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.