Analytic function

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In mathematics, an analytic function is a function that is locally given by a convergent power series. Analytic functions can be thought of as a bridge between polynomials and general functions. 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 it is equal to its Taylor series in some neighborhood of every point.

Contents

Definitions

Formally, a function ƒ is real analytic on an open set D in the real line if for any x0 in D one can write

in which the coefficients a0, a1, ... are real numbers and the series is convergent to ƒ(x) for x in a neighborhood of x0.

Alternatively, an analytic function is an infinitely differentiable function such that the Taylor series at any point x0 in its domain

converges to ƒ(x) for x in a neighborhood of x0. The set of all real analytic functions on a given set D is often denoted by Cω(D).

A function ƒ defined on some subset of the real line is said to be real analytic at a point x if there is a neighborhood D of x on which ƒ 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."

Examples

Most special functions are analytic (at least in some range of the complex plane). Typical examples of analytic functions are:

  • Any polynomial (real or complex) is an analytic function. This is because if a polynomial has degree n, any terms of degree larger than n in its Taylor series expansion will vanish, and so this series will be trivially convergent. Furthermore, every polynomial is its own Maclaurin series.

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