Rotating Fourier Series
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)!