the smoothing consequence of the formula

There is a quiet but profound side effect of Cauchy's integral formula: it is a smoothing operation. Even if you only assume f is holomorphic — which is a mild-looking condition — the formula represents f as a contour integral of f against the very smooth kernel 1 / (z - z_0), and integrating against a smooth kernel produces a smooth result. The roughness you might fear simply cannot survive.

Here is the mechanism in words. The value f(z_0) is an average of boundary values weighted by 1 / (z - z_0), and as z_0 moves around inside the contour, that kernel varies infinitely smoothly (it is itself holomorphic in z_0 away from the boundary). Differentiating with respect to z_0 just differentiates the smooth kernel, never the boundary data, so f inherits the kernel's infinite smoothness rather than any irregularity of its values. This is the same calculation that yields infinite differentiability, viewed as a smoothing statement: the integral launders f into something automatically C-infinity.

Why this matters: it is the structural reason a holomorphic function cannot have corners, kinks, or any of the mild misbehavior real functions tolerate. It is also a preview of a recurring theme — operators built from nice integral kernels (the Poisson integral, mollifiers, the heat kernel) all smooth their inputs. The honest framing: the smoothing is not adding information or 'improving' f; it reveals that f was already perfectly smooth, because holomorphy is a far more restrictive condition than it first appears.