structural:expansion
Definitions

Formally, a function f {\displaystyle f} is real analytic on an open set D {\displaystyle D} in the real line if for any x 0 ∈ D {\displaystyle x_{0}\in D} one can write

in which the coefficients a 0, a 1, … {\displaystyle a_{0},a_{1},\dots } are real numbers and the series is convergent to f (x) {\displaystyle f(x)} for x {\displaystyle x} in a neighborhood of x 0 {\displaystyle x_{0}}.

Alternatively, a real analytic function is an infinitely differentiable function such that the Taylor series at any point x 0 {\displaystyle x_{0}} in its domain

converges to f (x) {\displaystyle f(x)} for x {\displaystyle x} in a neighborhood of x 0 {\displaystyle x_{0}} pointwise. The set of all real analytic functions on a given set D {\displaystyle D} is often denoted by C ω (D) {\displaystyle {\mathcal {C}}^{\,\omega }(D)}, or just by C ω {\displaystyle {\mathcal {C}}^{\,\omega }} if the domain is understood.

A function f {\displaystyle f} defined on some subset of the real line is said to be real analytic at a point x {\displaystyle x} if there is a neighborhood D {\displaystyle D} of x {\displaystyle x} on which f {\displaystyle f} is real analytic.

The definition of a complex analytic function is obtained by replacing, in the definitions above, "real" with "complex" and "real line" with "complex plane". A function is complex analytic if and only if it is holomorphic i.e. it is complex differentiable. For this reason the terms "holomorphic" and "analytic" are often used interchangeably for such functions.