The title pretty much says it all. I am revisiting complex analysis for the first time since I "learned" some as an undergraduate. I am trying to wrap my head around why it should be the case that a function which is differentiable once should be differentiable twice. I know a proof (Use Cauchy integral formula and differentiate under the integral sign), but that proof doesn't do a whole lot to explain the magic.


Say you had never heard of complex numbers before, and someone told you that you had a function $f: \mathbb{R}^2 \to \mathbb{R}^2$ given by $f(x,y) = (u(x,y), v(x,y))$, which locally looks like a rotation/expansion (i.e. it satisfies the Cauchy-Riemann equations everywhere). Then why on Earth should this function be $C^\infty$? I would love to see a picture, or just a proof that makes this feel less like a magic trick. I hoped that "Visual Complex Analysis" would help me out here, but this seems to be the one theorem in the book which is not given a geometric motivation.



Is there an integration free proof (or heuristic) that once differentiable implies twice differentiable for complex functions?
I recalled a comment of Ahlfors at the beginning of Chapter 4 of his book "Complex Analysis" which asserted the existence of such a proof, and it led me to the paper "A proof of the power series expansion without Cauchy's formula" by E. H. Connell and P. Porcelli (Bull. Amer. Math. Soc. 67 (1961), 177-181), where they give just such a proof based on a topological theorem from G. T. Whyburn, "Topological analysis" stating that any holomorphic function is an open mapping. I have not checked the latter source to see that it is independent of integration, but it likely is; I can't get my hands on it at the moment since it is in a book rather than an article. In fact, the Connell–Porcelli article is actually an announcement of results and so omits some details. There is an article just by Connell, "On properties of analytic functions" (Duke Math. J. 28 1961 73–81) which allegedly gives the details and I haven't looked up.


**Edit:** After reading Gerald Edgar's answer, I should comment that I'm holding the Third Edition of Ahlfors.


A note on the proof. As you might expect, the topological content is that the complex plane has a basis of open balls such that even after removing finitely many points, they are connected. This implies that a function which is continuous on an open set and open away from a point is again open (contrast that with the absolute value function on the real line), and this trivially implies the maximum modulus principle if you know that holomorphic functions are open maps. From there you can just do elementary epsilon-delta reasoning (applied to the "difference quotient function" $(f(z) - f(z\_0))/(z - z\_0)$) to get that a continuous function which is holomorphic away from a point is also holomorphic at the point. You can replace "continuous" by "bounded" by doing this with $(z - z\_0) f(z)$.


Alas, it is at this point that the summary paper starts summarizing. However, they do go on to prove (completely) the existence of a power series development, which is stronger than mere twice-differentiability.


