Analytic function

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 real analytic functions and complex analytic functions, which have similarities as well as differences.

Contents

1 Definitions

2 Properties of analytic functions

3 Analyticity and differentiability

4 Real versus complex analytic functions

5 Analytic functions of several variables

Definitions

Formally, function f defined on an open set D in the real numbers is called analytic, if for any x₀ in D one can write f (x) as

<math>

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 <math>

in which the coefficients a₀, a₁, ... are real numbers and the series is convergent in a neighborhood of x₀.

The definition of a complex analytic function is obtained by replacing everywhere above "real" with "complex". For an in-depth article about complex analytic functions see holomorphic function.

Properties of analytic functions

The sum, product, and composition of analytic functions are analytic.
The reciprocal of a non-zero analytic function is analytic, and the inverse of an invertible analytic function with non-zero derivative is analytic.
Any polynomial is an analytic function. For a polynomial, the power series expansion contains only a finite number of non-zero terms.
Any analytic function is smooth.

A polynomial cannot be zero at too many points unless it is the zero polynomial (more precisely, the number of zeros is at most the degree of the polynomial). A similar but weaker statement holds for analytic functions. If the set of zeros of an analytic function f has an accumulation point inside its domain, then f is zero everywhere on the connected component containing the accumulation point.

More formally this can be stated as follows. If (r_n) is a sequence of distinct numbers such that f(r_n) = 0 for all n and this sequence converges to a point r in the domain of D, then f is identically zero on the connected component of D containing r.

Also, if all the derivatives of an analytic function at a point are zero, the same conclusion as above holds.

These statements imply that while analytic functions do have more degrees of freedom than polynomials, they are still quite inflexible.

Analyticity and differentiability

Any analytic function (real or complex) is differentiable, actually infinitely differentiable (that is, smooth). There exist smooth real functions which are not analytic, see the following example. The real analytic functions are much "fewer" than the real (infinitely) differentiable functions.

The situation is quite different for complex analytic functions. It can be proved that any complex function differentiable in an open set is analytic. Consequently, in complex analysis, the term analytic function is synonymous with holomorphic function.

Real versus complex analytic functions

Real and complex analytic functions have important differences (one could notice that even from their different relationship with differentiability). Complex analytic functions are more rigid in many ways.

According to Liouville's theorem, any bounded complex analytic function defined on the whole complex plane is constant. This statement is clearly false for real analytic functions, as illustrated by

Also, if a complex analytic function is defined in an open ball around a point x₀, its power series expansion at x₀ is convergent in the whole ball. This is not true for in general for real analytic functions. (Note that an open ball in the complex plane would be a disk, while on the real line it would be an interval.)

Any real analytic function on some open set on the real line can be extended to a complex analytic function on some open set of the complex plane. However, not any real analytic function defined on the whole real line can be extended to a complex function defined on the whole complex plane. The function f (x) defined in the paragraph above is a counterexample.