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.)

Expanding on my comment, this result can be translated into:
"A surface in $\mathbb{R}^3$ which satisfies the Maximum-Modulus principle is uniquely determined by specifying it's boundary"
To see this, write the holomorphic function $f(z)$ in terms of its real and imaginary parts:
$$ f(z) = f(x,y) = g(x,y) + ih(x,y)$$
Then since $f$ is holomorphic, the functions $g(x,y)$ and $h(x,y)$ are both real valued harmonic functions which satisfy the maximum modulus principle.  By interpreting the value of $g$ or $h$ as the height of a surface in $\mathbb{R}^3$, we can see that according to Cauchy's theorem such surfaces are uniquely determined by specifying their boundary.
