FOURIER SERIES

1. Theory
2. Exercises
3. Answers
4. Integrals

Table of contents

5. Useful trig results
6. Alternative notation
7. Tips on using solutions
Full worked solutions

1. Theory

? A graph of periodic function f (x) that has period L exhibits the same pattern every L units along the x-axis, so that f (x + L) = f (x) for every value of x. If we know what the function looks like over one complete period, we can thus sketch a graph of the function over a wider interval of x (that may contain many periods)

f(x)




x




PERIOD = L


? This property of repetition defines a fundamental spatial fre- quency k = 2? that can be used to give a first approximation to the periodic pattern f (x):

f (x) c c1 sin(kx + ?1) = a1 cos(kx) + b1 sin(kx),

where symbols with subscript 1 are constants that determine the am- plitude and phase of this first approximation

? A much better approximation of the periodic pattern f (x) can be built up by adding an appropriate combination of harmonics to this fundamental (sine-wave) pattern. For example, adding

c2 sin(2kx + ?2) = a2 cos(2kx) + b2 sin(2kx) (the 2nd harmonic)
c3 sin(3kx + ?3) = a3 cos(3kx) + b3 sin(3kx) (the 3rd harmonic)

Here, symbols with subscripts are constants that determine the am- plitude and phase of each harmonic contribution


One can even approximate a square-wave pattern with a suitable sum that involves a fundamental sine-wave plus a combination of harmon- ics of this fundamental frequency. This sum is called a Fourier series


Fundamental
Fundamental + 2 harmonics





x





Fundamental + 5 harmonics
Fundamental + 20 harmonics


PERIOD = L

? In this Tutorial, we consider working out Fourier series for func- tions f (x) with period L = 2?. Their fundamental frequency is then L = 1, and their Fourier series representations involve terms like

a1 cos x , b1 sin x a2 cos 2x , b2 sin 2x a3 cos 3x , b3 sin 3x

We also include a constant term a0/2 in the Fourier series. This allows us to represent functions that are, for example, entirely above the x?axis. With a sufficient number of harmonics included, our ap- proximate series can exactly represent a given function f (x)


f (x) = a0/2 + a1 cos x + a2 cos 2x + a3 cos 3x + ...
+ b1 sin x + b2 sin 2x + b3 sin 3x + ...


A more compact way of writing the Fourier series of a function f (x), with period 2?, uses the variable subscript n = 1, 2, 3, . . .
?

f (x) = a0 + . [a
2 n
n=1

cos nx + bn

sin nx]

? We need to work out the Fourier coefficients (a0, an and bn) for given functions f (x). This process is broken down into three steps



STEP ONE


a0 =

1 ¸
f (x) dx ?
2?
1 ¸

STEP TWO an =


f (x) cos nx dx ?
2?


STEP THREE bn =

1 ¸
f (x) sin nx dx ?

2?
where integrations are over a single interval in x of L = 2?


? Finally, specifying a particular value of x = x1 in a Fourier series, gives a series of constants that should equal f (x1). However, if f (x) is discontinuous at this value of x, then the series converges to a value that is half-way between the two possible function values


"Vertical jump"/discontinuity in the function represented



f(x)





x



Fourier series converges to half-way point

2. Exercises
Click on Exercise links for full worked solutions (7 exercises in total).
Exercise 1.
Let f (x) be a function of period 2? such that
. 1, ?? < x < 0

f (x) =

0, 0 < x < ? .


a) Sketch a graph of f (x) in the interval ?2? < x < 2?

b) Show that the Fourier series for f (x) in the interval ?? < x < ? is
1 2 . 1 1 .

2 ? ?

sin x + 3 sin 3x + 5 sin 5x + ...


c) By giving an appropriate value to x, show that

?
4 = 1 ?

1 1
3 + 5 ?

1
7 + . . .


Let f (x) be a function of period 2? such that
. 0, ?? < x < 0

f (x) =

x, 0 < x < ? .


a) Sketch a graph of f (x) in the interval ?3? < x < 3?

b) Show that the Fourier series for f (x) in the interval ?? < x < ? is
? 2 . 1 1 .

4 ? ?

cos x + 32 cos 3x + 52 cos 5x + ...

. 1 1 .
+ sin x ? 2 sin 2x + 3 sin 3x ? ...
c) By giving appropriate values to x, show that

2



f (x) =

. x, 0 < x < ? ?, ? < x < 2? .


a) Sketch a graph of f (x) in the interval ?2? < x < 2?

b) Show that the Fourier series for f (x) in the interval 0 < x < 2? is
3? 2 . 1 1 .

4 ? ?

cos x + 32 cos 3x + 52 cos 5x + . . .

. 1 1 .
? sin x + 2 sin 2x + 3 sin 3x + . . .
c) By giving appropriate values to x, show that

2


f (x) = x
2


over the interval 0 < x < 2?.


a) Sketch a graph of f (x) in the interval 0 < x < 4?

b) Show that the Fourier series for f (x) in the interval 0 < x < 2? is
? . 1 1 .
2 ? sin x + 2 sin 2x + 3 sin 3x + . . .
c) By giving an appropriate value to x, show that

?
4 = 1 ?

1 1
3 + 5 ?

1 1
7 + 9 ? . . .



f (x) =

. ? ? x, 0 < x < ?
0, ? < x < 2?


a) Sketch a graph of f (x) in the interval ?2? < x < 2?

b) Show that the Fourier series for f (x) in the interval 0 < x < 2? is

? 2 . 1
+ cos x +

1
cos 3x +

.
cos 5x + . . .

4 ? 32 52
1 1 1
+ sin x + 2 sin 2x + 3 sin 3x + 4 sin 4x + . . .
c) By giving an appropriate value to x, show that

? = 1 + 1
8 32

1
+ 52 + . . .


f (x) = x in the range ? ? < x < ?.


a) Sketch a graph of f (x) in the interval ?3? < x < 3?

b) Show that the Fourier series for f (x) in the interval ?? < x < ? is
. 1 1 .
2 sin x ? 2 sin 2x + 3 sin 3x ? . . .
c) By giving an appropriate value to x, show that


? = 1
4


1 1
3 + 5 ?


1
7 + . . .


Let f (x) be a function of period 2? such that

f (x) = x2 over the interval ? ? < x < ?.


a) Sketch a graph of f (x) in the interval ?3? < x < 3?

b) Show that the Fourier series for f (x) in the interval ?? < x < ? is
?2 . 1 1 .

3 ? 4

cos x ? 22 cos 2x + 32 cos 3x ? . . .

c) By giving an appropriate value to x, show that


? = 1 + 1
6 22


1 1
+ +
32 42


+ . . .

3. Answers

The sketches asked for in part (a) of each exercise are given within the full worked solutions – click on the Exercise links to see these solutions

The answers below are suggested values of x to get the series of constants quoted in part (c) of each exercise
1. x = ? ,
2. (i) x = ? , (ii) x = 0,
3. (i) x = ? , (ii) x = 0,
4. x = ? ,
5. x = 0,
6. x = ? ,
7. x = ?.

4. Integrals
Formula for integration by parts: ¸ b u dv dx = [uv]b ? ¸ b du v dx

a dx

a a dx




[g (x)]

g (x)












sinh x
cosh x











sin a

ln . a .







2 2
2 a 2
a2

? 2 2
a a2

5. Useful trig results
When calculating the Fourier coefficients an and bn , for which n = 1, 2, 3, . . . , the following trig. results are useful. Each of these results, which are also true for n = 0, ?1, ?2, ?3, . . . , can be deduced from the graph of sin x or that of cos x




? sin n? = 0





????????????


?


?? ?


sin(x)




x
? ?? ??


??





? cos n? = (?1)n





????????????


?


?? ?

cos(x)




x
? ?? ??


??



sin(x)
?

cos(x)
?




????????????


?? ?

x
? ?? ??


????????????


?? ?

x
? ?? ??


?? ??


?
? sin n ? = ?


0 , n even
1 , n = 1, 5, 9, ...

?
? cos n ? = ?


0 , n odd
1 , n = 0, 4, 8, ...

2 ? ?1 , n = 3, 7, 11, ...

2 ? ?1 , n = 2, 6, 10, ...



Areas cancel when when integrating over whole periods


???sin(x)
+ + + x

? ¸ sin nx dx = 0
2?
? ¸ cos nx dx = 0
2?

??? ???

?? ?

??

? ?? ??


? For a waveform f (x) with period L = 2?

f (x) = a0 + . [a
2 n
n=1


cos nkx + bn


sin nkx]

The corresponding Fourier coefficients are



STEP ONE


a0 =

2 ¸
f (x) dx L
L
2 ¸

STEP TWO an =


f (x) cos nkx dx L
L


STEP THREE bn =

2 ¸
f (x) sin nkx dx L

L
and integrations are over a single interval in x of L

k = 2? ?

n?x k

2L = L and nkx = L
?

f (x) = a0 + . .a
2 n
n=1

cos n?x + b
L n

sin n?x .
L

The corresponding Fourier coefficients are



STEP ONE


a0 =

1 ¸
f (x) dx L
2L
1 ¸





n?x

STEP TWO an =


f (x) cos dx L L
2L


STEP THREE bn =

1 ¸
f (x) sin
L
2L

n?x
dx L

and integrations are over a single interval in x of 2L

? For a waveform f (t) with period T = 2?

f (t) = a0 + . [a
2 n
n=1


cos n?t + bn


sin n?t]

The corresponding Fourier coefficients are



STEP ONE


a0 =

2 ¸
f (t) dt T
T
2 ¸

STEP TWO an =


f (t) cos n?t dt T
T


STEP THREE bn =

2 ¸
f (t) sin n?t dt T

T
and integrations are over a single interval in t of T