0. Complex Numbers $\mathbb{C}$
We admit the set of complex numbers, denoted by $\mathbb{C}$, which consists of all numbers of the form $a + bi$, where $a$ and $b$ are real numbers and $i = \sqrt{-1}$ is the imaginary unit satisfying $i^2 = -1$. Complex numbers can be represented in the complex plane, where the real part $a$ is plotted on the horizontal axis and the imaginary part $b$ is plotted on the vertical axis.
We need to see the concept of number here, in the complex plane is a
two-dimensional space, and it is a field, i.e. given the set of real
numbers $\mathbb{R}$ is a field. The set of complex numbers should be
one likewise.
We will refer to the complex number as a ordered pair
in the Argand-Gauss plane, where the ordinate is imaginary and the coordinate
is real, therefore, forming an ordered pair $(a, b)$.
First, we need to verify that $\mathbb{C}$ is a field, but
need a whole of properties of complex numbers.
We let z = (a,b) and w = (c,d) be two complex numbers.
So it is easy to verify that $\mathbb{C}$ is a field.
a. \[(a, b) + (c, d) = (a + c, b + d) \in \mathbb{C} \]
b. \[ (a, b) (c, d) = (ac - bd, ad + bc) \in \mathbb{C} \]
c. \[ (a, b)^{-1} = \frac{a}{a^2 + b^2}, \frac{-b}{a^2 + b^2} \in \mathbb{C} \]
d. \[\bar{z} = (a, -b)\]
e.
Mappings in $\mathbb{C}$
A mapping is $f$ an injective function, meaning that each element in the domain is mapped to a unique element in the codomain. In other words, no two elements in the domain $D_f$ are mapped to the same element in the codomain $C_f$. An important mapping is a bijective mapping $g$, such that $g^{-1}$ is admitted as the inverse mapping of $g$. Meaning that we can reverse the effects that the function $g$ has applied to a set of elements.
A mapping in $\mathbb{C}$, occurs from one plane to another plane, or domain (a subset of $\mathbb{C}$). The mapping $f(z) = w^2$ has $f(z)$ be the point f(z) be expanded by $f'(z) = 2w around the point $f(z_0) = 0$. It is how it leaves or arrives at the point $f(z_0) = 0$, and in what direction, and intensity.
Are two mappings of the same family of solutions. But they differ in the
number of poles.
Conformal Maps
Conformal maps are transformations in which the angles between the derivative and the curve is preserved. It guarantees that the the sum of two points are also a point on the curve.