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Last updated: Sept 4, 2004.

Practice Problems on Fourier Series

It may be useful for your work to recall the following integrals : u cos u du = cos u + u sin u+C;

u sin u du = sin u u cos u+C;

cos mx cos nx dx =

0, when m =n,, when m= n.

sin mx sin nx dx =

0, when m =n,, when m= n.

cos mx sin nx dx = 0 for all mand n.

Problem 1. Find the period of the given periodic function:

(a) cos 2x (b) sin 3x (c) sinx

3

(d) cot 3x (e) 3 sin 5x (f) 3 sin x+ cos 2x(g) 5 cos 3x+ 2cos 2x (h) 5 cos

x

3+ 2 cos2x (i) |cosx|

(j) cos2x (k) cos3x (l) cos4x

Problem 2. For a given 2-periodic function,(i) sketch several periods of its graph;(ii) find its Fourier series.

(a) f(x) = 3, < x .

(b) f(x) =

2, < x

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(d) f(x) = cos x, < x .

(e) f(x) =

2+x, x

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Problem 6. For the functions in Problem 5,

(i) give the odd extension to the full period L < x < L;(ii) for the functions in (a) and (b), find the Fourier sineseries.

Problem 7. Consider the 2 periodic function given on interval (, ) bythe formulas:

f(x) =

1, < x 2 ,1, 2 < x

2 ,

1, 2 < x .

The Fourier Series of this function is f(x) =

k=04(1)k+1

(2k+1) cos((2k+ 1) x).

(a) Determine the antiderivative F(x) for f(x) with F(0) = 0. Sketchthe graphs off(x) and F(x).

(b) Use the integration theorem to find the Fourier series for F(x).(c) Use the integration theorem again to find the Fourier series for the

second antiderivative off(x).(d) What is the condition(s) when the integration theorem is applicable?

Problem 8. Consider the 2 periodic function given on interval (, ) bythe formula f(t) =et +et. The Fourier series off(t) is

f(

t) =

e e

1 +

k=1

2(1)k

1 +k2 cos(kt

).

(a) Use the differentiation theorem to find the Fourier series for f(t) =

et et.(b) Sketch the graphs off(t) and f(t).(c) Can one use the differentiation theorem to find the Fourier series of

f(t)? Why?

Answers

Problem 1.(a) . (b) 23 . (c) 6. (d)

3 . (e) 25 . (f ) 2. (g) 2. (h) 6. (i) . (j) .

(k) 2. (l) .

Problem 2.(a) a0= 6, an= bn = 0 for n 1.

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