Point information from distributed information.

Cauchy's Integral Formula is a central result of complex analysis that links the value of a holomorphic function inside a closed curve to its values along the boundary. In effect, the boundary contains enough information to reconstruct what happens within. The formula is strikingly powerful: it leads directly to derivatives of every order, Taylor expansions, and many of the elegant rigidity results that make complex analysis feel almost enchanted. It also provides a practical method for evaluating certain contour integrals that otherwise look forbidding, across theory and practice. It is used primarily in complex analysis, contour integration, and mathematical physics.