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.

Definitions

Formally, function f defined on an open set D in the real numbers is called analytic, if for any x_{0_{in D one can write f (x) as}}

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

in which the coefficients a₀, a₁, ... are real numbers and the series being convergent in a neighborhood of x_{0_.}

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.

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.

Also, if all the derivatives of an analytic function at a point are zero, then that function is zero. In particular, if an analytic function is zero in the neighborhood of a point, it is zero everywhere.

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

Real versus complex analytic functions

Real and complex analytic functions share all the properties we described so far. However, they also have important differences. Complex analytic functions are more rigid in many ways.

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 differentiable function is analytic. Consequently, in complex analysis, the term analytic function is synonymous with holomorphic function.

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

f(x)={\frac {1}{x^{2}+1}}.

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.

Analytic functions of several variables

One can define analytic functions in several variables by means of power series in those variables (see power series). Analytic functions of several variables have some of the same properties as analytic functions of one variable. However, especially for complex analytic functions, new and interesting phenomena show up in many dimensions.