MATH 300 -- Directed Readings: RAAA II
February 1 -- A Fourier Series Example

Example 1)

For the function on the interval
we saw that the Fourier coefficients were
, and

if
using and for integral m!

Because Maple will not assume that is an integer unless we

tell it to, let's start by doing that:

> assume(m,integer);

Then we compute the value of the Fourier coefficient using the

int (integrate) command:

> a[m] := (1/Pi)*int(x^2*cos(m*x),x=-Pi..Pi);

Similarly,

> b[m] := (1/Pi)*int(x^2*sin(m*x),x=-Pi..Pi);

Here is the partial sum of the series for m = 0..k :

> FS := k -> Pi^2/3 + sum(a[m]*cos(m*x),m=1..k);

(This is set up in Maple as function that takes the integer k to the k th partial

sum.)

Plots of the partial sums of the Fourier series, together with

> plot({x^2,FS(2)},x=-Pi..Pi);

> plot({x^2,FS(5)},x=-Pi..Pi);

> plot({x^2,FS(10)},x=-Pi..Pi);

The absolute error if we approximate by the partial sum of the Fourier series:

> plot(abs(x^2 - FS(10)),x=-Pi..Pi);

This shows good agreement! (The absolute error is always less than .4).