This document provides information on Fourier series and Fourier coefficients over various intervals. It defines the Fourier series representation of a periodic function f(x) over a general interval (–l, l) and the interval (–π, π). It also gives the formulas for the Fourier coefficients in the cases when f(x) is even or odd. The document concludes with a table of common Fourier series representations and a list of frequently used formulas for computing Fourier coefficients.

1. Elementary Concepts of Fourier Series and
Table of Fourier Coefficients

2. MAT2002 - ADDE
Module 1
Page 1
Fourier series over a general interval: Let f (x) be a periodic function, with period 2l
satisfying the following Dirichlet conditions on each interval (–l, l):
(a) )
(x
f has only a finitely many finite discontinuities;
(b) )
(x
f has only at most finite number of maxima or minima.
Then )
(x
f can be represented by the Fourier series
  0
1
cos sin ,
2
n n
n
a n x n x
f x a b
l l

  
 
   
  
    
 
   
 
where
 
0
1
,
l
l
a f x dx
l 
   
1
cos
n
l
l
n x
a f x dx
l l


 
  
 
 for 1, 2, 3,
n  
and
 
1
sin
l
n
l
n x
b f x dx
l l


 
  
 
 for 1, 2, 3,
n  .
are called the Fourier coefficients of f.
 At a point of finite discontinuity: If f is discontinuous at some interior point c of
(–l, l), then the Fourier series converges to the average
( 0) ( 0)
2
f c f c
  
of left
hand and the right hand limits of f at c
 At the end points: The Fourier series converges to the average
( 0) ( 0)
2
f l f l
   
of left hand limit of f at the right end point l and the right hand limit of f at the left end
point
The nth harmonic of f is given by
cos sin
n n n
n x n x
H a b
l l
 
   
 
   
   
for 1, 2, 3,
n  .
Case (a): f (x) is an even function: The Fourier coefficients of f are
 
0
0
2
,
l
a f x dx
l
   
0
2
cos
n
l n x
a f x dx
l l

 
  
 
 , 0
n
b  for 1, 2, 3,
n  
Case (b): f (x) is an odd function: The Fourier coefficients of f are
0 0,
a  0,
n
a   
0
2
sin
l
n
n x
b f x dx
l l

 
  
 
 for 1, 2, 3,
n  .

3. MAT2002 - ADDE
Module 1
Page 2
Fourier series over ( , )
  : Let f (x) be a periodic function, with period 2 on ( , )
  .
The Fourier series of f (x) is
   
0
1
cos sin
2
n n
n
a
f x a nx b nx


  

where
 
0
1
,
a f x dx



   
1
cos
n
a f x nx dx



  for 1, 2, 3,
n  
and
 
1
sin
n
b f x nx dx



  for 1, 2, 3,
n  
The nth harmonic of f is given by
cos sin
n n n
H a nx b nx
  for 1, 2, 3,
n  .
Case (a): f (x) is an even function: The Fourier coefficients of f are
 
0
0
2
,
a f x dx


   
0
2
cos
n
a f x nx dx


  , 0
n
b  for 1, 2, 3,
n  .
Case (b): f (x) is an odd function: The Fourier coefficients of f are
0 0,
a  0,
n
a   
0
2
sin
n
b f x nx dx


  for 1, 2, 3,
n  .

4. MAT2002 - ADDE
Module 1
Page 1
Table of Fourier Coefficients
Interval Fourier Series of ( )
f x Fourier coefficients
(0, 2l),
l > 0
0
th harmonic
1
cos sin
2
n x n x
l l
n
n n
n
a
a b
   
π π
   
   
=
∞
 
+ +
∑  



( )
0
2
0
1
,
l
a f x dx
l
= ∫
( )
2
0
1
cos ,
n x
n l
l
a f x dx
l
 
π
 
 
= ∫
( )
2
0
1
sin ,
n x
l
l
n
b f x dx
l
π
 
 
 
= ∫
for 1, 2, 3,
n = 
(0, 2π) ( )
0
th harmonic
1
cos sin
2
n
n n
n
a
a nx b nx
=
∞
+ +
∑

( )
0
2π
0
1
,
π
a f x dx
= ∫
( )
2π
0
1
cos ,
π
n
a f x nx dx
= ∫
( )
2π
0
1
sin ,
π
n
b f x nxdx
= ∫
for 1, 2, 3,
n = 
(–l, l),
l > 0
0
th harmonic
1
cos sin
2
n x n x
l l
n
n n
n
a
a b
   
π π
   
   
=
∞
 
+ +
∑  



( )
0
1
,
l
l
a f x dx
l −
= ∫
( )
1
cos ,
n x
n l
l
l
a f x dx
l
 
π
 
 
−
= ∫
( )
1
sin ,
n x
l
l
n
l
b f x dx
l
π
 
 
 
−
= ∫
for 1, 2, 3,
n = 
Even f
(–l, l),
l > 0
( )
0
th harmonic
1
cos /
2
n
n
n
a
a n x l
=
∞
+ π
∑




( )
0
0
2
,
l
a f x dx
l
= ∫
( ) ( )
0
2
cos / ,
n
l
a f x n x l dx
l
= π
∫
0,
n
b = for 1, 2, 3,
n = 
Odd f
(–l, l),
l > 0
( )
th harmonic
1
sin /
n
n
n
a n x l
=
∞
π
∑




0 0,
a = 0,
n
a =
( ) ( )
0
2
sin / ,
n
l
b f x n x l dx
l
= π
∫
for 1, 2, 3,
n = 
(–π, π) ( )
0
th harmonic
1
cos sin
2
n
n n
n
a
a nx b nx
=
∞
+ +
∑

( )
0
π
π
1
,
π
a f x dx
−
= ∫
( )
π
π
1
cos ,
π
n
a f x nx dx
−
= ∫
( )
π
π
1
sin ,
π
n
b f x nxdx
−
= ∫
for 1, 2, 3,
n = 
Even f
(–π, π)
0
th harmonic
1
cos
2
n
n
n
a
a nx
=
∞
+ ∑ 



( )
0
π
0
2
,
π
a f x dx
= ∫
( ) ( )
π
0
2
cos ,
π
n
a f x nx dx
= ∫
0,
n
b = for 1, 2, 3,
n = 
Odd f
(–π, π) th harmonic
1
sin
n
n
n
a nx
=
∞
∑ 



0 0,
a = 0,
n
a =
( )
0
2
sin ,
n
b f x nx dx
π
=
π ∫ 1, 2, 3,
n = 

5. MAT2002 - ADDE
Module 1
Page 2
Formulas frequently used in computing the Fourier coefficients:
1. Leibnitz rule of integration:
a. Version 1 d d
U V UV V U
= −
∫ ∫
:
b. Version 2 1 2 3 4
d ' '' ''' ,
UV x UV U V U V U V
= − + − + ⋅⋅⋅
∫
: where ', '', ''',
U U U ⋅⋅⋅
are the successive derivatives of U, and 1 2 3 4
, , , ,
V V V V ⋅⋅⋅ are the successive
integrals of V
2.
( )
0
2 , if is even
( )
0, if is even
a
f x dx f
f x
f


= 


∫
3. cos
sin px
p
px dx = −
∫ , sin
cos px
p
px dx =
∫
4. [ ]
2 2
sin sin cos
Ax
Ax e
A B
e Bx dx A Bx B Bx
+
= −
∫ ,
5. [ ]
2 2
cos cos sin
Ax
Ax e
A B
e Bx dx A Bx B Bx
+
= +
∫
6. Property of Absolute value function:
x x
= − if 0
x < , x x
= if 0
x >
7. sin0 0,cos0 1 cos2
= = = π
8. sin 0,cos ( 1)n
n n
π = π = − for all n
9. (2 1) 1
2
sin ( 1) , 1,2,3,...
k k
k
− π −
  =
− =
  for all n
10. ( )
2
1, if 1(mod4)
sin
1, if 3(mod4)
n
n
n
π
≡

= 
− ≡

11. ( )
2
cos 0
nπ
= for all odd values of n
12. 1
2
sin cos [sin( ) sin( )]
A B A B A B
⋅ = + + −
13. 1
2
cos sin [sin( ) sin( )]
A B A B A B
⋅ = + − −
14. 1
2
cos cos [cos( ) cos( )]
A B A B A B
⋅ = + + −
15. 1
2
sin sin [cos( ) cos( )]
A B A B A B
⋅ = − − −
16. 1
2
sin cos sin 2
A A A
⋅ =
17. 2 2 2 2
cos2 2cos 1 cos sin 1 2sin
A A A A A
= − = − = −