analyticity

Analyticity is a property of a function, typically a complex-valued one, describing its expression as a power series locally. In complex analysis, a function is analytic on a domain if it is complex differentiable at every point and, in fact, is holomorphic: around each point there exists a neighborhood in which the function can be written as a convergent power series. Equivalently, the function is determined by its Taylor series on each such neighborhood.

Analytic functions have strong consequences. They are infinitely differentiable, satisfy the Cauchy-Riemann equations, and obey the identity theorem: if two analytic functions agree on a set with a limit point within a connected domain, they agree everywhere on that domain. The values of an analytic function in a region often determine its values elsewhere through analytic continuation, though continuation can fail at singularities.

Isolated singularities are a central feature of complex analyticity. Points where a function fails to be analytic can sometimes be removed, yielding a removable singularity; otherwise they may be poles or more complicated essential singularities. Analyticity also extends to several complex variables, where a function is analytic if it is holomorphic in each variable in a neighborhood, with analogous radius-of-convergence properties.

Examples illustrate the concept. Entire functions such as the exponential, sine, and cosine are analytic everywhere on the complex plane. The function 1/z is analytic on the plane minus zero but has a pole at zero and is not analytic there. In real analysis, a function can be real-analytic (locally given by a real power series) but still differ from merely smooth functions, as some smooth functions are not analytic.