Les guides pour les nuls

Index

• Complex Numbers
  • Euler’s Equation
  • Useful Identities
• Transforms
  • Laplace Transform
  • Fourier Transform
  • Z-transform

Complex Numbers

We learn in high school complex numbers are a simple mathematical definition where $j = \sqrt{-1}$. This seemingly innocuous number has a hidden rotational property to it.

\[j = j, \quad j^2 = -1, \quad j^3 = -j, \quad j^4 = 1, \quad j^5 = j\]

Realize that when plotted on a complex plane, the act of multiplying a complex number by $j$ becomes a $90 \degree$ CC rotation. This is an insanely useful property that is leveraged heavily in electrical and controls engineering.

showing how j can be seen as a rotation operator in complex plane

Euler’s Equation

Largely considered the most beautiful equation in math, the Euler Equation is as follows

\[\]

Useful Identities

Transforms

Laplace Transform

The essence of a Laplace transform lies in how it captures both natural exponentials and frequencies of a system. The Laplace transform is the cornerstone of dynamical systems and control theory where transients are involved. It converts a function in the time domain to the complex domain and vice versa through its inverse.

This is the Laplace Transform $\mathcal{L} | f(t) \rightarrow F(s)$.

\[\mathcal{L}([f(t)]) = \int_0^\infty f(t) e^{-st} dt = F(s)\] \[\mathcal{L}^{-1}([F(s)]) = \frac{1}{2\pi}\int_{\sigma -j\omega}^{\sigma + j\omega} F(s) e^{st} ds\]

Fourier Transform

Z-transform