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Assigned: March 1, 2011Due: March 9, 2011

MATH 480: Homework 5

SPRING 2011

Fourier Series:

1. (a) Find the Fourier sine series for f(x) = 1 x defined on the interval 0 x 1.

(b) In MATLAB, plot the first 20 terms and the first 200 terms of the sine seriesin the interval 3 x 3.

(c) To what value does the series converge at x = 0?

2. (a) Find the Fourier cosine series for f(x) = 1x defined on the interval 0 x 1.

(b) In MATLAB, plot the first 20 terms and the first 200 terms of the cosine seriesin the interval 3 x 3.

(c) To what value does the series converge at x = 0?3. (a) Find the Fourier series for

f(x) =

0 if 1 x < 0

1 x2 if 0 < x 1

defined on the interval 1 x 1.

(b) In MATLAB, plot the first 20 terms and the first 200 terms of the Fourier seriesin the interval 3 x 3.

(c) To what value does the series converge at x = 0?

4. The Fourier series of the function f(x) = cos(ax) on the interval [, ], when a isnot an integer, is given by

cos(ax) =2a sin(a)

1

2a2+n=1

(1)n+1

n2 a2cos(nx)

for x .

(a) Differentiate both sides of this equation with respect to x, differentiating theseries term by term, to find the Fourier series for sin(ax):

sin(ax) = 2sin(a) n=1

(

1)

n

nn2 a2 sin(nx)

for < x < .

(b) Explain why this method for computing the Fourier series is valid.

(c) If you know the Fourier series for sin(ax) given in (a), why can you not dif-ferentiate it term by term with respect to x to derive the Fourier series forcos(ax).

1

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Assigned: March 1, 2011Due: March 9, 2011

Problem # 1 (b)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% HW # 5: pr1b.m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

clear all; clf;

x=-3:1e-3:3;

NN=20;

FS=0;

for n=1:NN

Bn=2/(n*pi);

FS=FS+Bn*sin(n*pi*x);

end

figure(1); clf(1)

subplot(2,1,1), plot(x,FS);

xlabel(x)

ylabel([First ,num2str(NN), terms of Fourier sine series])

title([First ,num2str(NN), ...

terms of the Fourier sine series for the function f(x)=1-x defined on 0

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Assigned: March 1, 2011Due: March 9, 2011

3 2 1 0 1 2 31.5

1

0.5

0

0.5

1

1.5

x

First20termsofFouriersineseries

First 20 terms of the Fourier sine series for the function f(x)=1x defined on 0

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Assigned: March 1, 2011Due: March 9, 2011

terms of the Fourier cosine series for the function f(x)=1-x defined on 0

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Problem # 3 (b)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% HW # 5: pr3b.m

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

clear all; clf;

x=-3:1e-3:3;

NN=20;

FS=1/3;

for n=1:NN

An=2*(-1)^(n+1)/(n^2*pi^2);

Bn=(2*(-1)^(n+1)+n^2*pi^2+2)/(n^3*pi^3);

FS=FS+An*cos(n*pi*x)+Bn*sin(n*pi*x);end

figure(1);clf(1)

subplot(2,1,1), plot(x,FS);

xlabel(x)

ylabel([First ,num2str(NN), terms,of Fourier series])

title([First ,num2str(NN), ...

terms of the Fourier series for the function f(x)=1-x defined on 0

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Assigned: March 1, 2011Due: March 9, 2011

print -depsc2 pr3b_graph.eps

3 2 1 0 1 2 30.2

0

0.2

0.4

0.6

0.8

1

1.2

x

First20terms,o

fFourierseries

First 20 terms of the Fourier series for the function f(x)=1x defined on 0

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