APPLIED MATHEMATICS/MAT538

FOURIER SERIES

Fourier Series is an infinite series representation of periodic functions in terms of the

trigonometric sine and cosine functions. Fourier series have many applications in
mathematics, physics and engineering. For example, they are sometimes essential in solving
problems in heat conduction, wave propagation etc, that involve partial differential equations,
and in electrical circuits of engineering systems.

Periodic non-sinusoidal function

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Square wave function

Start with sin x

sin 3x 
Take
3

sin 3x 
Add to make sin x 
3

sin 5 x 
Now take
5

sin 3x  sin 5x 

Add to make sin x  
3 5

Using 20 sine wave gives

sin 3x  sin 5x  sin 39 x 
sin x    .. 
3 5 39

Using 100 sine wave gives

sin 3x  sin 5x  sin 199 x 
sin x    .. 
3 5 199

A square wave =
sin 3x  sin 5x 
sin x    ..  infinitely..
3 5

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To represent a periodic non-sinusoidal function f x of period 2L defined in the interval

 L, L
General form :

f x = a0  a1 cos x  a2 cos2 x   a3 cos3x   .....  b1 sin x  b2 sin 2 x   b3 sin 3x   ...


 nx    nx 
= a0   an cos    bn sin   where a0 , an , bn are Fourier coefficients
n 1  L  n1  L 

Note : Obtaining the Fourier coefficients for a given functions f x is our main task.

 f x  dx
1
General expressions : a0 
2L L

 nx 
L

 f x  cos
1
an   dx
L L
L 

 nx 
L

 f x  sin 
1
bn   dx
L L
L 

Note :

 nx   nx 
 If a function is even, f  x  f x , and sin   is odd, then f x  sin   is
 L   L 
odd. (the product of an even and odd function is an odd function). Therefore bn  0
for all n.
 nx   nx 
 If a function is odd, f  x   f x , and cos  is even, then f  x  cos  is
 L   L 
odd. (the product of an even and odd function is an odd function). Therefore an  0
for all n.

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Steps

 Check whether f x is even or odd

 If :
o f x is odd, all the Fourier coefficients a n for n  1, 2, 3, 4... are 0 (zero)
o f x is even, all the Fourier coefficients bn for n  1, 2, 3, 4... are 0 (zero)
 Compute the remaining Fourier coefficients. It is a good strategy to use Integration by
Parts, successively integrating sin nx and cosnx and differentiating f x
 Replace the expressions for the Fourier coefficients a n and bn in

 nx    nx 
f x   a0   an cos    bn sin  
n 1  L  n 1  L 

Examples

Let f x be a periodic function of period 2 such that f x  

x
E1 over the
2

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

  
 sin x  sin 2 x   sin 3x   ...
1 1
2  2 3 

0    x  0
E2 Let f x be a periodic function of period 2 such that 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   
cos x  2 cos3x   2 cos5 x   ...  sin x  sin 2 x   sin 3x   ...
1 1 1 1
 
4  3 5   2 3 

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E3 Let f x be a periodic function of period 2 such that f x    2  x 2

for x    ,   .

Supposing that f x has a convergent trigonometric Fourier series,

2 2    4 
show that  2  x 2     2  1 cosnx
n

3 n 1  n 

E4 Show that the trigonometric Fourier series of that f x  3x


6
for x    ,   is given by    1 sin nx
n

n 1 n 

x 0  x  
E5 Let f x be a periodic function of period 2 such that f  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    
 cos x  2 cos3x   2 cos5 x   ...  sin x  sin 2 x   sin 3x   ...
1 1 1 1
4  3 5   2 3 

E6 Let f x be a periodic function of period 2 such that f x   x 2 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  
 4cos x  2 cos2 x   2 cos3x   ...
1 1
3  2 3 

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