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Concept# Analytic function

Summary

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. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about converges to the function in some neighborhood for every in its domain. It is important to note that it's a neighborhood and not just at some point , since every differentiable function has at least a tangent line at every point, which is its Taylor series of order 1. So just having a polynomial expansion at singular points is not enough, and the Taylor series must also converge to the function on points adjacent to to be considered an analytic function. As a counterexample see the Fabius function.
Formally, a function is real analytic on an open set in the real line if for any one can write
in which the coefficients are real numbers and the series is convergent to for in a neighborhood of .
Alternatively, a real analytic function is an infinitely differentiable function such that the Taylor series at any point in its domain
converges to for in a neighborhood of pointwise. The set of all real analytic functions on a given set is often denoted by .
A function defined on some subset of the real line is said to be real analytic at a point if there is a neighborhood of on which 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.
Typical examples of analytic functions are
The following elementary functions:
All polynomials: if a polynomial has degree n, any terms of degree larger than n in its Taylor series expansion must immediately vanish to 0, and so this series will be trivially convergent.

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