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Fourier Series of Polygons

A. Robert
Journal / Anthology

American Mathematical Monthly
Year: 1994
Volume: 101
Issue: 5
Page range: 420-428

[A] complex Fourier series represents a 2pi-periodic map [which] can be viewed as a closed parametrized curve t -> f(t) \in C in the complex plane. In general, this curve wll have multiple points (namely, f is not one-to-one). It is our purpose to illustrate this point of view. In particular, we intend to show that, for n >= 2, f_n(t) = Sum[(e^ikt)/k^2, {k \[Congruent] 1(n)] is the Fourier series of a regular n-gon in the complex plane.

*Mathematics > Calculus and Analysis > Complex Analysis