First, let me state Cauchy's Integral formula:
Let $U$ be an open region in the complex plane and $D = \{z : |z-z_0| \leq R\}$ a disk in $U$. If $f : U \to \mathbb C$ is holomorphic and $\gamma$ is the boundary of $D$, then for all points $a$ inside the disk
$$ f(a) = \frac 1{2πi}\oint_\gamma \frac{f(z)}{z-a}\ dz$$
and Cauchy's Integral Formula for the derivatives:
$$f^{(n)}(a) = \frac{n!}{2πi}\oint_\gamma \frac{f(z)}{(z-a)^{n+1}}\ dz$$
The majority of books prove the second formula by differentiation of the first formula and then proceeding by induction. However, it seems like Wikipedia wants to suggest a different proof that I couldn't find anywhere else, but they only give hints:
Since ${\displaystyle 1/(z-a)}$ can be expanded as a power series in the variable ${\displaystyle a}$:
${\displaystyle {\frac {1}{z-a}}={\frac {1+{\frac {a}{z}}+\left({\frac {a}{z}}\right)^{2}+\cdots }{z}}}$
— it follows that holomorphic functions are analytic, i.e. they can be expanded as convergent power series. In particular f is actually infinitely differentiable
So I took the time to try to understand what they are actually suggesting. What follows is my attempt to turn Wikipedia's hint into a proof, however, it's incomplete because I don't know how to justify the red-marked interchange of integral and series:
Let $a$ be a point inside $D$ and $w$ a point in the largest open disk around $a$ contained in $D$. Then, $|w-a| < |z-a|$ for all $z$ on $\gamma$, so $\left|\frac{w-a}{z-a}\right|<1$ and then
$$\sum_{n=0}^\infty \frac{(w-a)^n}{(z-a)^{n+1}} = \frac{1}{z-a} \frac{1}{1-\frac{w-a}{z-a}} = \frac 1{(z-a)-(w-a)} = \frac 1{z-w} $$
Applying Cauchy's Integral formula:
$$\begin{align} f(w) &= \frac 1{2πi}\oint_\gamma \frac{f(z)}{z-w}\ dz \\ &= \frac 1{2πi}\oint_\gamma \sum_{n=0}^\infty f(z)\frac{(w-a)^n}{(z-a)^{n+1}}\ dz \\ &\color{red}{=} \sum_{n=0}^\infty \frac 1{2πi}\oint_\gamma f(z)\frac{(w-a)^n}{(z-a)^{n+1}}\ dz \\ &= \sum_{n=0}^\infty \left(\frac 1{2πi}\oint_\gamma \frac{f(z)}{(z-a)^{n+1}}\ dz\right)(w-a)^n \end{align}$$
but this is a power series in $w$ around $a$. And we know that power series are infinitely-differentiable in their radius of convergence with
$$ f(w) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(w-a)^n $$ therefore $$ f^{(n)}(a) = \frac{n!}{2πi}\oint_\gamma \frac{f(z)}{(z-a)^{n+1}}\ dz $$
Is the red-marked interchange valid? Apart from the interchange, is everything alright?
Best Answer
I felt like elaborating more so here's a more complete answer. Verify the following for yourself:
I think this power series argument is very slick because it completely bypasses the differentiation under the integral sign issue, and it directly proves holomorphic implies analytic, and furthermore that the power series expansion is valid on the whole disc of holomorphicity (if you differentiate under the integral sign, all you manage to show is that $f$ is infinitely complex differentiable, but the existence of a power series expansion still needs to be proved. I just checked and this whole argument takes about 2-3 pages in Ahlfors's book, but merely under 1 page in Cartan's book).
On the other hand, one has to be slightly careful about the radii of the various discs; this is one of those "open set chasing games". Regarding this, I have one issue with your proposed proof:
I'm not sure that this is good enough. Sure, this way, for each $w$ we can expand as a series $\frac{1}{z-w}=\sum_{n=0}^{\infty}\frac{(w-a)^n}{(z-a)^{n+1}}$, but I doubt the convergence is uniform (because the Weierstrass M-test only guarantees uniform convergence on compact subsets of the unit disc), so we need to ensure $\sup_{z,w}\left|\frac{(w-a)^n}{(z-a)^{n+1}}\right|<1$, and based on the current way you've phrased it I doubt this is true. So, as written, your series and integral interchange is unjustified.
Here's how I'd phrase the argument (essentially what Cartan does, except he WLOG assumes $z_0=0$) to prove holomorphic implies analytic (and hence easily "read off" the formula for the derivatives).
Some remarks I should make: when I made the series expansion above, because we ensured \begin{align} \sup\limits_{\substack{|w-z_0|\leq r\\ |z-z_0|=r_0}}\left|\frac{w-z_0}{z-z_0}\right| &\leq \frac{r}{r_0}<1, \end{align} the geometric series converges absolutely and uniformly with respect to both $w$ and $z$. So, after integrating with respect to $z$, the convergence of the series to $f(w)$ is still uniform with respect to $w$; so everything really is as nice as it can be. Lastly, in hindsight, we can invoke Cauchy's theorem once again to apply a homotopy to the path $\gamma_0$ without changing any of the integral values. Thus, we do not have to assume our path is a circle of a given radius and having center $z_0$.