analytic function

In mathematical analysis, an analytic function is a function that is locally represented by a convergent power series. More precisely, a real or complex function is analytic at a point if, in some neighborhood of that point, it is equal to a power series centered there. Analytic functions are therefore locally determined by their coefficients, or equivalently by their derivatives at the center of the expansion. In other words, an analytic function is a function that is locally represented by a convergent Taylor series.

Analytic functions occur in both real analysis and complex analysis, in slightly different ways. A real or complex analytic function is necessarily smooth, having derivatives of all orders. But a smooth real function need not be analytic. By contrast, a complex function on an open set is analytic if and only if it is holomorphic, that is, complex differentiable at every point of the set. For this reason, in complex analysis the terms analytic function and holomorphic function are often used interchangeably. The terms complex analytic and real analytic distinguish between these cases. In signal processing, a complex analytic function is sometimes called an analytic signal.