LECTURE NOTES - FOURIER SERIES
Introduction:
Fourier series is an infinite series representation of periodic function in terms of sine and
cosine function. Fourier series is very powerful method to solve ODE and PDE particularly
with periodic functions.
Applications of Fourier series:
Fourier series are very useful in electrical engineering problems where it is applied for force,
potential, electromagnetic force in electricity.
Fourier series is applied in engineering problems like mechanical vibrations, electric circuits,
heat flow, theory of sound etc, where the periodic functions are quite common occurrence.
Such periodic functions are complicated .So it is important to represent these periodic
functions in terms of simple functions of sine and cosine.
Periodic Functions:
A function f(x) is said to be periodic if f(x + p) =f(x) for all X and ‘p’ being a positive
constant. The minimum value of p is called the period of f(x).
Eg:
The period of sinx and cosx is 2π. The period of tanx, cotx is π.
Dirichlet’s Conditions:
A function f(x) can be developed as Fourier series
ao
2
∞
+ ∑∞
n=1 a n cosnx + ∑n=1 bn sinnx where
ao , an and bn are constants provided
f(x) is periodic, single valued and finite.
f(x) has a finite no of finite discontinuities and no infinite discontinuity.
f(x) has at most a finite no of maxima and minima.
Euler’s formula for the Fourier coefficients:
If a function f(x) defined in c<x<c+2π can be expanded as the infinite trigonometric series
a
∞
f(x) = 2o + ∑∞
n=1 a n cosnx + ∑n=1 bn sinnx
1
c+2π
Where ao = ∫c
π
1
c+2π
f(x)dx, an = ∫c
π
1
c+2π
f(x)cosnxdx, bn = ∫c
π
f(x)sinnxdx
Fourier Series:
The infinite trigonometric series f(x) =
ao
2
∞
+ ∑∞
n=1 a n cosnx + ∑n=1 bn sinnx is called the
Fourier series of f(x) in the interval c<x<c+2π provided the coefficient are given by Euler’s
formulas.
Parseval’s Identity:
If f(x) is a periodic function c<x<c+2π with period 2π then
∞
1 c+2π
ao 2 1
∫
f(x)dx =
+ ∑(a2n + b2n )
2π c
2
2
n=1
Problems
1.1 Obtain the Fourier series expansion for the function f(x) = x 2 with period 2π, in the
interval 0<x< 2π.
Solution:
a
∞
The Fourier Series is f(x) = 2o + ∑∞
n=1 a n cosnx + ∑n=1 bn sinnx
Given f(x) = x 2
1
2π
1
1 x3
2π
ao = π ∫0 f(x)dx =π ∫0 x 2 dx =π [ 3 ]
2π
8π2
=
0
3
1 2π
1 2π 2
an = ∫ f(x)cosnxdx = ∫ x cosnxdx
π 0
π 0
1
sinnx
−cosnnx
n
n2
=π [x 2 (
1
) − (2x) (
sinnx
cosnnx
n
n2
=π [x 2 (
) + (2x) (
sinnx
2π
n3
0
)+ 2(
sinnx
)+ 2(
n3
)]
2π
)]
0
4
an =n2
1 2π
1 2π
bn = ∫ f(x)sinnxdx = ∫ x 2 sinnxdx
π 0
π 0
1
−cosnx
−sinnx
n
n2
=π [x 2 (
1
) − (2x) (
)+ 2(
cosnx
n3
2π
)]
0
cosnx
sinnx
cosnx
2π
n
n2
n3
0
=π [−x 2 (
) + (2x) (
) + 2(
)]
−4π
bn =
n
Hence the obtained Fourier Series is
∞
∞
n=1
n=1
4π2
4
4π
f(x) =
+ ∑ 2 cosnx − ∑ sinnx
3
n
n
1.2 Express f(x) = (π − x)2 as a Fourier Series in 0<x<2π and deduce the sum of
∑
1
1
1
1
1
ii) 12 − 22 + 32 − 42 + ⋯ + iii)
n2
1
12
1
1
+ 32 + 52 + ⋯ + ∞
1
iv) ∑ n4
Solution:
The Fourier Series is f(x) =
ao
2
∞
+ ∑∞
n=1 a n cosnx + ∑n=1 bn sinnx __________(1)
Given f(x) = (π − x)2
1
2π
1
2π
1 (π−x)3
ao = ∫0 f(x)dx = ∫0 (π − x)2 dx = [
π
π
π
−3
2π
]
0
=
2π2
3