Complex Numbers Introduction

Complex analysis provides a powerful framework for studying mathematical functions, including functions of a real variable, by extending them to complex domains. Complex analysis deals with many of the same concepts as real analysis, such as limits, continuity, derivatives, and integrals, but it also includes new concepts that are unique to complex functions. In complex analysis, we delve into the intricate interplay between complex functions and the geometric properties of their domain. This field holds great significance in mathematics.

Moreover, complex analysis uncovers captivating phenomena, such as the rich interplay between analytic functions, singularities and the behavior of functions at infinity. The study of residue and contour integration offers elegant techniques for evaluating complex integrals and solving differential equations, with applications ranging from physics to engineering. 

Another fundamental aspect of complex analysis is the study of analytic continuation, which allow us to extend the domain of function beyond their initial definition. The concept plays a pivotal role in understanding the behavior of complex functions and their connection to other areas of mathematics, such as number theory and algebraic geometry. 

Some other important applications of complex analysis are Cauchy integral, Laurent series, Power Series, Singularity, etc. These all are useful applications of complex analysis.

In this analysis, we will delve into the depth of complex analysis, exploring its key concepts, theorems, and techniques. We will investigate the behavior of complex functions, uncover geometric interpretations, and examine their applications in different fields. By delving into the intricacies of complex analysis, we aim to gain a deeper understanding of this fascinating branch of mathematics and appreciate its profound impact on various scientific endeavors.

                                Complex Analysis

Why Complex Numbers?

Solving equations like:

                               `x^2 + 1 = 0`                  or                `x^2 - 10x + 4 = 0`

give solutions that cannot be represented on real line i.e they are not real numbers. Such solutions are complex numbers. These numbers have rules and representations different from those of real numbers. Similarly, the functions of these numbers are complex functions can be handled with complex calculus rather than real calculus.

Complex Numbers:

A number of the form 𝑧 = 𝑎 + 𝑏𝑖, where 𝑎, 𝑏 ∈ ℛ , 𝑖 = √−1 , is called a complex number.

This can also be written as: 𝑧 = (𝑎, 𝑏) 

‘𝑎’is called real part of ‘𝑧’. i.e 𝑅𝑒(𝑧) = 𝑎 

‘𝑏’is called imaginary part of ‘𝑧’. i.e 𝐼𝑚(𝑧) = 𝑏

e.g 

a = 2 + 3𝑖 ⇒ 𝑅𝑒( a ) = 2,      𝐼𝑚( a ) = 3 

b = 4 − 9𝑖 ⇒ 𝑅𝑒( b ) = 4,      𝐼𝑚( b ) = 9 

Pure Imaginary Number: 

A real constant multiple of ‘𝑖’ is called pure imaginary number e.g 𝑧 = 6𝑖.

Equality of complex numbers: 

Two complex numbers, x = 𝑎 + 𝑏𝑖 and y = c + d𝑖 are equal if and only if: 𝑅𝑒(x) = 𝑅𝑒(y) and 𝐼𝑚(x) = 𝐼𝑚(y).

Geometrical Representation of a complex number

A complex number 𝑧 = 𝑥 + 𝑦𝑖 = (𝑥, 𝑦) can be represented geometrically as shown in the following figure:

The 𝑥𝑦-plane in which complex numbers are represented is called Complex Plane or Argand Diagram. The x-axis represented the real axis and the y-axis represented the imaginary axis on which imaginary numbers are plotted. This is a very beautiful representation of complex numbers on a plane because before this we considered imaginary numbers not to exist on a plane. By using complex planes, we can easily study complex numbers and their behavior.

Argand diagrams are useful for visualizing complex numbers and their properties such as addition, subtraction, multiplication, and division. They can also be used to represent functions of complex variables, such as the complex exponential function and complex logarithmic function.

Argand diagrams are widely used in mathematics, physics, and engineering as well as in other fields where complex numbers are important.

Geometrically, the complex plane provides a powerful tool for understanding the properties and relationships of complex numbers. It facilitates the visualization of operations, mapping and geometric interpretations, of complex functions, contributing to a deeper understanding of complex analysis concepts. For example, the addition of complex numbers corresponds to vector addition in the complex plane, where the real and imaginary parts are added separately. Similarly, the multiplication of complex numbers involves scaling and rotation of vectors.

Modulus of a complex number: 

The modulus or absolute value of a complex number 𝑧 = 𝑥 + 𝑦𝑖 is denoted by |𝑧| and is given by: 

|𝑧| = `sqrt(x^2 + y^2)` = 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 a vector.

Arithmetic Operations on Complex Numbers

Let `z_1` = (a, b) & `z_2` = (c, d)

Addition:

Adding two complex numbers,

`z_1 + z_2` = (a, b) + (c, d) = (a + c, b + d)

Subtraction:

Subtracting two complex numbers,

`z_1 - z_2` = (a, b) - (c, d) = (a - c, b - d)

Multiplication:

Multiplicating two complex numbers

`z_1 . z_2` = (a, b).(c, d) = (ac, bd)

Division:

Dividing two complex numbers:

`\frac{z_1}{z_1}=\frac{a + bi}{c + di}`

`=\frac{a + bi}{c + di}*\frac{c - di}{c - di}`

`=\frac{(ac + cd) + (bc - ad)i}{c^2 + d^2}`   

Note: If `i` is in the denominator then we rationalize to remove `i` from the denominator.

Properties of Complex Numbers 

Commutative laws

Commutative law for multiplication and addition                          

                           `z_1 + z_2 =  z_2 + z_1, z_1 . z_2 = z_2 . z_1`

Associative laws

Associative law for multiplication and addition                         

                           `z_1 + (z_2 + z_3) = (z_1 + z_2) + z_3`

                           `z_1(z_2 . z_3) = (z_1 . z_2)z_3`

Distribution laws

                          

                           `z_1(z_2 + z_3) = z_1 z_2 + z_1 z_3`

Conjugate of a Complex number: 

If 𝑧 = 𝑥 + 𝑦𝑖 is a complex number then the complex conjugate of ‘𝑧’ is given by: 

                                                                     𝑧̅ = 𝑥 − 𝑦𝑖

Note: Only `i` term change its sign and the real part remains same after solving conjugate.

Let's have some examples of conjugate.

Example # 1:

Conjugate of z = `4 - 3i`

Solution:

𝑧̅ = 4 + 3`i`

Example # 2:

Conjugate of z = `-2 + 9i`

Solution:

𝑧̅ = `- 2 - 9i`