One contour integral knows every disk value
Cauchy's integral formula says a holomorphic function on a disk is fixed by its values on the boundary circle: f(a) equals a contour integral of f(z)/(z−a) against dz over 2πi. The Cauchy integral theorem powers the proof and implies holomorphic functions are analytic power series.
The statement packs a surprising claim: complex differentiation behaves as well as integration does, surviving uniform limits in a way that fails for real functions. The same idea yields an integral for every derivative at a, with n factorial in front and the denominator raised to the power n plus one, a version sometimes called Cauchy's differentiation formula. Because the factor 1 over z minus a can be unrolled into a geometric series in a over z, the integral hands back a convergent power series, which is why every holomorphic function is analytic.
The circle is not essential. Any closed rectifiable curve can take its place, and, as with the integral theorem, the function need only be holomorphic in the open region the path encloses and continuous up to its edge. The proof first uses the integral theorem to shrink the contour to a tiny circle around a, then parametrises that circle and bounds the gap between the integral and f(a) by the largest change in f on the circle, which vanishes as the radius goes to zero.
Boundary data cannot be chosen freely. Some continuous functions on the circle, fed into the formula, return zero at every interior point, so they match no holomorphic function inside. In fact the real part alone on the boundary pins down the whole function up to an imaginary constant, and a Möbius transformation combined with the Stieltjes inversion formula rebuilds it.
In practice, when an integrand has several singularities inside a contour, the Cauchy–Goursat theorem splits the integral into small circles around each pole. Each piece is rewritten so that the remaining factor is analytic on its circle, and then the formula evaluates it directly.
Source: Cauchy's integral formula
