Wolfram Demonstrations Project

This Demonstration shows Fourier series approximations to functions representing the

roots of unity, for

. The "number of vertices" is equal to

, with the roots of unity shown in blue. Each (gray) vector, representing the progressing Fourier series, is displayed along with the (orange) circle within which it rotates. The approximation is the point at the end of the final vector, indicated with a red dot. The "number of terms" equals the number of (gray) vectors/(orange) circles. You can select the angle that the vector from the origin makes with the ray

(

) with the "angle" control. The checkbox lets you plot the partial Fourier sum (green), as the angle is swept from

Contributed by: Walden Freedman
After work by: Grant Sanderson (3Blue1Brown)

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Let

be the

-periodic function defined by:

for

. It can be shown that the Fourier series for

is given by

where

and

is the ceiling function. This Demonstration shows the partial sums with between one and 50 terms.

With a smaller number of vertices and terms, you can see how the circular motions combine. Try a slowed-down animation with a smaller number of vertices, say 2, 3 or 4, and 5, 6 or 7 terms. With more terms, the approximation improves in accuracy.

This Demonstration is inspired by the videos on Fourier Series by Grant Sanderson of 3Blue1Brown. There are other related animations online, such as: https://isaacvr.github.io/coding/fourier_transform.

Snapshot 1: with

, we see that for large values of

, the first term with

dominates and the green curve approximates the unit circle

Snapshot 2: With

and an odd number of terms (7 in this case), we get an approximation between two "official" partial sums. The parametrization is not strictly real valued as it would be with an even number of terms; the number of vertices of the star-like figures one sees at

is equal to the odd number of terms.