1. {
Welcome to Our
Presentation
Fourier
Transformation &
Fourier Series

2. Outline
 History
 Fourier Series
 Odd and Even Function
 Half Range Fourier Sine & Cosine series
 Complex Form of Fourier Series
 Fourier Transformation
 Relationship Between Fourier & Laplace
Transformation
 Applications
Fourier Transformation & Fourier Series

3. The Fourier series is named in hono of Jean-Baptiste Joseph
Fourier (1768–1830), who made important contributions to the
study of trigonometric series, after preliminary investigations
by Leonhard Euler, Jean le Ron d'Alembe , and Daniel
Bernoulli. Fourier introduced the series for the purpose of
solving the heat equation in a metal plate.
Early ideas of decomposing a periodic function into the sum
of simple oscillating functions date back to the 3rd century BC,
when ancient astronomers proposed an empiric model of
planetary motions, based on deferent and epicycles.
The heat equation is a partial differential equation. Prior to
Fourier's work, no solution to the heat equation was known in
the general case, although particular solutions were known if
the heat source behaved in a simple way, in particular, if the
heat source was a sine or cosine wave. These simple solutions
are now sometimes called eigensolution .
History

4. Let F(x) satistfy the following condition :
1. F(x) is defined in the interval c<x<c+2l
2. F(x) and 𝐹′(x) are sectionally continuous in
c<x<c+2l
3. F(x+2l) =F(x) , i.e F(x) is periodic with period 2l.
Then at every point of continuity , we have
F(x)=
𝑎0
2
+ 𝑛=1
∞ ( 𝑎 𝑛 𝑐𝑜𝑠
𝑛π 𝑥
𝑙
+ 𝑏 𝑛 𝑠𝑖𝑛
𝑛π 𝑥
𝑙
) …….(1)
𝑎 𝑛=
1
𝑙 𝑐
𝑐+2𝑙
𝐹(𝑥) cos
𝑛π 𝑥
𝑙
dx ………….(2)
b 𝑛=
1
𝑙 𝑐
𝑐+2𝑙
𝐹(𝑥) sin
𝑛π 𝑥
𝑙
dx
At a point of discontinuity, the left side of (1) is replaced by
1
2
(F(x + 0) + F(x - 0) ),
Fourier Series

5. the mean value at the discontinuity.
The series (1) with coefficients (2) is called the Fourier series of F(x).
For many problems, 𝑐 = 0 𝑜𝑟 − 𝑙. In case 𝑙 = π, F(x) has period 2π and
(1) and (2) are simplified.
The above conditions are often called Dirichlet conditions and are
sufficient (but not necessary) conditions for convergence of Fourier
series.
Fourier Series

6. A function F(x) is called odd if F(-x) = -F(x). Thus x^3, x^5 - 3x^3 + 2x,
sin x, tan 3x are odd functions.
A function F(x) is called even if F(-x) = F(x). Thus x^4, 2x^6 - 4x^2 + 5,
cos x, 𝑒 𝑥+𝑒−𝑥 are even functions .The functions portrayed graphically
Figures (1) and (2)are odd and even respectivel
but that of Fig (3) is neither odd nor even.
Odd and Even Function

7. In the Fourier series corresponding to an odd function, only sine
terms can be present.
In the Fourier series corresponding to an even function, only cosine
terms (and possibly a constant which we shall consider a cosine
term) can be present.
In the Fourier series corresponding to an odd function, only sine
terms can be present. In the Fourier series corresponding to an odd
function, only sine terms can be present. In the Fourier series
corresponding to an even function, only cosine terms (and possibly
a constant which we shall consider a cosine term) can be present.
In the Fourier series corresponding to an odd function, only sine
terms can be present.
Odd and Even Function

8. Problem:
(a) Find the Fourier coefficients corresponding to the
function
0 −5 < 𝑥 < 0
F(x)= Period=10
3 0 < 𝑥 < 5
(b) Write the corresponding Fourier series.
PROBLEM

9. Solution:
(a)
Period=2𝑙 𝑎𝑛𝑑 𝑙 = 5 𝑐ℎ𝑜𝑜𝑠𝑒 𝑡ℎ𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 𝑐 𝑡𝑜 𝑐 + 2𝑙 𝑎𝑠 −
5 𝑡𝑜 5 , 𝑠𝑜 𝑡ℎ𝑎𝑡
Then
𝑎 𝑛=
1
𝑙 𝑐
𝑐+2𝑙
𝐹(𝑥) cos
𝑛π 𝑥
𝑙
dx=
1
5 −5
5
𝐹(𝑥) cos
𝑛π 𝑥
5
dx
=
1
5
{ −5
0
(0) cos
𝑛π 𝑥
5
dx + 0
5
(3) cos
𝑛π 𝑥
5
dx }
=
3
5 0
5
𝑐os
𝑛π 𝑥
5
dx
=
3
5
(
5
𝑛𝜋
sin
𝑛π 𝑥
5
)0
= 0 if n ≠ 0
PROBLEM

10. If n=0 , 𝑎 𝑛=𝑎0=
3
5 0
5
𝑐os
0π 𝑥
5
dx =
3
5 0
5
𝑑x =3
b 𝑛=
1
𝑙 𝑐
𝑐+2𝑙
𝐹(𝑥) sin
𝑛π 𝑥
𝑙
dx =
1
5 −5
5
𝐹(𝑥) sin
𝑛π 𝑥
5
dx
=
1
5
{ −5
0
(0) sin
𝑛π 𝑥
5
dx+ 0
5
(0) sin
𝑛π 𝑥
5
dx}
=
3
5 0
5
𝑠in
𝑛π 𝑥
5
dx
=
3
5
(-
5
𝑛𝜋
cos
𝑛𝜋𝑥
5
) =
3(1−cos 𝑛𝜋)
𝑛𝜋
PROBLEM

11. (b) The corresponding Fourier series is
𝑎0
2
+ 𝑛=1
∞ ( 𝑎 𝑛 𝑐𝑜𝑠
𝑛π 𝑥
𝑙
+ 𝑏 𝑛 𝑠𝑖𝑛
𝑛π 𝑥
𝑙
)
=
3
2
+ 𝑛=1
∞ 3(1−cos 𝑛𝜋𝑥)
𝑛𝜋
𝑠𝑖𝑛
𝑛π
5
)
=
3
2
+
6
𝜋
(sin
𝜋𝑥
5
+
1
3
sin
3𝜋𝑥
5
+
1
5
sin
5𝜋𝑥
5
+……..)
PROBLEM

12. A half range Fourier sine or cosine series is a series in which only sine terms
or only cosine terms are present respectively. When a half range series
corresponding to a given function is desired, the function is generally
defined in the interval (0, 𝑙)[which is half of the interval(−𝑙, 𝑙), thus
accounting for the name half range] and then the function is specified as
odd or even, so that it is clearly defined in the other half of the interval,
namely(−𝑙, 0). In such case, we have
𝑎 𝑛=0 , b 𝑛=
2
𝑙 0
𝑙
𝐹(𝑥) sin
𝑛π 𝑥
𝑙
dx for half range sine series ……(3)
b 𝑛=0 , 𝑎 𝑛=
2
𝑙 0
𝑙
𝐹(𝑥) cos
𝑛π 𝑥
𝑙
dx for half range cosine series
Half Range Fourier Sine & Cosine
series

13. In complex notation, the Fourier series (1) and coefficients (2) can
be written as
F(x)= 𝑛=−∞
∞ 𝑐 𝑛 𝑒
𝑖𝑛𝜋𝑥
𝑙 ……………..(1)
Where 𝑐 = −𝑙
𝑐 𝑛=
1
2𝑙 −𝑙
𝑙
𝐹(𝑥) 𝑒−
𝑖𝑛𝜋𝑥
𝑙 ……………………..(2)
Complex Form of Fourier Series

14. If
𝑓 λ =
−∞
∞
𝑒 𝑖λ 𝑢
𝐹 𝑢 𝑑𝑢 … … … … … … … . 1
Then F 𝑢 =
1
2𝜋 −∞
∞
𝑒 𝑖λ 𝑢
𝑓(λ)𝑑λ………………………..(2)
The function f (λ) is called the Fourier transform of F(u) and is
sometimes written f(λ) =F{F(u)}. The function F(u) is the
inverse Fourier transform of f(λ) and is written
F(u) = 𝐹−1 { f (λ) }. We also call (2) an inversion formula
corresponding to (1). Note that the constants preceding the
integral signs can be any constants whose product
is Type equation here
1
2𝜋
If they are each taken as
1
2𝜋
we
obtain the so-called symmetric form .
FOURIER TRANSFORMS

15. Consider the function
𝑒−𝑥𝑡
𝜑(t) t > 0
F(t)= 0 t<0
we see that the Fourier transform of F(t) is
F{f(t)}= 0
∞
𝑒− 𝑥+𝑖𝑦 𝑡 𝜑(t )dt = 0
∞
𝑒−𝑠𝑡 𝜑(t) dt ………(1)
where we have written s = x + iy . The right side of (1) is the Laplace
transform of 𝜑(t) and the result indicates a relationship of Fourier and
Laplace transforms.
RELATIONSHIP OF FOURIER AND
LAPLACE TRANSFORMS

16. The Fourier series has many such applications in electrica
engineering , vibration analysis , acoustics , optics , signal
processing , image processing , quantum mechanics ,
econometrics , thin-walled shell theory etc.
 Electrical engineers design complex power systems.
 The mechanical system vibrates at one or more of its natural
frequencies and damps down to motionlessness.
APPLICATION

17.  The application of acoustics is present in almost all aspects
of modern society with the most obvious being the audio
and noise control industries.
 Thin-shell structures are also called plate and shell
structures.
APPLICATION

18. REFERENCES
 LECTURE OF Dr. Md.Golam Hossain
 LAPLACE TRANSFORMATION
 WIKIPEDIA
 MATHEMATICS BLOG

19. ANY QUESTION ???
THANK YOU ALL