Complex Numbers

Basics

A complex number has two components: The real component and the imaginary components, typically denoted by \(\displaystyle i\) or \(\displaystyle j\).
Here \(\displaystyle i\) represents the imaginary number \(\displaystyle i^2 = -1\)

There are a few common operations on complex numbers:
Consider a complex number \(\displaystyle c = a + bi\),

The conjugate of is \(\displaystyle c^* = a - bi\), also sometimes denoted as \(\displaystyle \bar{c}\).
The squared norm is \(\displaystyle |c|^2 = c * c^* = a^2 + b^2\). Or \(\displaystyle |c| = \sqrt{a^2 + c^2}\).
The angle is \(\displaystyle \angle c = \arctan(b, a)\).
The exponential representation is \(\displaystyle c = |c| e^{i \theta} = |c| (\cos(\theta) + i \sin(\theta))\) where \(\displaystyle \theta = \angle c\).
- In this representation, \(\displaystyle \theta\) is known as the argument and \(\displaystyle |c|\) is the modulus or absolute value.
- This analagous to polar coordinates representation for 2d vectors (angle and magnitude).
- Note that the word amplitude is overloaded and can refer to either the complex number, the absolute value, or the angle depending on context.
Addition and multiplication are as usual, but with \(\displaystyle i^2 = -1\).

Euler's formula

Euler's formula states: \[ e^{ix} = \cos(x) + i \sin(x) \]

Properties

The conjugate is \(\displaystyle e^{-ix}\) since cosine is symmetric and sine is odd (i.e. \(\displaystyle sin(-x) = -sin(x)\))

Euler's identity

Euler's identity states: \[ e^{i \pi} + 1 = 0 \]

Resources

https://web.stanford.edu/~boyd/ee102/complex-primer.pdf Brief review of complex numbers