# Rotating Fourier Series

**Chapter 1. Introduction**

**Complex numbers** are more difficult than “**normal**“, i.e. **real** ones. Addition, subtraction, multiplication and division is more complicated. You can agree here. Then **Complex Functions** should be even worse! Well, how to imagine **exp(z), sin(z), tg(z) **or** log(z)**?

Fortunately, we only need to know about the **exponential** function **exp(z)**. In addition, it is limited to the domain **z** of the imaginary axis **z=jωt** and not to the entire complex plane **z=x+jωt**. So we are interested in the “**simplified**” function **exp(jωt)**, which has quite a nice interpretation. It is a **vector** of length **1** rotating with angular velocity** ω**. And its **projection** on the real axis of the complex plane **Re z** is just a real function **cos(ωt)**!

The** Fourier Series** and the **Fourier** and **Laplace** **Transform** decompose the **real** function **f(t)** into **sine** and **cosine** components. Approaching this problem as a real function is hardly intuitive. It is completely different with rotating **exp(jωt)** vectors.