FOURIER
SERIES
1.0 INTRODUCTION
A FOURIER SERIES is an infinite trigonometric series of
a PERIODIC FUNCTION which is continuous or have
finite discontinuities. The fourier series of a 𝑓(𝑥) of a
period 2L where L denotes the half period which is
given by:
𝒂𝟎
𝝅𝒙
𝟐𝝅𝒙
𝟑𝝅𝒙
𝝅𝒙
𝟐𝝅𝒙
𝟑𝝅𝒙
𝒇 𝒙 =
+ 𝒂𝟏 𝒄𝒐𝒔
+ 𝒂𝟐 𝒄𝒐𝒔
+ 𝒂𝟑 𝒄𝒐𝒔
+ ⋯ + 𝒃𝟏 𝒔𝒊𝒏
+ 𝒃𝟐 𝒔𝒊𝒏
+ 𝒃𝟑 𝒔𝒊𝒏
+⋯
𝟐
𝑳
𝑳
𝑳
𝑳
𝑳
𝑳
Summation Notation of Fourier Series
∞
𝒂𝟎
𝝅𝒏𝒙
𝝅𝒏𝒙
𝒇 𝒙 =
+ ෍ 𝒂𝒏 𝒄𝒐𝒔
+ 𝒃𝒏 𝒔𝒊𝒏(
)
𝟐
𝑳
𝑳
𝒏=𝟏
where 𝒂𝟎 , 𝒂𝟏 , 𝒂𝟐 , 𝒂𝟑 … 𝒃𝟏 , 𝒃𝟐 , 𝒃𝟑 are known as fourier series constants
Note: Fourier Series are only applicable for periodic and continuous functions as implied
by its definition. In special cases where there is a discontinuity it should only be finite
such that we can still take the fourier series of that function 𝒇 𝒙
1.1 FINDING THE FOURIER SERIES CONSTANT
The fourier series constants can be calculated using the general equation and some special
cases.
General Formula:
𝑐+2𝐿
1
𝑎0 = න
𝐿 𝑐
𝑓 𝑥 𝑑𝑥
1 𝑐+2𝐿
𝜋𝑛𝑥
𝑎𝑛 = න
𝑓 𝑥 cos(
)𝑑𝑥
𝐿 𝑐
𝐿
1 𝑐+2𝐿
𝜋𝑛𝑥
𝑏𝑛 = න
𝑓 𝑥 sin(
)𝑑𝑥
𝐿 𝑐
𝐿
Special Case 1:
• If 𝒇(𝒙) is an even function then 𝑏𝑛 = 0 and the
fourier series coefficients are obtained as follows
2 𝐿
𝑎0 = න 𝑓 𝑥 𝑑𝑥
𝐿 0
2 𝐿
𝜋𝑛𝑥
𝑎𝑛 = න 𝑓 𝑥 cos(
)𝑑𝑥
𝐿 0
𝐿
Special Case 2:
• If 𝒇(𝒙) is an odd function then 𝑎0 = 0 and 𝑎𝑛 = 0 in
such case 𝑏𝑛 can be obtained using
2 𝐿
𝜋𝑛𝑥
𝑏𝑛 = න 𝑓 𝑥 sin(
)𝑑𝑥
𝐿 0
𝐿
1.2 ODD AND EVEN FUNCTIONS
EVEN FUNCTIONS
ODD FUNCTIONS
A function is said to be even if 𝑓 −𝑥 = 𝑓(𝑥). Graphically the graph of
𝑓(𝑥) is said to be even if its symmetrical about the vertical y-axis.
(Example: 𝑥 2 , 𝑐𝑜𝑠𝑥, sin2 𝑥)
A function is said to be odd if 𝑓 −𝑥 = −𝑓(𝑥). Graphically the graph
of 𝑓(𝑥) is said to be odd if its symmetrical about the origin.
(Example: x, 𝑠𝑖𝑛𝑥, 𝑥 3 )
1.3 TRIGONOMETRIC IDENTITIES
Recall
on
Trigonometric
Product and Sum identities
1.
2.
3.
Identities
1
𝑠𝑖𝑛𝐴𝑐𝑜𝑠𝐵 = sin 𝐴 − 𝐵 + sin 𝐴 + 𝐵
2
1
𝑐𝑜𝑠𝐴𝑐𝑜𝑠𝐵 = cos 𝐴 − 𝐵 + cos 𝐴 + 𝐵
2
1
𝑠𝑖𝑛𝐴𝑠𝑖𝑛𝐵 = cos 𝐴 − 𝐵 − cos 𝐴 + 𝐵
2
Recall on Integration
Summary on Evaluation of sine and
cosine values
Sine Values
sin 0 = 0
sin 𝜋𝑛 = sin −𝜋𝑛 = 0
Cosine Values
cos 0 = 1
cos 𝜋𝑛 = cos −𝜋𝑛 = −1 𝑛
sin 2𝜋𝑛 = 0
−sin 𝜋𝑛 = 0
cos 2𝜋𝑛 = 1
−cos 𝜋𝑛 = −1 𝑛+1
Odd and Even “n” representation
1.
𝜋
1
𝜋
sin(𝑛𝑥)
𝑑𝑥
=
−
cos(𝑛𝑥)
‫׬‬−𝜋
−𝜋
𝑛
2.
𝜋
1
𝜋
‫׬‬−𝜋 cos(𝑛𝑥) 𝑑𝑥 = 𝑛 sin(𝑛𝑥) −𝜋
=0
=0
If 𝑛 = 1,2,3,4,5,6, … , then 2𝑛 − 1
numbers
represents odd
If 𝑛 = 1,2,3,4,5,6, … , then 2𝑛 represents even numbers
1.4 GRAPHS OF SOME BASIC FUNCTIONS
1.5 EXAMPLES
1
1. 𝑓 𝑥 = ቊ −𝜋<𝑥<0
0 0<𝑥<𝜋
a) Graph the function
b) Determine its Fourier Series
𝜋
1
1
1
c) Prove that by assigning value to x will result to = 1 − + − + ⋯
4
0
2. 𝑓 𝑥 = ቊ −𝜋<𝑥<0
𝑥 0<𝑥<𝜋
a) Graph the function
b) Determine its Fourier Series
−𝜋 < 𝑥 < 0
𝜋−𝑡
3. 𝑓 𝑥 = ቐ 𝜋 + 𝑡
0<𝑥<𝜋
𝑓(𝑡 + 2𝜋)
a) Graph the function
b) Determine its Fourier Series
3
5
7