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.)

An exercise that was a big help to me was to compute the integral of 1/z on the closed path consisting of the square from (1,-1) to (1,1) to (-1,1) to (-1,-1) to (1,-1). It illustrates how the sides with varying real differ from the sides with varying imaginary values.
