Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not hold generally for real analytic functions. A function is analytic if and only if its Taylor series about x0 converges to the function in some neighborhood for every x0 in its domain. In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not hold generally for real analytic functions. A function is analytic if and only if its Taylor series about x0 converges to the function in some neighborhood for every x0 in its domain. 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, an 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 C^{,omega }(D)} . 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.