**Rotating Fourier Series **

1. Introduction

1. Introduction

**Complex numbers** are more difficult than”normal”, read** real** numbers. Adding, subtracting, multiplying and dividing is more complicated. You can agree here, Then **Complex Functions** should be even worse! Because how to imagine **exp(z)**,** sin(z)**, **tg(z) **or** log(z)**?

Fortunately, we only need to know about the exponential complex function **exp(z). **In addition, limited to the domain 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 lenght **1**, rotating at angular velocity **ω**.And its projection on the** real axis** of the **complex plane** is the real function **cos**(**ωt)**!

The **Fourier Series** as well as the** Fourier** and **Laplace Transform** decompose the real function **f(t)** into **sinusoidal** and **cosine** components. The approach to this problem as real function is not very intuitive. It is completely different with the rotating vectors **exp(jωt)**!