Cauchy's Integral Formula expresses the value of an analytic function (and, by differentiation, all of its derivatives) at an interior point of a domain as a contour integral of the function divided by powers of (w - z) around the domain's boundary; it follows from Cauchy's Theorem and the mean value property. This leads to the Cauchy Estimates bounding derivative magnitudes and to Liouville's Theorem, which states every bounded entire function is constant — a sharp contrast with real analysis, where boundedness and smoothness do not imply constancy. These results belong to complex analysis and underpin further consequences such as the fundamental theorem of algebra.
Cauchy's Integral Formula expresses the value of an analytic function (and, by differentiation, all of its derivatives) at an interior point of a domain as a contour integral of the function divided …