsilon$ arguments that Bak/Newman uses.)

My question is this: Is there a general feeling about whether or not one approach is better than the other in terms of any/all of the following criteria: Beauty; Ease of instruction; Deeper understanding of the subject; Usefulness in applications; (Other criteria)?

Any thoughts are more than welcome.