Analytic Function

Introduction

An analytic function is a function that is locally given by a convergent power series. In complex analysis, an analytic function is a complex-valued function that is complex differentiable at every point in its domain.

Definition

Formally, a function f of a complex variable z is analytic in a region R of the complex plane if, for every z0 in R, there exists a neighborhood of z0 in which f(z) can be represented by a power series that converges uniformly to f(z) in that neighborhood.

A visual representation of an analytic function.

Properties

Analytic functions have many properties that make them a central object of study in analysis and other branches of mathematics. These properties include:

Differentiability: Analytic functions are infinitely differentiable, meaning they have derivatives of all orders. This property is a consequence of the power series representation of analytic functions.

Holomorphicity: In complex analysis, analytic functions are often referred to as holomorphic functions. A function is holomorphic if it is complex differentiable at every point in its domain. This is a stronger condition than real differentiability, as it requires the function to be differentiable in all directions in the complex plane.

Harmonicity: If a function is analytic, then its real and imaginary parts are harmonic functions, i.e., they satisfy Laplace's equation.

Conformality: Analytic functions preserve angles locally. In other words, if two curves intersect at a point in the domain of an analytic function, then their images under the function will intersect at the same angle.

Analyticity and Differentiability

In real analysis, differentiability and analyticity are distinct concepts: a function can be differentiable without being analytic. However, in complex analysis, the two concepts coincide: a function is analytic if and only if it is complex differentiable. This is a consequence of the Cauchy-Riemann equations, which provide a necessary and sufficient condition for a function to be analytic.

Analytic Functions and Power Series

A key property of analytic functions is that they can be represented locally by a power series. This property is a consequence of the Cauchy integral formula, which provides a way to compute the coefficients of the power series representation of an analytic function.

Examples of Analytic Functions

There are many examples of analytic functions in mathematics. Some of the most common include:

Polynomials: All polynomials are analytic, as they can be represented by a finite power series.

Exponential function: The exponential function e^z is analytic everywhere in the complex plane.

Trigonometric functions: The sine and cosine functions are analytic everywhere in the complex plane.

Logarithmic function: The natural logarithm function ln(z) is analytic everywhere in the complex plane except at the origin.

Non-Examples of Analytic Functions

While many functions are analytic, there are also many that are not. For example, the absolute value function |z| is not analytic at the origin, as it is not differentiable there. Similarly, the function f(z) = z* if z is real and 0 otherwise is not analytic at any point, as it is not differentiable anywhere.

Analytic Continuation

Analytic continuation is a technique in complex analysis for extending the domain of an analytic function. Given an analytic function defined on a subset of the complex plane, one can often extend the function to a larger domain in such a way that the extended function is still analytic.