In chapter $V$ of Palka, "Consequences of the Local Cauchy Integral Formula,"
3.1. If a function $f$ is analytic in an open set $U$, then $f'$ is analytic in $U$. In particular, $f$ belongs to the class $C^1(U)$.
I was surprised to read this because I thought we'd established this fact long ago with the Cauchy-Riemann equations. Isn't that the whole thing about complex differentiability – that a function is infinitely differentiable whenever it's differentiable once? Obviously there is something fundamental I'm misunderstanding here about the difference between analyticity and differentiability, because I don't understand why these two statements are different.
Can someone please explain to me what is different about these two statements in detail, and in particular, the motivation for why this is (as Palka notes) a "striking theoretical consequence"?
Best Answer
Palka's book does indeed cover the Cauchy-Riemann equations (and simple consequences of the Cauchy-Riemann equations) in chapter 3, and defines (in chapter 3) an analytic function to be one which is complex-differentiable. However, it is only in chapter 5, where he treats consequences of the local Cauchy integral formula, that he manages to prove that the analyticity of $f$ implies that $f'$ is also analytic (and hence all derivatives are also analytic).
It is a well-known fact that if a function is analytic in a domain, then so are all the derivatives, but this fact requires proof - it is not an immediate consequence of the Cauchy-Riemann equations, and in Palka's development, it requires the power of the Cauchy integral formula to prove it.
To answer the question you raise at the end (why is it a striking theoretical consequence): If a function is defined to be analytic if it is differentiable in a domain, then there would appear to be no 'a priori' reason why the derivative $f'$ should also be differentiable (for example nothing similar happens for differentiability of real functions), and it must have seemed to be a pretty striking consequence when it was first discovered.