More on the Properties of Analytic Functions

Analytic functions, which can be real or complex, are functions that are given locally by a convergent power series. While both types are infinitely differentiable, more properties apply to complex analytic functions than they do to real analytic functions. Let’s review the common Properties of Analytic Functions and how you can apply them to problem solving.

Types of Analytic Function

A function is considered analytic if its Taylor series about x=0 converges to a function around every x0 value in its domain. As mentioned, there are two types of analytic functions:
• Real analytic function
• Complex analytic function

Properties of Analytic Function
There are many properties of analytic functions, some of which only apply to complex analytic functions. The basic ones you should know about are:
• The compositions, sums, and products of an analytic function are analytic.
• The limits of uniformly convergent sequences in analytics functions are also analytic functions.
• If the reciprocal of an analytic function is nowhere zero, it is analytic. This also applies to the inverse of invertible functions where the derivative is nowhere zero.
• All analytic functions are infinitely differentiable.
• A bounded entire function is a constant function.
• All non-constant polynomials have roots, i.e., there is a z0 where p(z0) = 0.
• Where there is an analytic function f(z) defined on U, then f(z)|, which is the modulus of the function, cannot reach its maximum on U.
• A zero in an analytic function is an isolated point unless the function is identically zero.
• Where an analytic function f(z) is defined on a disk D, then there exists an F(z) analytic function defined on the disk so that F′(z) = f(z). This function is a primitive of the original analytic function f(z), and, as a result, ∫C f(z) dz =0; for all closed curves in D.
• If C is a curve that connects two points r0 and r1 in the domain of an analytic function f(z) and there is an analytic function F(z), then ∫C F’(z) = F(z1) – F(z0)
• Where there is an analytic function f(z) and a point z0, which is one the domain U of the function, then the function [f(z)-f(z0)]/ [z – z0] is also analytic on U.

Analytic functions in mathematics are defines as functions created through the convergent power series. There are two different types of analytic functions: complex and real functions. Generally, both of these functions are infinitely differentiable but complex functions have more properties than real ones. Understanding the properties of analytic functions can help you solve statistical problems.