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 is real analytic on an open set D in the real line if for any x₀ in D one can write

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 is convergent 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

T(x) = \sum_{n=0}^{\infin} \frac{f^{(n)}(x_0)}{n!} (x-x_0)^{n} is convergent for x close enough to x₀ and its value equals f(x).

The definition of a complex analytic function is obtained by replacing everywhere above "real" with "complex", and "real line" with "complex plane".

Examples

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.

The exponential function is analytic. Any Taylor series for this function converges not only for x close enough to x₀ (as in the definition) but for all values of x (real or complex).

The trigonometric functions, logarithm, and the power functions are analytic on any open set of their domain.

The absolute value function is not analytic because it is not differentiable. Piecewise defined functions (functions given by different formulas in different regions) are in general not analytic.

The complex conjugate function, $z\to \overline z,$ is not complex analytic, although its restriction to the real line is real analytic.

Properties of analytic functions

The sums, products, and compositions of analytic functions are analytic.
The reciprocal of an analytic function that is nowhere zero, is analytic, as is the inverse of an invertible analytic function whose derivative is nowhere zero.
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 function is constant on the corresponding connected component.

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

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

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

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 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 every 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.