A real-parameter (r(theta) = sum(r_k cos(k theta))) Fourier series can only draw a "wiggly circle" figure with one point on each radial ray from the origin.
A compex parameter (z(theta) = sum(e^(z_ theta))) can draw more squiggly figures (epicycles) -- the pen can backtrack as the drawing arm rotates, as each parameter can move a point somewhere on a small circle around the point computed from the previous parameter (and recursively).
A real-parameter (r(theta) = sum(r_k cos(k theta))) Fourier series can only draw a "wiggly circle" figure with one point on each radial ray from the origin.
A compex parameter (z(theta) = sum(e^(z_ theta))) can draw more squiggly figures (epicycles) -- the pen can backtrack as the drawing arm rotates, as each parameter can move a point somewhere on a small circle around the point computed from the previous parameter (and recursively).
Obligatory 3B1B https://m.youtube.com/watch?v=r6sGWTCMz2k
Since a complex parameter is 2 real parameters, we should compare the best 4-cosine curve to the best 2-complex-exponential curve.