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KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
1
Prepared by K.V.B. Rajakumar
I YEAR - II SEMESTER
Unit – III
FOURIER TRANSFORMS AND FOURIER SERIES
Objectives:
To introduce
Fourier transform of a given function and the corresponding inverse.
Fourier sine and cosine transform of a given function and their corresponding inverses.
Finite Fourier transforms of a given function and their corresponding inverses.
Syllabus:
Fourier Transforms: Fourier integral theorem (without proof) – Fourier sine and cosine
integrals – Sine and cosine transforms – Properties – inverse transforms – Finite Fourier
transforms.
Outcomes:
Students will be able to
Find the Fourier transform of the given function in infinite cases.
Find the Fourier sine and cosine transforms of the given function in infinite cases.
Learning Material
Fourier Transforms are widely used to solve Partial differential equations and in various
boundary value problems of Engineering such as vibration of strings, Conduction of heat,
Oscillation of an elastic beam, transmission lines so on..
1 FOURIER TRANSFORMS 2-31
2 FOURIER SERIES 32-44
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
2
Prepared by K.V.B. Rajakumar
Fourier Integral Theorem:
If f(x) satisfies Dirichlet’s conditions for expansion of Fourier series in (-C, C) an ( )
f x dx
converges, then
1( ) ( )
2
f x f t cosα(t - x) dt dα is known as Fourier Integral of f(x).
Fourier Sine Integral: If f(x) satisfies Dirichlet’s conditions for expansion of Fourier series
in (-C, C) and ( )
f x dx converges, if f(t) is odd function then
0 0
2( ) ( )
f x f t sinαt sinα x dt dα .
Fourier Cosine Integral: If f(x) satisfies Dirichlet’s conditions for expansion of Fourier
series in (-C, C) and ( )
f x dx converges, if f (t) is even function then
0 0
2( ) ( )
f x f t Cosαt Cosα x dt dα .
Complex form or exponential form of Fourier Integral: The complex form of Fourier
integral is known as
i ( t x)1( ) ( )
2
f x f t e dt dα
Note:
2 20
2 20
a-axe Cosbxdx =a +b
b-axe Sin xdx =a +b
Examples:
1. Using Fourier integral, prove that 2 2
0
2
-ax a Cosαxe dα
a
The Fourier Cosine integral of f(x) is 0 0
2( ) ( )
f x f t Cosαt Cosα x dt dα
Let us take ( )-a t
f t = e , a > 0
0 0
2
-ax -a te e Cosαt dt Cosα x dt dα
0 0
2
-ax -a te e Cosαt Cosα x dt dα
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
3
Prepared by K.V.B. Rajakumar
2 20
2
aCosα x dα
a
2 20
2 20
2a
2
-ax
Cosα xdα
a
aCosα xdα=e
a
2. Using Fourier integral, prove that 2 2
02
-axSin αx
dα= ea
The Fourier Sine integral of f(x) is 0 0
2( ) ( )
f x f t sinαt sinα x dt dα
Let us take ( )-a t
f t = e , a > 0
0 0
2
-ax -a te e Sinαt Sinα x dt dα
0 0
2
-ax -a te e Sinαt dt Sinα x dt dα
2 20
2
Sinα x dα
a
2 20
2 20
2
2
-ax
Sinα xdα
a
Sinα xdα=e
a
3. Using Fourier integral, prove that 2
4
0
2
2 4
-ax
Cosαxe Cosx dα
The Fourier Cosine integral of f(x) is 0 0
2( ) ( )
f x f t Cosαt Cosα x dt dα
Let us take ( )- t
f t = e Cost then
0 0
2
- x - t
e Cos x e Cost Cosαt Cosα x dt dα
0 0
2
- x - te Cos x e Cost Cosαt dt Cosα x dα
We have 2 Cos C Cos D = Cos (C+D) + Cos(C-D)
0 0
21 1-
- x - te Cos x Cos + t +Cos t e dt Cosα x dα
0 0 0
21 1-
- x - t - te Cos x e Cos + t dt + e Cos tdt Cosα x dα
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
4
Prepared by K.V.B. Rajakumar
2 2
0
2 1 1
1 1 1 1
- xe Cos x Cosα x dα
2
4
0
2
2 4
-ax
Cosαxe Cosx dα
4. Using Fourier integral, prove that 3
4
04 2
-axSin αx
dα= e Cos x
0 0
2( ) ( )
f x f t sinαt sinα x dt dα
Let us take ( )- t
f t = e Cost
0 0
2( )
-axe Cos x f t sinαt sinα x dt dα
0 0
2
- t-axe Cos x e Cost sinαt sinα x dt dα
0 0
2
- t-axe Cos x Sinα x e Cost sinα t dt dα
0 0
2
2 S S 1 S 1
2 S S 1 S 1
- t-axe Cos x Sinα x e sinα t Cosα t dt dα
inα t Cosα t inα t inα t
inα t Cosα t inα t inα t
0 0
2S 1 S 1
- t-axe Cos x Sinα x e inα t inα t dt dα
0 0
2S 1 S 1
- t-axe Cos x Sinα x e inα t inα t dt dα
0 0 0
1S 1 S 1
- t - t-axe Cos x = Sinα xdα e inα t e inα t
2 2
0
1 1 1
1 1 1 1
-axe Cos x Sinα x dα
3
4
0
1 2
4
-axe Cos x Sinα x dα
3
4
04 2
-axSin αx
dα= e Cos x
5. Using Fourier integral, prove that
0,,
sin2
0
2222
22
badba
xabee bxax
Since the integrand on R.H.S contains sine term, we use Fourier sine integral formula.
We know that fouries sine integral for f(x) is given by
00
sin)(sin2
)( ptdtdptfpxxf
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
5
Prepared by K.V.B. Rajakumar
Replacing p with λ we get
00
sin)(sin2
)(
tdtdtfxxf
Here f(x) = e-ax
- e- bx
→ f(t) = e-at
- e-bt
Substituting (2) in (1) , we get
dtdteexxf btat
00
sinsin2
)(
dttbb
etta
a
exxf
btat
0
22
0
22
0
cossincossinsin2
)(
d
baxxf
2222
0
sin2
)(
d
ba
abxxf
2222
22
0
sin2
)(
d
ba
xabee bxax
0
2222
22 sin2
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
6
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
7
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
8
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
9
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
10
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
11
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
12
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
13
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
14
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
15
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
16
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
17
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
18
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
19
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
20
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
21
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
22
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
23
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
24
Prepared by K.V.B. Rajakumar
Problem: Find the Fourier transform of f(x) defined by 1, | |
( )0, | |
if x af x
if x a
And hence evaluate
0
sindp
p
p and
dpp
pxapcossin
Sol: We have F[f(x)]=
dxxfeipx )( =
a
ipx dxxfe )( +
a
a
ipx dxxfe )( +
a
ipx dxxfe )(
=
a
a
ipx dxe =p
apsin2
By the inversion formula, we know that f(x)= 2
1
dppFe ipx )(
2
1
dpp
ape ipx sin2
= 1, | |
0, | |
if x a
if x a
2
1
dpp
appx
sin2cos -
2
1
dpp
appx
sin2sin =
1, | |
0, | |
if x a
if x a
Since the second integral is an odd function,
dpp
pxapcossin= ,
1, | |
0, | |
if x a
if x a
Put x=0, we get, 2
1
dpp
apsin2=
1, 0
0, 0
if a
if a
0
sindp
p
p =
2
, a>0
= 0, a<0
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
25
Prepared by K.V.B. Rajakumar
And put x=0 and a=1 then we get
0
sindp
p
p=
2
Problem: Find the fourier sine transform of e-ax
, a>0 and hence deduce that
0
22
sindp
pa
pxp
Sol: xfFs =
0
sin)( dxpxxf =
0
sin dxpxe ax=
22 pa
p
By the inversion formula, we know that f(x)=
2
0
sin dppxxfFs
=
2
0
22sin dppx
pa
p
0
22 2
sin axedppa
pxp
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
26
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
27
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
28
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
29
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
30
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
31
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
32
Prepared by K.V.B. Rajakumar
FOURIER SERIES
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
33
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
34
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
35
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
36
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
37
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
38
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
39
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
40
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
41
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
42
Prepared by K.V.B. Rajakumar
KALLAM HARANADHAREDDY INSTITUTE OF TECHNOLOGY (Affiliated to JNTUK, Kakinada; Approved by AICTE, New Delhi)
NH-5,Chowdavaram, Guntur-522019
MATHEMATICS – III Common to all Branches
43
Prepared by K.V.B. Rajakumar