Analytic Function

Analytic functions are necessary in many applications of mathematics to solve problems in physics, fluid dynamics, and electrical engineering. There are two types of analytic functions, namely, real analytic functions and complex analytic functions. Commonly, the properties of both real and complex analytic functions differentiate.

1.0What is an Analytic Function?

An analytic function is a function that is differentiable at every point in a region. An analytic function is a branch of mathematics dealing with complex numbers and functions of complex variables. It is defined as an f(z), where z is the complex variable, and the function is determined to be differentiable at any point in its domain. Such functions are smooth, well-behaved, and may represent power series. 

Analytic Function Example: Examples of analytical functions include trigonometric functions like Sin(x), Cos(x), etc. 

Types of Analytic Function: 

As mentioned earlier, analytic functions are of two types: 

1. Real Analytic Function: A real analytic function is a real-valued function of any real variable that can be represented by a convergent power series around any point in its domain. For example, ex is a real analytic function that can be written in a power series. 
2. Complex Analytic Function: A complex analytic function, or holomorphic function, is a function of a complex variable differentiable at every point in an area of the complex plane. It obeys Cauchy-Riemann equations, which describe smooth and differentiable behaviour. For example, z²  because it satisfies Cauchy-Riemann equations. 

2.0How Do You Know if a Function is Analytic?

To determine whether a function is analytical or not, two conditions must be satisfied:

1. Differentiability: This step is the same for both types of Analytic functions. For a function to be analytical, it needs to be differentiable at every point in its region. 
2. Taylor’s Equation: For the Real analytic function, it should satisfy the expansion of Taylor’s series equation. 
3. Cauchy-Riemann Equations: For complex analytic functions, these equations must be true at a certain point to be analytical. 

3.0Analytic Function Complex Analysis

Functions that are analytic at every point in their domain can be described using power series and have specific properties. Such functions are studied thoroughly in analytic function complex analysis, a branch of mathematics studying functions of complex variables.

4.0Analytic Functions of a Complex Variable

A complex variable analytic function is a critical component of understanding functions' behaviour in the complex plane within the realm of complex analysis. A function f(z) is said to be analytic provided that f(z), where z is a complex number, satisfies the Cauchy-Riemann equations:

$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$

Here f(z)=u(x, y)+i v(x, y), where u(x, y) \& v(x, y) are the real and imaginary parts, respectively?  

5.0Solved Problems: 

Problem 1: Check if the function

$f(z)=z^2+\bar{z}$ (where

$\bar{Z}$ is the complex conjugate of z) is an analytical function. 

Solution: Express f(z) as

$f(z)=z^2+\bar{z}=(x+iy{)}^2+(x-iy),\text{ here }z=x+iy$

Expanding, 

$f(z)=\left(x^2-y^2+2ixy\right)+(x-iy)$

So, u(x,y) = x²-y²+x and v(x,y) = 2xy-y. 

Equating in Cauchy-Reimen Equation: 

$\frac{\partial u}{\partial x}=2x+1,\frac{\partial v}{\partial y}=2x-1$

Hence,

$\frac{\partial u}{\partial x}\ne \frac{\partial v}{\partial y}$

the given equation is not an analytic function. 

Problem 2: Verify if the function f(z)=x^3-3xy^2+i(3x^2y-y^3)is analytic. 

Solution: Let z = x + iy here, the function can be written as: 

f(z)=u(x,y)+iv(x,y)

Where u(x,y) = x³ – 3xy² and v(x,y) = 3x²y – y³. 

For the equation to be analytic, it must satisfy the Cauchy-Riemann equation, that is: 

$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$

Now, by calculating the partial derivatives: 

$\frac{\partial u}{\partial x}=3x^2-3y^2,\frac{\partial u}{\partial y}=-6xy$

$\frac{\partial v}{\partial x}=6xy,\frac{\partial v}{\partial y}=3x^2-3y^2$

Here, we can clearly see that: 

$\frac{\partial u}{\partial x}=3x^2-3y^2=\frac{\partial v}{\partial y},\frac{\partial v}{\partial y}=-6xy=-\frac{\partial v}{\partial x}$

Hence, f(z) is an analytic function. 

Problem 3: Find whether the given function f(z) = x² − iy² is analytic or not.

Solution: Let z = x+iy here, the function can be written as: 

f(z)=u(x,y)+iv(x,y)

Where u(x,y) = x² and v(x,y) = -y². 

For function f(z) to be analytic, it needs to satisfy the Cauchy-Riemann equations, that is: 

$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$

Partial derivatives of the function: 

$\frac{\partial u}{\partial x}=2x,\frac{\partial u}{\partial y}=-2y$

$\frac{\partial v}{\partial y}=0,\frac{\partial v}{\partial x}=0$

$\frac{\partial v}{\partial x}\ne \frac{\partial v}{\partial y},\frac{\partial v}{\partial y}\ne -\frac{\partial v}{\partial x}$

Since the equation is satisfied, the given function u(x,y)= x²–y² is not analytic. 

Problem 4: Show that the function f(z) = e^z is analytic and find its Taylor series expansion around z=0.

Solution: 

Analyticity: The function f(z) = e^z is known to be analytic everywhere in the complex plane because it is infinitely differentiable, and its Taylor series converges to the function for all z.

Taylor Series Expansion: The Taylor series expansion of f(z) = e^z around z = 0 is given by:

$f(z)=e^z=\sum \frac{z^n}{n!}$

This series converges with all values of z in the function of the complex plane. 

6.0Properties of Analytic function

Some of the most common properties of analytical function are as follows: 

• Differentiable everywhere in a neighbourhood
• Satisfies the Cauchy-Riemann equations.
• It can be expressed as a power series.
• It is continuous and infinitely differentiable.
• Closed under arithmetic operations (addition, multiplication, division).
• Holomorphic (complex differentiable).
• Analytic under composition: if f(z) and g(z) are analytic, so is f(g(z)).
• Liouville's Theorem: bounded entire functions are constant.
• Maximum Modulus Principle: maximum modulus attained at the boundary.
• Integration preserves analyticity.