Analytic function - Definition and Overview

In mathematics, an analytic function is one that is locally given by a convergent power series.

Complex analysis teaches us that if a function f of one complex variable is differentiable in some open disk D centered at a point c in the complex field, then it necessarily has derivatives of all orders in that same open neighborhood, and the power series

<math>\sum_{n=0}^\infty {f^{(n)}(c) \over n!} (z-c)^n<math>

converges to f(z) at every point within D. For a proof of this result, see proof that holomorphic functions are analytic. That is an important respect in which complex functions are better-behaved than real functions; see an infinitely differentiable function that is not analytic. Consequently, in complex analysis, the term analytic function is synonymous with holomorphic function.

A complex analytic function of several complex variables is defined to be analytic and holomorphic at a point if it is locally expandable (within a polydisk, a cartesian product of disks, centered at that point) as a convergent power series in the variables. This condition is stronger than the Cauchy-Riemann equations; in fact it can be stated

A function of several complex variables is holomorphic if and only if it satisfies the Cauchy-Riemann equations and is locally square-integrable.

For real variables, even just one, smoothness does not suffice to ensure analyticity. A real function of a real variable given locally by a power series is a real analytic function.