Complex Numbers

Complex numbers are defined in the form

$$\mathbb{C}=\{a+bi|a,b\in \mathbb{R}\}$$

Where $i^2=-1$. This form is known as the standard form.

We define $\mathbb{C}$ to denote all complex numbers.

We say $a$ and $b$ are the real and imaginary parts of the complex number, respectively.

$$Re(z)=aIm(z)=b$$

Properties of Complex Numbers

Sums and Products

Let $z=a+bi$, $w=c+di$, $z,w\in \mathbb{C}$. Then,

• The sum of $z$ and $w$ is

  $$z+w=(a+c)+(b+d)i$$

• The product of $z$ and $w$ is

  $$z\cdot w=(ac-bd)+i(ad+bc)$$

Complex Conjugates and Norm

Let $z=a+bi$ be a complex number.

We define it’s complex conjugate, $\bar{z}=z-bi$, which is the complex number whose product with $z$ yields a non-negative real number.

$$z\cdot \bar{z}=a^2+b^2$$

Complex conjugates have the following two properties:

• Sum of Conjugates: $\overline{z+w}=\bar{z}+\bar{w}$
• Product of Conjugates: $\overline{z\cdot w}=\bar{z}\cdot \bar{w}$

The norm of complex number $z$ is defined as

$$|z|=\sqrt{a^2+b^2}=\sqrt{z\cdot \bar{z}}$$

Geometry of Complex Numbers

Complex numbers can be plotted on a plane known as the complex plane. This is a plane where the horizontal axis consists of all real numbers ($1,2,\dots$) and the vertical axis consists of all complex numbers ($i,2i,\dots$).

Geometric Representation

Let $z=a+bi$ be a complex number on the plane. Then, if $\theta$ is the angle between $z$ and the horizontal axis,

$$z=a+bi=|z|\cos (\theta )+|z|i\sin (\theta )=|z|(\cos (\theta )+i\sin (\theta ))$$

Theorem: Complex Exponential

For every $\theta \in \mathbb{R}$, we have

$$e^{i\theta }=\cos (\theta )+i\sin (\theta )$$

We can use this to find $\sin (\theta )$ and $\cos (\theta )$ as

$$\sin (\theta )=\frac{e^{i\theta }-e^{-i\theta }}{2i}\cos (\theta )=\frac{e^{i\theta }+e^{-i\theta }}{2}$$

Using this theorem, we can represent all complex numbers in the polar form

$$z=r\cdot e^{i\theta }0\le \theta \le 2\pi r\in \mathbb{R}$$

Theorem: Properties of Complex Geometries

Let $x$ and $y$ be two real numbers and $n$ be an integer. Then,

1. $e^{ix}{e}^{xy}=e^{i(x+y)}$
2. De Moivre’s Formula: $(e^{ix}{)}^n=e^{inx}$

Example: Applying Complex Geometric Properties

Evaluate

$$\int e^x\cos (x)dx$$

We observe that $\cos (x)$ is equivalent to the real part of the equation $e^{ix}$, so we can rewrite the answer as the real part of the equation

$$\int e^x\cdot e^{ix}dx=\int e^{(1+i)x}dx=\frac{1}{1+i}{e}^{(1+i)x}$$

We can then convert and solve for the real part of the equation.