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
= − = − = −
