# Gibbs Phenomenon and Fourier Series expansion

4. In this problem, you will devise a computer experiment to investigate Gibb's phenomenon, which is the presence of spurious oscillations in the graph of a truncated Fourier series near the places where the full Fourier series is discontinous.

Choose any function you like that demonstrates Gibb's phenomenon. Your goal is to answer these two questions:

(a) You should find that the amount of overshoot only depends on the height of the discontinuity of your function. Expressed as a ratio to the height of the discontinuity, what is the approximate amount of overshoot/undershoot?

(b) What happens to the amount of overshoot/undershoot as you increadse the number of terms in your truncated Fourier series?

*Please see attachment for complete directions

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#### Solution Preview

Hi there!

Here is my solution.

I used Maple instead of Mathematica, so the code is in an additional Word file.

Make sure to check out:

http://www.sosmath.com/fourier/fourier3/gibbs.html

The Gibbs' phenomenon states that as the overshoot in the truncated Fourier series representation of a function in the discontinuity point converges to 0.09 of "bump" of the function at the discontinuity point.

For example look at the function:

The function looks like:

This function is even; therefore we have to worry only about the coefficients of ...

#### Solution Summary

The Gibbs phenomenon occurs at points where the derivative of the function is discontinuous.

This assignment demonstrates the Gibbs phenomenon with graphs and numerical analysis.