Analytic Function - Definitions

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

$\begin{align} f(x) & = \sum_{n=0}^\infty a_{n} \left( x-x_0 \right)^{n} \\ & = a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + a_3 (x-x_0)^3 + \cdots \end{align}$

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 (in the mean-square sense). 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."

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