Complex numbers are an extension of the real numbers, allowing for the manipulation of quantities that cannot be represented by real numbers alone. Complex numbers are used in various fields of mathematics, engineering, and physics to solve problems that real numbers cannot address.

A complex number is a number that can be expressed in the form \(a+bi \), where \(a\) and \(b\) are real numbers, and \(i\) is an imaginary unit with the property \(i^2=-1 \). In this expression, \(a\) is called the real part of the complex number, and b is called the imaginary part.

Mathematically, the set of complex numbers is denoted as \(∁\), and can be visualized as a two-dimensional plane called the complex plane (or Argand plane), with the real part on the horizontal axis and the imaginary part on the vertical axis.

**1. Arithmetic operations:**

**Addition and subtraction of complex numbers are performed component-wise:**

**Addition:**

\((a+bi)+(c+di)=(a+c)+(b+d)i \)

**Subtraction:**

\((a+bi)–(c+di)=(a–c)+(b–d)i \)

**Multiplication and division of complex numbers involve more manipulation:**

**Multiplication:**

\( (a+bi) \cdot (c+di)=(ac–bd)+(ad+bc)i \)

**Division:**

\( \frac{a+bi}{c+di} = \frac{(a+bi) (c-di)}{c^2 + d^2} = \frac{(ac+bd)+(bc-ad)i}{c^2 + d^2} \)

**2. Complex conjugate:**

The complex conjugate of a complex number \( a + bi\) is defined as \( a – bi\). It is denoted as \( \overline{a+bi} \). The conjugate has the property that when multiplied with the original complex number, it yields a real number:

\( (a+bi)(a–bi)= a^2 + b^2 \).

**3. Modulus and argument:**

The modulus (or magnitude) of a complex number \(a+bi \) is the distance from the origin to the point in the complex plane, and is calculated as \( |a+bi| = \sqrt{a^2 + b^2} \). The argument (or angle) of a complex number is the angle formed between the positive real axis and the line connecting the origin to the point in the complex plane, typically measured in radians. It is denoted as \( \arg (a+bi)= \theta \), and can be computed using the arctangent function: \( \theta = \arctan (\frac{b}{a}) \) (keeping in mind the quadrant where the complex number lies).

**4. Polar and exponential forms:**

A complex number can also be represented in polar form, as \( a+bi = r( \cos \theta +i\ \sin \theta) \), where \( r=|a+bi| \) and \( \theta = \arg (a+bi) \). This can be further simplified using Euler's formula, \( e^{i \theta} = \cos \theta + i \sin \theta \), giving the exponential form: \( a+bi = re^{i \theta } \).

**5. Powers and roots of complex numbers:**

Using the exponential form of a complex number, it is easy to calculate its powers and roots.

Let \(z=re^{i \theta } \) be a complex number, then:

**Powers:**

\( z^n = (re^{i \theta } )^n = r^n e^{ in \theta } \), where \(n\) is a positive integer.

**Roots:**

The \(n\)-th root of \(z\) is given by

\(w_k = r^{ \frac{1}{n} }\ e^{ \frac{i( \theta + 2 k \pi ) }{n} } \), for \( k = 0,1,2,…,n-1 \).

This formula yields \(n\) distinct roots.

**6. De Moivre's theorem:**

De Moivre's theorem is a powerful result that relates the powers of complex numbers to trigonometric functions. For any complex number in polar form, \( z = r( \cos \theta +i \sin \theta ) \), and a positive integer \(n\), De Moivre's theorem states:

\( ( \cos \theta + i \sin \theta )^n = \cos n \theta + i \sin n \theta \)

**7. Complex functions:**

Complex functions are functions that take complex numbers as inputs and produce complex numbers as outputs. They can be written as

\( f(z)=f(x+yi)=u(x,y)+iv(x,y) \), where \( u(x,y) \) and \( v(x,y) \) are real-valued functions of two real variables, representing the real and imaginary parts of the function, respectively.

**8. Complex analysis:**

Complex analysis is a branch of mathematics that studies complex functions and their properties, such as differentiability, analyticity, and integration. Some important results in complex analysis are:

**Cauchy-Riemann equations:** A necessary (but not sufficient) condition for a complex function to be differentiable at a point is that its real and imaginary parts satisfy the Cauchy-Riemann equations:

\( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \).

**Analytic functions:** Functions that are differentiable in a neighborhood of a point are called analytic functions. Analytic functions have many desirable properties, such as infinite differentiability and the ability to be represented as a convergent power series.

**Cauchy's integral theorem and Cauchy's integral formula:** These theorems concern the integration of complex functions around closed contours and provide powerful tools to evaluate complex integrals and derive important results in complex analysis.