Analytic Function: Definition, Examples, Properties

The complex analytic functions are one of the main objects to study in Complex Analysis. Here we learn the definition of analytic functions with examples, properties, and solved problems.

Key concept to study analytic functions is complex differentiation.

ℂ: = The set of complex numbers.

Definition of Analytic Function

A function f is called an analytic function at a point a∈ℂ if it is differentiable at every point of some neighbourhood of a.

Analytic functions are also known as holomorphic functions.

For example, f(z) = z² is an example of an analytic function as it is differentiable everywhere in the whole complex plane.

Read Also: Derivative of Complex Functions

Example

Show that f(z) = zⁿ is an analytic function.

Proof:

The derivative of the function f(z) = zⁿ at a point z₀ is given by

f$’$(z0) = limh→0 $\dfrac{f(z_0+h)-f(z_0)}{h}$ ⇒ f$’$(z0) = limh→0 $\dfrac{(z_0+h)^n-z_0^n}{h}$

Let t = z₀+h. Then t→z₀ as h→0. Thus,

f$’$(z₀) = $\lim\limits_{t \to z_0}\dfrac{t^n-z_0^n}{t-z_0}$ = $nz_0^{n-1}$.

This shows that f(z) = zⁿ is differentiable at z₀ with derivative f$’$(z₀) = $nz_0^{n-1}$. As z₀ is an arbitrary point in ℂ, it follows that zⁿ is everywhere differentiable. In other words, the function f(z) = zⁿ is analytic in the whole complex plane. This makes zⁿ an entire function.

Non-Example

The function f(z) = $\overline{z}$ is nowhere analytic in ℂ.

Properties of Analytic Functions

An analytic function satisfies the following properties.

1. The sum and the product of two analytic functions are also analytic. If f and g are analytic, then f+g and fg are also analytic.
2. If f(z) and g(z) are analytic functions, then the quotient $\dfrac{f(z)}{g(z)}$ is also analytic, provided that $g(z) \neq 0$.
3. If an entire function (analytic everywhere) is bounded, then it is constant. This is known as Liouville’s theorem.
4. Zeros of an analytic functions are isolated.
5. If f(z) is an analytic function at z₀, then it satisfies the Cauchy-Riemman equations at that point.
6. If f(z) is analytic at z₀, then it can be expressed as a power series in a neighbourhood of z₀: $f(z) = \sum_{n=0}^\infty a_n (z-z_0)^n$ where $a_n=\dfrac{f^{(n)}(z_0)}{n!}$.

Solved Problems

Q1: Show that f(z) = |z|2 is nowhere analytic.

Answer:

The function f(z) = |z|² is differentiable at z=0 only and nowhere else. Thus, any neighbourhood of 0 will always contain points where |z|² is not differentiable. So we conclude that the function f(z) = |z|² is nowhere analytic in ℂ.

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