Analytic function

Short description: Type of function in mathematics

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 [math]\displaystyle{ x_0 }[/math] converges to the function in some neighborhood for every [math]\displaystyle{ x_0 }[/math] in its domain. It is important to note that it is a neighborhood and not just at some point [math]\displaystyle{ x_0 }[/math], 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 [math]\displaystyle{ x_0 }[/math] to be considered an analytic function. As a counterexample see the Weierstrass function or the Fabius function.

Definitions

Formally, a function [math]\displaystyle{ f }[/math] is real analytic on an open set [math]\displaystyle{ D }[/math] in the real line if for any [math]\displaystyle{ x_0\in D }[/math] one can write

[math]\displaystyle{ 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 + \cdots }[/math]

in which the coefficients [math]\displaystyle{ a_0, a_1, \dots }[/math] are real numbers and the series is convergent to [math]\displaystyle{ f(x) }[/math] for [math]\displaystyle{ x }[/math] in a neighborhood of [math]\displaystyle{ x_0 }[/math].

Alternatively, a real analytic function is an infinitely differentiable function such that the Taylor series at any point [math]\displaystyle{ x_0 }[/math] in its domain

[math]\displaystyle{ T(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(x_0)}{n!} (x-x_0)^{n} }[/math]

converges to [math]\displaystyle{ f(x) }[/math] for [math]\displaystyle{ x }[/math] in a neighborhood of [math]\displaystyle{ x_0 }[/math] pointwise.^{[lower-alpha 1]} The set of all real analytic functions on a given set [math]\displaystyle{ D }[/math] is often denoted by [math]\displaystyle{ \mathcal{C}^{\,\omega}(D) }[/math].

A function [math]\displaystyle{ f }[/math] defined on some subset of the real line is said to be real analytic at a point [math]\displaystyle{ x }[/math] if there is a neighborhood [math]\displaystyle{ D }[/math] of [math]\displaystyle{ x }[/math] on which [math]\displaystyle{ f }[/math] 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.^[1]

Examples

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. Furthermore, every polynomial is its own Maclaurin series.
- 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.
Most special functions (at least in some range of the complex plane):

Typical examples of functions that are not analytic are

The absolute value function when defined on the set of real numbers or complex numbers is not everywhere analytic because it is not differentiable at 0.
Piecewise defined functions (functions given by different formulae in different regions) are typically not analytic where the pieces meet.
The complex conjugate function z → z* is not complex analytic, although its restriction to the real line is the identity function and therefore real analytic, and it is real analytic as a function from [math]\displaystyle{ \mathbb{R}^{2} }[/math] to [math]\displaystyle{ \mathbb{R}^{2} }[/math].
Other non-analytic smooth functions, and in particular any smooth function [math]\displaystyle{ f }[/math] with compact support, i.e. [math]\displaystyle{ f \in \mathcal{C}^\infty_0(\R^n) }[/math], cannot be analytic on [math]\displaystyle{ \R^n }[/math].^[2]

Alternative characterizations

The following conditions are equivalent:

[math]\displaystyle{ f }[/math] is real analytic on an open set [math]\displaystyle{ D }[/math].
There is a complex analytic extension of [math]\displaystyle{ f }[/math] to an open set [math]\displaystyle{ G \subset \mathbb{C} }[/math] which contains [math]\displaystyle{ D }[/math].
[math]\displaystyle{ f }[/math] is smooth and for every compact set [math]\displaystyle{ K \subset D }[/math] there exists a constant [math]\displaystyle{ C }[/math] such that for every [math]\displaystyle{ x \in K }[/math] and every non-negative integer [math]\displaystyle{ k }[/math] the following bound holds^[3] [math]\displaystyle{ \left| \frac{d^k f}{dx^k}(x) \right| \leq C^{k+1} k! }[/math]

Complex analytic functions are exactly equivalent to holomorphic functions, and are thus much more easily characterized.

For the case of an analytic function with several variables (see below), the real analyticity can be characterized using the Fourier–Bros–Iagolnitzer transform.

In the multivariable case, real analytic functions satisfy a direct generalization of the third characterization.^[4] Let [math]\displaystyle{ U \subset \R^n }[/math] be an open set, and let [math]\displaystyle{ f: U \to \R }[/math].

Then [math]\displaystyle{ f }[/math] is real analytic on [math]\displaystyle{ U }[/math] if and only if [math]\displaystyle{ f \in C^\infty(U) }[/math] and for every compact [math]\displaystyle{ K \subseteq U }[/math] there exists a constant [math]\displaystyle{ C }[/math] such that for every multi-index [math]\displaystyle{ \alpha \in \Z_{\geq 0}^n }[/math] the following bound holds^[5]

[math]\displaystyle{ \sup_{x \in K} \left | \frac{\partial^\alpha f}{\partial x^\alpha}(x) \right | \leq C^{|\alpha|+1}\alpha! }[/math]

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. (See also the Lagrange inversion theorem.)
Any analytic function is smooth, that is, infinitely differentiable. The converse is not true for real functions; in fact, in a certain sense, the real analytic functions are sparse compared to all real infinitely differentiable functions. For the complex numbers, the converse does hold, and in fact any function differentiable once on an open set is analytic on that set (see "analyticity and differentiability" below).
For any open set [math]\displaystyle{ \Omega \subseteq \mathbb{C} }[/math], the set A(Ω) of all analytic functions [math]\displaystyle{ u\ :\ \Omega \to \mathbb{C} }[/math] is a Fréchet space with respect to the uniform convergence on compact sets. The fact that uniform limits on compact sets of analytic functions are analytic is an easy consequence of Morera's theorem. The set [math]\displaystyle{ \scriptstyle A_\infty(\Omega) }[/math] of all bounded analytic functions with the supremum norm is a Banach space.

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 ƒ has an accumulation point inside its domain, then ƒ is zero everywhere on the connected component containing the accumulation point. In other words, if (r_n) is a sequence of distinct numbers such that ƒ(r_n) = 0 for all n and this sequence converges to a point r in the domain of D, then ƒ is identically zero on the connected component of D containing r. This is known as the identity theorem.

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

As noted above, any analytic function (real or complex) is infinitely differentiable (also known as smooth, or [math]\displaystyle{ \mathcal{C}^{\infty} }[/math]). (Note that this differentiability is in the sense of real variables; compare complex derivatives below.) There exist smooth real functions that are not analytic: see non-analytic smooth function. In fact there are many such functions.

The situation is quite different when one considers complex analytic functions and complex derivatives. It can be proved that any complex function differentiable (in the complex sense) 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). Analyticity of complex functions is a more restrictive property, as it has more restrictive necessary conditions and complex analytic functions have more structure than their real-line counterparts.^[6]

According to Liouville's theorem, any bounded complex analytic function defined on the whole complex plane is constant. The corresponding statement for real analytic functions, with the complex plane replaced by the real line, is clearly false; this is illustrated by

[math]\displaystyle{ f(x)=\frac{1}{x^2+1}. }[/math]

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 open ball (holomorphic functions are analytic). This statement for real analytic functions (with open ball meaning an open interval of the real line rather than an open disk of the complex plane) is not true in general; the function of the example above gives an example for x₀ = 0 and a ball of radius exceeding 1, since the power series 1 − x² + x⁴ − x⁶... diverges for |x| ≥ 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 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 ƒ(x) defined in the paragraph above is a counterexample, as it is not defined for x = ±i. This explains why the Taylor series of ƒ(x) diverges for |x| > 1, i.e., the radius of convergence is 1 because the complexified function has a pole at distance 1 from the evaluation point 0 and no further poles within the open disc of radius 1 around the evaluation point.