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n=0:p;
x=(2*pi/p)*n;
a=cos((2*pi/p)*n'*k);
c=((-1).^k)./(k.^2+1);
y=a*c';
plot(x,y)
axis([0 2*pi -1 1.2])

Draw in your notebook the approximate shape of the resulting curve for dif-
ferent values of M.

In-Class Exercises
Pb. 8.8 For different values of the cutoff, plot the resulting curves for the
functions given by the following Fourier series:

∞
yx() = 8 2 ∑   1   cos(( k 2 − 1) x)
1 π k 2 (  − 1) 
2
=
k 1
∞  (− k 1− 
yx() = 4  ∑ 1)  cos(( k 2 − 1) x)
2 π k 2 (  − 1)
k 1=
2 ∞ 1
yx() = ∑ sin(( k 2 − 1) x)
3 π k 2 ( − 1)
k 1=
Pb. 8.9 The purpose of this problem is to explore the Gibbs phenomenon.
This phenomenon occurs as a result of truncating the Fourier series of a dis-
continuous function. Examine, for example, this phenomenon in detail for
the function y (x) given in Pb. 8.8.
3
The function under consideration is given analytically by:

 05. for 0 < x < π
yx() = 
3 − for π < x < 2π
 05.

a. Find the value where the truncated Fourier series overshoots the
value of 0.5. (Answer: The limiting value of this first maximum is
0.58949).
b. Find the limiting value of the first local minimum. (Answer: The
limiting value of this first minimum is 0.45142).

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