Intuitive explanation of Cauchy's Integral Formula in Complex Analysis There is a theorem that states that if $f$ is analytic in a domain $D$, and the closed disc {$  z:|z-\alpha|\leq r$} contained in $D$, and $C$ denotes the disc's boundary followed in the positive direction, then for every $z$ in the disc we can write:
$$f(z)=\frac{1}{2\pi i}\int\frac{f(\zeta)}{\zeta-z}d\zeta$$
My question is:
What is the intuitive explanation of this formula? (For example, but not necessary, geometrically.)
(Just to clarify - I know the proof of this theorem, I'm just trying to understand where does this exact formula come from.)

Even though I am not totally satisfied by my argument, I would say it is about average values on the curve of integration that is dedicated to analytic functions. 
Lets look at a circle $C$ around $z$: If we choose points $\zeta_1,\ldots,\zeta_n$ on $C$ such that they span a regular $n$-sided polygon the average of their sum is $z$, that is
$$\sum_{k=1}^n \zeta_k\cdot\frac{1}{n}= z.$$ 
This is far from Cauchy integral formula, however, since
$$1 = \frac{1}{2\pi i}\int_C \frac{1}{\zeta-z}d\zeta$$
we can write the Cauchy formula in the form 
$$\frac{1}{2\pi i}\int_C\frac{f(\zeta)-f(z)}{\zeta-z}d\zeta=0.$$
Next, if we think of $f$ as a power series around $z$, we can interpret the formula in text as saying that:
"If we remove the constant term $f(z)$ from $f(\zeta)$ and divide out the factor $\zeta-z$, the remaining will have average 0". 
