INTEGRAL TRANSFORMS · B.A./B.Sc.(Mathematics)–3rd Year–6th Sem. E-CONTENT BY DR.N.A.P ANDE ॥॥॥॥ °ीह ïर ° °ीीहह ï ïरर °ीह ïर: :: : ॥॥॥॥

Paper No. 12(B.A.) / 17(B.Sc.) Paper No. 12(B.A.) / 17(B.Sc.) Integral Transforms Integral Transforms ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–3rd Year–6th Sem. B.A./B.Sc.(Mathematics)–3rd Year–6th Sem.

॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

INTEGRAL TRANSFORMS

Dr.N.A.Pande Associate Professor

Department of Mathematics & Statistics, Yeshwant Mahavidyalaya, Nanded – 431602

Maharashtra, INDIA



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Paper Details

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Paper Details • University : Swami Ramanand Teerth Marathwada University, Nanded, India

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• Course : B.A./B.Sc.

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• Course : B.A./B.Sc. • Subject : Mathematics

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• Course : B.A./B.Sc. • Subject : Mathematics • Year : 3rd

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• Course : B.A./B.Sc. • Subject : Mathematics • Year : 3rd • Semester : 6th

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• Course : B.A./B.Sc. • Subject : Mathematics • Year : 3rd • Semester : 6th • Paper No. : 12(B.A.) / 17(B.Sc.)

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• Course : B.A./B.Sc. • Subject : Mathematics • Year : 3rd • Semester : 6th • Paper No. : 12(B.A.) / 17(B.Sc.) • Syllabus Effective From : 2015-16

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• Course : B.A./B.Sc. • Subject : Mathematics • Year : 3rd • Semester : 6th • Paper No. : 12(B.A.) / 17(B.Sc.) • Syllabus Effective From : 2015-16 • Paper Code : MT-305

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• Course : B.A./B.Sc. • Subject : Mathematics • Year : 3rd • Semester : 6th • Paper No. : 12(B.A.) / 17(B.Sc.) • Syllabus Effective From : 2015-16 • Paper Code : MT-305 • Marks : 40 (University) + 10 (Internal) = 50

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Syllabus

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Syllabus • Unit-I : Laplace Transforms

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Syllabus • Unit-I : Laplace Transforms

� Introduction, Laplace Transform, Important Formulae, Properties of Laplace Transforms, Laplace Transforms of the derivative of f(t), Laplace Transforms of derivative of order n, Laplace Transform of integral of f(t), Laplace Transform of t·f(t) (Multiplication by t), Laplace Transform of 1/t·f(t) (Division by t), Unit Step Function, Second Shifting Theorem, Impulse Function, Periodic Functions, Convolution Theorem, Evaluation of Integrals, Formulae of Laplace Transform, Properties of Laplace Transform.

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Syllabus

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Syllabus • Unit-II : Inverse Laplace Transforms

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Syllabus • Unit-II : Inverse Laplace Transforms

� Inverse Laplace Transforms, Important formulae, Multiplication by s, Division by s, First Shifting Property, Second Shifting Property, Inverse Laplace Transform of Derivatives, Inverse Laplace Transform of Integrals, Partial Fractions Method, Inverse Laplace Transform by Convolution, Solution of Differential Equation by Laplace Transforms, Solution of Simultaneous Differential Equations by Laplace Transforms

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Syllabus

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Syllabus • Unit-III : : Fourier Transforms

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Syllabus • Unit-III : : Fourier Transforms

� Introduction, Integral Transforms, Fourier Integral Theorem, Fourier sine and Cosine Integrals, Fourier Complex Integral, Fourier transforms, Fourier sine and cosine transforms, Properties of Fourier Transforms.

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Text Book & Scope

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Text Book & Scope • Recommended Text Book :

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�Title : Advanced Engineering

Mathematics

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Mathematics

�Author : H. K. Dass

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Mathematics


�Publisher : S. Chand & Company Ltd.

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Mathematics



� ISSN : 81-219-0345-9

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Mathematics



� ISSN : 81-219-0345-9

• Scope :

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



Mathematics



� ISSN : 81-219-0345-9

• Scope : �Unit–I : 13.1 to 13.15, and 13.17 to 13.19

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



Mathematics



� ISSN : 81-219-0345-9


�Unit–II : 13.20 to 13.31

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



Mathematics



� ISSN : 81-219-0345-9


�Unit–II : 13.20 to 13.31

�Unit–III : 14.1 to 14.8

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Reference Books

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Reference Books • Operational Mathematics, By R. V. Churchill, John Wiley & Sons, New

• York, 1985.

• An Introduction to Laplace Transforms and Z-Transforms, By Grove A.C., Printice Hall, 1991.

• Guide to the Laplace Transforms and Applications, By Doetsch G. Springer, Ver. 1963.

• Introduction to Theory and Application of Laplace Transforms, By Doetsch G., Springer Verg. 1990.

• Complex Analysis, By Alhlfors L.V., Mc-Graw Hill, 3rd ed., 1979.

• Schaum’s Outline of Laplace Transforms, By Murray Spigel, Schame Outline Series, Mc-Graw Hill, 2012.

• The Laplace Transforms Theory and Applications, By Joel. L. Schiff, Springer, 2008.

• Fast Fourier Transforms : Algorithms and Applications, By K. R. Rao, D. N. Kim, J. J. Hwang, Springer 2010.

• Fourier Transform, By S. Bochner and K. Chandrasekehran, University Press, 1949.

• Fourier and Laplace Transform, By R.J. Becrends, H. G. Morsche, J. C. Vande Berg and E. M. Vande Vrie, Cambridge Press, 2003.

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Introduction

UNIT-I

8



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Introduction • Laplace transforms help in solving the differential equations with boundary conditions without finding the general solution and the values of the arbitrary constants.

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8



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Introduction • Laplace transforms help in solving the differential equations with boundary conditions without finding the general solution and the values of the arbitrary constants.

• If f(t) is defined and continuous for all non-negative values, then Laplace transform of f(t) is denoted by L[f(t)] or F(s) and is given by

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8

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

−−−−========0

)()( dttfesFtfL st



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Important Formulae •

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9



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


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9

(((( ))))s

L1

1 ====



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-I

9

(((( ))))s

L1

1 ====

(((( )))) asas

eL at >>>>−−−−

==== for ,1



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


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9

(((( ))))s

L1

1 ====

(((( )))) asas

eL at >>>>−−−−

==== for ,1

(((( )))) 0 for ,sin22

>>>>++++

==== sas

aatL



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-I

9

(((( ))))s

L1

1 ====

(((( )))) asas

eL at >>>>−−−−

==== for ,1

(((( )))) 0 for ,sin22

>>>>++++

==== sas

aatL

(((( )))) 0 for ,cos22

>>>>++++

==== sas

satL



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-I

9

(((( ))))s

L1

1 ====

(((( )))) asas

eL at >>>>−−−−

==== for ,1

(((( )))) 0 for ,sin22

>>>>++++

==== sas

aatL

(((( )))) 0 for ,cos22

>>>>++++

==== sas

satL

(((( )))) 22

22 for ,sinh as

as

aatL >>>>

−−−−====



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-I

9

(((( ))))s

L1

1 ====

(((( )))) asas

eL at >>>>−−−−

==== for ,1

(((( )))) 0 for ,sin22

>>>>++++

==== sas

aatL

(((( )))) 0 for ,cos22

>>>>++++

==== sas

satL

(((( )))) 22

22 for ,sinh as

as

aatL >>>>

−−−−====

(((( )))) 22

22 for ,cosh as

as

satL >>>>

−−−−====



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-I

10



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-I

10

(((( ))))1

1|++++++++====

n

n

s

ntL



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-I

10

(((( ))))1

1|++++++++====

n

n

s

ntL

(((( )))) (((( )))) 0 for ,sin22 >>>>

++++−−−−==== s

bas

bbteL at



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-I

10

(((( ))))1

1|++++++++====

n

n

s

ntL

(((( )))) (((( )))) 0 for ,sin22 >>>>

++++−−−−==== s

bas

bbteL at

(((( )))) (((( )))) 0 for ,cos22 >>>>

++++−−−−−−−−==== s

bas

asbteL at



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-I

10

(((( ))))1

1|++++++++====

n

n

s

ntL

(((( )))) (((( )))) 0 for ,sin22 >>>>

++++−−−−==== s

bas

bbteL at

(((( )))) (((( )))) 0 for ,cos22 >>>>

++++−−−−−−−−==== s

bas

asbteL at

(((( )))) (((( ))))22

22 for ,sinh asbas

bbteL at >>>>

−−−−−−−−====



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-I

10

(((( ))))1

1|++++++++====

n

n

s

ntL

(((( )))) (((( )))) 0 for ,sin22 >>>>

++++−−−−==== s

bas

bbteL at

(((( )))) (((( )))) 0 for ,cos22 >>>>

++++−−−−−−−−==== s

bas

asbteL at

(((( )))) (((( ))))22

22 for ,sinh asbas

bbteL at >>>>

−−−−−−−−====

(((( )))) (((( ))))22

22 for ,cosh asbas

asbteL at >>>>

−−−−−−−−−−−−====



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-I

10

(((( ))))1

1|++++++++====

n

n

s

ntL

(((( )))) (((( )))) 0 for ,sin22 >>>>

++++−−−−==== s

bas

bbteL at

(((( )))) (((( )))) 0 for ,cos22 >>>>

++++−−−−−−−−==== s

bas

asbteL at

(((( )))) (((( ))))22

22 for ,sinh asbas

bbteL at >>>>

−−−−−−−−====

(((( )))) (((( ))))22

22 for ,cosh asbas

asbteL at >>>>

−−−−−−−−−−−−====

(((( )))) (((( )))) 1

1|++++−−−−

++++==== n

nat

as

nteL



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Special Functions

UNIT-I

11



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Special Functions • Unit Step Function : If a is any constant, the unit step function, denoted by u(t − a) is given by

UNIT-I

11



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-I

11

(((( ))))

≥≥≥≥<<<<

====−−−−at

atatu

if ,1

if ,0



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• The graph of unit step function is

UNIT-I

11

(((( ))))

≥≥≥≥<<<<

====−−−−at

atatu

if ,1

if ,0



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• The graph of unit step function is

UNIT-I

11

(((( ))))

≥≥≥≥<<<<

====−−−−at

atatu

if ,1

if ,0



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Special Functions

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12



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Special Functions • Functions with conditional definitions can be represented by using unit step function :

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12



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• If

UNIT-I

12

(((( )))) <<<<<<<<

====otherwise ,0

if , btaEtf



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• If

• then, f(t) = E(u(t − a) − u(t − a))

UNIT-I

12

(((( )))) <<<<<<<<

====otherwise ,0

if , btaEtf



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• If

• then, f(t) = E(u(t − a) − u(t − a))

• The graph such difference of 2 unit step functions is

UNIT-I

12

(((( )))) <<<<<<<<

====otherwise ,0

if , btaEtf



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• If

• then, f(t) = E(u(t − a) − u(t − a))

• The graph such difference of 2 unit step functions is

UNIT-I

12

(((( )))) <<<<<<<<

====otherwise ,0

if , btaEtf

(((( ))))(((( ))))s

eatuL

as−−−−

====−−−−



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Special Functions

UNIT-I

13



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Special Functions • For a large force acting for a short time, the product of force and time is called impulse.

UNIT-I

13



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• Unit Impulse Function : The limiting value :

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13



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



UNIT-I

13

(((( ))))

→→→→++++<<<<<<<<====−−−−

otherwise ,0

0 as if ,1

εεεδ

ataat



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• The hard core value :

UNIT-I

13

(((( ))))

→→→→++++<<<<<<<<====−−−−

otherwise ,0

0 as if ,1

εεεδ

ataat



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




UNIT-I

13

(((( ))))

→→→→++++<<<<<<<<====−−−−

otherwise ,0

0 as if ,1

εεεδ

ataat

(((( )))) ====∞∞∞∞

====−−−−otherwise ,0

if , atatδ



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• The area under the graph of unit impulse function is 1.

UNIT-I

13

(((( ))))

→→→→++++<<<<<<<<====−−−−

otherwise ,0

0 as if ,1

εεεδ

ataat

(((( )))) ====∞∞∞∞

====−−−−otherwise ,0

if , atatδ



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Special Functions

UNIT-I

14



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Special Functions • The area under the graph of unit impulse function is 1, i.e.,

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14



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-I

14

(((( )))) 1====−−−−∫∫∫∫∞∞∞∞

0

δ dtat



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-I

14

(((( )))) 1====−−−−∫∫∫∫∞∞∞∞

0

δ dtat

(((( )))) )()( afdtattf ====−−−−∫∫∫∫∞∞∞∞

0

δ



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-I

14

(((( )))) 1====−−−−∫∫∫∫∞∞∞∞

0

δ dtat

(((( )))) )()( afdtattf ====−−−−∫∫∫∫∞∞∞∞

0

δ

(((( )))) )(')( afdtattf −−−−====−−−−∫∫∫∫∞∞∞∞

0

δ'



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-I

14

(((( )))) 1====−−−−∫∫∫∫∞∞∞∞

0

δ dtat

(((( )))) )()( afdtattf ====−−−−∫∫∫∫∞∞∞∞

0

δ

(((( )))) )(')( afdtattf −−−−====−−−−∫∫∫∫∞∞∞∞

0

δ'

(((( ))))(((( )))) aseatL −−−−====−−−−δ



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Properties of Laplace Transforms

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15



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Properties of Laplace Transforms • Linearity Property :

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15



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


�L[af1(t) + bf2(t)] = aL[f1(t)] + bL[f2(t)].

UNIT-I

15



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• First Shifting Theorem :

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15



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• First Shifting Theorem : �L[eatf(t)] = F(s − a)

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15



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• Laplace Transform of Derivative :

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15



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• Laplace Transform of Derivative : �L[f(n)(t)] = snF(s) - sn-1f(0) - sn-2f’(0)- ⋯⋯⋯⋯ -f(n-1)(0).

UNIT-I

15



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• Laplace Transform of Derivative : �L[f(n)(t)] = snF(s) - sn-1f(0) - sn-2f’(0)- ⋯⋯⋯⋯ -f(n-1)(0).

• Laplace Transform of Integral :

UNIT-I

15

)(1

)(0

sFs

dttfLt

====

∫∫∫∫



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-I

16



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Properties of Laplace Transforms • Laplace Transform of Multiplication by t :

UNIT-I

16



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-I

16

[[[[ ]]]] )()1()( sFds

dtftL

n

nnn −−−−====



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• Laplace Transform of Division by t :

UNIT-I

16

[[[[ ]]]] )()1()( sFds

dtftL

n

nnn −−−−====



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



UNIT-I

16

[[[[ ]]]] )()1()( sFds

dtftL

n

nnn −−−−====

∫∫∫∫∞∞∞∞

====

s

dssFt

tfL )(

)(



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Second Shifting Theorem :

UNIT-I

16

[[[[ ]]]] )()1()( sFds

dtftL

n

nnn −−−−====

∫∫∫∫∞∞∞∞

====

s

dssFt

tfL )(

)(



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Second Shifting Theorem : L[u(t − a)f(t − a)] = e−asF(s)

UNIT-I

16

[[[[ ]]]] )()1()( sFds

dtftL

n

nnn −−−−====

∫∫∫∫∞∞∞∞

====

s

dssFt

tfL )(

)(



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Second Shifting Theorem : L[u(t − a)f(t − a)] = e−asF(s)

L[u(t − a)f(t)] = e−asL[f(t + a)] (alternate form)

UNIT-I

16

[[[[ ]]]] )()1()( sFds

dtftL

n

nnn −−−−====

∫∫∫∫∞∞∞∞

====

s

dssFt

tfL )(

)(



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-I

17



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Properties of Laplace Transforms • Laplace Transform of Periodic Function with Period T :

UNIT-I

17



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-I

17

sT

T

st

e

dttfe

tfL −−−−

−−−−

−−−−====∫∫∫∫

1

)(

)]([ 0



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• Convolution Theorem :

UNIT-I

17

sT

T

st

e

dttfe

tfL −−−−

−−−−

−−−−====∫∫∫∫

1

)(

)]([ 0



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• Convolution Theorem : If L[f1(t)] = F1(s) and L[f2(t)] = F2(s), then

UNIT-I

17

sT

T

st

e

dttfe

tfL −−−−

−−−−

−−−−====∫∫∫∫

1

)(

)]([ 0



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



UNIT-I

17

sT

T

st

e

dttfe

tfL −−−−

−−−−

−−−−====∫∫∫∫

1

)(

)]([ 0

)()()()( 21

0

21 sFsFdtxtfxfLt

====

−−−−∫∫∫∫



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Introduction

UNIT-II

18



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Introduction • If f(t) is defined and continuous for all non-negative values and Laplace transform of f(t) is L[f(t)] or F(s), then inverse Laplace transform of F(s) is f(t).

UNIT-II

18



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• We use indirect form of inverse Laplace transform.

UNIT-II

18



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• That is

UNIT-II

18



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• That is • L[f(t)] = F(s) ⇔ L−1[F(s)] = f(t).

UNIT-II

18



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-II

19



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-II

19

111 ====

−−−−

sL



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-II

19

111 ====

−−−−

sL

aseas

L at >>>>====

−−−−−−−− for ,

11



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-II

19

111 ====

−−−−

sL

aseas

L at >>>>====

−−−−−−−− for ,

11

0 for ,sin22

1 >>>>====

++++−−−− sat

as

aL



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-II

19

111 ====

−−−−

sL

aseas

L at >>>>====

−−−−−−−− for ,

11

0 for ,sin22

1 >>>>====

++++−−−− sat

as

aL

0 for ,cos22

1 >>>>====

++++−−−− sat

as

sL



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-II

19

111 ====

−−−−

sL

aseas

L at >>>>====

−−−−−−−− for ,

11

0 for ,sin22

1 >>>>====

++++−−−− sat

as

aL

0 for ,cos22

1 >>>>====

++++−−−− sat

as

sL

22

22

1 for ,sinh asatas

aL >>>>====

−−−−−−−−



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-II

19

111 ====

−−−−

sL

aseas

L at >>>>====

−−−−−−−− for ,

11

0 for ,sin22

1 >>>>====

++++−−−− sat

as

aL

0 for ,cos22

1 >>>>====

++++−−−− sat

as

sL

22

22

1 for ,sinh asatas

aL >>>>====

−−−−−−−−

22

22

1 for ,cosh asatas

sL >>>>====

−−−−

−−−−



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-II

20



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-II

20

n

nt

s

nL ====

++++++++

−−−−1

1 1|



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-II

20

n

nt

s

nL ====

++++++++

−−−−1

1 1|

(((( )))) 0 for ,sin22

1 >>>>====

++++−−−−−−−− sbte

bas

bL at



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-II

20

n

nt

s

nL ====

++++++++

−−−−1

1 1|

(((( )))) 0 for ,sin22

1 >>>>====

++++−−−−−−−− sbte

bas

bL at

(((( )))) 0 for ,cos22

1 >>>>====

++++−−−−−−−−−−−− sbte

bas

asL at



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-II

20

n

nt

s

nL ====

++++++++

−−−−1

1 1|

(((( )))) 0 for ,sin22

1 >>>>====

++++−−−−−−−− sbte

bas

bL at

(((( )))) 0 for ,cos22

1 >>>>====

++++−−−−−−−−−−−− sbte

bas

asL at

(((( ))))22

221 for ,sinh asbte

bas

bL at >>>>====

−−−−−−−−−−−−



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-II

20

n

nt

s

nL ====

++++++++

−−−−1

1 1|

(((( )))) 0 for ,sin22

1 >>>>====

++++−−−−−−−− sbte

bas

bL at

(((( )))) 0 for ,cos22

1 >>>>====

++++−−−−−−−−−−−− sbte

bas

asL at

(((( ))))22

221 for ,sinh asbte

bas

bL at >>>>====

−−−−−−−−−−−−

(((( ))))22

221 for ,cosh asbte

bas

asL at >>>>====

−−−−−−−−−−−−−−−−



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-II

20

n

nt

s

nL ====

++++++++

−−−−1

1 1|

(((( )))) 0 for ,sin22

1 >>>>====

++++−−−−−−−− sbte

bas

bL at

(((( )))) 0 for ,cos22

1 >>>>====

++++−−−−−−−−−−−− sbte

bas

asL at

(((( ))))22

221 for ,sinh asbte

bas

bL at >>>>====

−−−−−−−−−−−−

(((( ))))22

221 for ,cosh asbte

bas

asL at >>>>====

−−−−−−−−−−−−−−−−

(((( ))))nat

n teas

nL ====

−−−−++++

++++−−−−

11 1|



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Inverse Laplace Transform Properties

UNIT-II

21



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Inverse Laplace Transform Properties • Linearity Property :

UNIT-II

21



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


�L−1[aF1(s) + bF2(s)] = aL−1[F1(s)] + bL

−1 [F2(s)].

UNIT-II

21



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


�L−1[aF1(s) + bF2(s)] = aL−1[F1(s)] + bL

−1 [F2(s)].

• First Shifting Theorem :

UNIT-II

21



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


�L−1[aF1(s) + bF2(s)] = aL−1[F1(s)] + bL

−1 [F2(s)].

• First Shifting Theorem : �L−1[F(s + a)] = e−atf(t)

UNIT-II

21



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


�L−1[aF1(s) + bF2(s)] = aL−1[F1(s)] + bL

−1 [F2(s)].


• Inverse Laplace Transform of multiplication by s :

UNIT-II

21

)()0()()]([1 tftfdt

dssFL δ++++====−−−−



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


�L−1[aF1(s) + bF2(s)] = aL−1[F1(s)] + bL

−1 [F2(s)].


• Inverse Laplace Transform of multiplication by s :

• Inverse Laplace Transform of division by s :

UNIT-II

21

)()0()()]([1 tftfdt

dssFL δ++++====−−−−

∫∫∫∫====

−−−−t

dttfs

sFL

0

1 )()(



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-II

22



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Inverse Laplace Transform Properties • Inverse Laplace Transform of Derivative :

UNIT-II

22



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-II

22

)()(1 ttfsFds

dL −−−−====

−−−−



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• Inverse Laplace Transform of Integral :

UNIT-II

22

)()(1 ttfsFds

dL −−−−====

−−−−



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



UNIT-II

22

)()(1 ttfsFds

dL −−−−====

−−−−

t

tfdssFL

s

)()(1 ====

∫∫∫∫∞∞∞∞

−−−−



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Second Shifting Theorem :

UNIT-II

22

)()(1 ttfsFds

dL −−−−====

−−−−

t

tfdssFL

s

)()(1 ====

∫∫∫∫∞∞∞∞

−−−−



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Second Shifting Theorem : L−1[e−asF(s)] = f(t − a)u(t − a).

UNIT-II

22

)()(1 ttfsFds

dL −−−−====

−−−−

t

tfdssFL

s

)()(1 ====

∫∫∫∫∞∞∞∞

−−−−



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-II

23



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Inverse Laplace Transform Properties • If f(t) is periodic with period T and L[f(t)] = F(s), then :

UNIT-II

23



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-II

23

(((( )))) (((( ))))sFedttfeL sT

T

st −−−−−−−−−−−− −−−−====

∫∫∫∫ 1)(0

1



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• Convolution Theorem :

UNIT-II

23

(((( )))) (((( ))))sFedttfeL sT

T

st −−−−−−−−−−−− −−−−====

∫∫∫∫ 1)(0

1



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



UNIT-II

23

(((( )))) (((( ))))sFedttfeL sT

T

st −−−−−−−−−−−− −−−−====

∫∫∫∫ 1)(0

1



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



UNIT-II

23

(((( )))) (((( ))))sFedttfeL sT

T

st −−−−−−−−−−−− −−−−====

∫∫∫∫ 1)(0

1

[[[[ ]]]] ∫∫∫∫ −−−−====−−−−t

dtxtfxfsFsFL0

21211 )()()()(



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Solution of Differential Equations

UNIT-II

24



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Solution of Differential Equations • Differential equations with boundary/initial conditions can be solved by coupled use of Laplace and inverse Laplace transforms

UNIT-II

24



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• Systems of simultaneous differential equations with similar conditions can also be solved by these techniques.

UNIT-II

24



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Steps in obtaining solutions :

UNIT-II

24



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Steps in obtaining solutions : �Take Laplace transform on both sides of equation(s).

UNIT-II

24



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




�Substitute boundary or initial values.

UNIT-II

24



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





�Rearrange the terms.

UNIT-II

24



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





�Rearrange the terms.

�Take inverse Laplace transform to get solution(s).

UNIT-II

24



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Introduction

UNIT-III

25



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Introduction • Integral transforms help in solving the differential equations with boundary conditions without finding the general solution and the values of the arbitrary constants.

UNIT-III

25



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• There are various integral transforms.

UNIT-III

25



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• There are various integral transforms. • The choice of a particular integral transform in solving a differential equation depends upon its own nature and the boundary or initial conditions prescribed for it.

UNIT-III

25



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• There are various integral transforms. • The choice of a particular integral transform in solving a differential equation depends upon its own nature and the boundary or initial conditions prescribed for it.

• In every integral transform, the function is multiplied by a function called kernel and integrated between specific limts.

UNIT-III

25



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Various Integral Transforms

UNIT-III

26



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Various Integral Transforms 1. Laplace Transform :

UNIT-III

26



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

26

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

−−−−========0

)()( dxxfesFxfL sx



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


There is no explicit formula for its inverse.

UNIT-III

26

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

−−−−========0

)()( dxxfesFxfL sx



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



2. Fourier Complex Transform :

UNIT-III

26

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

−−−−========0

)()( dxxfesFxfL sx



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




UNIT-III

26

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

−−−−========0

)()( dxxfesFxfL sx

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

∞∞∞∞−−−−

======== dxxfesFxfF isx )(2

1)(

π



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




Inverse Fourier complex transform is given by

UNIT-III

26

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

−−−−========0

)()( dxxfesFxfL sx

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

∞∞∞∞−−−−


1)(

π



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




Inverse Fourier complex transform is given by

UNIT-III

26

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

−−−−========0

)()( dxxfesFxfL sx

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

∞∞∞∞−−−−


1)(

π

∫∫∫∫∞∞∞∞

∞∞∞∞−−−−

−−−−==== dssFexf isx )(2

1)(

π



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

27



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Various Integral Transforms 3. Fourier Sine Transform :

UNIT-III

27



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

27

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

========0

sin)(2

)( sxdxxfsFxfF ssπ



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


Inversion :

UNIT-III

27

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

========0

sin)(2

)( sxdxxfsFxfF ssπ



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


Inversion :

UNIT-III

27

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

========0

sin)(2

)( sxdxxfsFxfF ssπ

∫∫∫∫∞∞∞∞

====0

sin)(2

)( sxdssFxf sπ



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


Inversion :

4. Fourier Cosine Transform :

UNIT-III

27

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

========0

sin)(2

)( sxdxxfsFxfF ssπ

∫∫∫∫∞∞∞∞

====0

sin)(2

)( sxdssFxf sπ



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


Inversion :


UNIT-III

27

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

========0

sin)(2

)( sxdxxfsFxfF ssπ

∫∫∫∫∞∞∞∞

====0

sin)(2

)( sxdssFxf sπ

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

========0

cos)(2

)( sxdxxfsFxfF ccπ



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


Inversion :


Inversion :

UNIT-III

27

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

========0

sin)(2

)( sxdxxfsFxfF ssπ

∫∫∫∫∞∞∞∞

====0

sin)(2

)( sxdssFxf sπ

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

========0

cos)(2

)( sxdxxfsFxfF ccπ



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


Inversion :


Inversion :

UNIT-III

27

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

========0

sin)(2

)( sxdxxfsFxfF ssπ

∫∫∫∫∞∞∞∞

====0

sin)(2

)( sxdssFxf sπ

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

========0

cos)(2

)( sxdxxfsFxfF ccπ

∫∫∫∫∞∞∞∞

====0

cos)(2

)( sxdssFxf cπ



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

28



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Various Integral Transforms 5. Hankel Transform :

UNIT-III

28



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

28

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

========0

)()()( dxxfsxxJsFxfH n



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


Inversion :

UNIT-III

28

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

========0




॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


Inversion :

UNIT-III

28

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

========0


∫∫∫∫∞∞∞∞

====0

)()()( dssFsxsJxf n



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


Inversion :

6. Hilbert Transform :

UNIT-III

28

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

========0


∫∫∫∫∞∞∞∞

====0

)()()( dssFsxsJxf n



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


Inversion :


UNIT-III

28

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

========0


∫∫∫∫∞∞∞∞

====0

)()()( dssFsxsJxf n

(((( )))) ∫∫∫∫∞∞∞∞

∞∞∞∞−−−− −−−−==== dx

xs

xfsF

)(1

π



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


Inversion :


Inversion :

UNIT-III

28

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

========0


∫∫∫∫∞∞∞∞

====0

)()()( dssFsxsJxf n

(((( )))) ∫∫∫∫∞∞∞∞

∞∞∞∞−−−− −−−−==== dx

xs

xfsF

)(1

π



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


Inversion :


Inversion :

UNIT-III

28

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

========0


∫∫∫∫∞∞∞∞

====0

)()()( dssFsxsJxf n

(((( )))) ∫∫∫∫∞∞∞∞

∞∞∞∞−−−− −−−−==== dx

xs

xfsF

)(1

π

∫∫∫∫∞∞∞∞

∞∞∞∞−−−− −−−−−−−−==== ds

xs

sFxf

)(1)(

π



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

29



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Various Integral Transforms 7. Mellin Transform :

UNIT-III

29



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

29

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

−−−−========0

1 )()( dxxfxsFxfM s



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



UNIT-III

29

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

−−−−========0

1 )()( dxxfxsFxfM s



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Out of the seven integral transforms introduced here, two do not have explicit formulae for their inverse transforms.

UNIT-III

29

[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞

−−−−========0

1 )()( dxxfxsFxfM s



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Various Integrals

UNIT-III

30



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Various Integrals 1. Fourier Integral Theorem :

UNIT-III

30



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

30

(((( ))))∫∫∫∫ ∫∫∫∫∞∞∞∞ ∞∞∞∞

∞∞∞∞−−−−

−−−−====0

cos)(1

)( dtduxtutfxfπ



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


2. Fourier Sine Integral :

UNIT-III

30

(((( ))))∫∫∫∫ ∫∫∫∫∞∞∞∞ ∞∞∞∞

∞∞∞∞−−−−

−−−−====0

cos)(1

)( dtduxtutfxfπ



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



If f(x) is an odd function, then it has Fourier sine integral given by

UNIT-III

30

(((( ))))∫∫∫∫ ∫∫∫∫∞∞∞∞ ∞∞∞∞

∞∞∞∞−−−−

−−−−====0

cos)(1

)( dtduxtutfxfπ



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



If f(x) is an odd function, then it has Fourier sine integral given by

UNIT-III

30

(((( ))))∫∫∫∫ ∫∫∫∫∞∞∞∞ ∞∞∞∞

∞∞∞∞−−−−

−−−−====0

cos)(1

)( dtduxtutfxfπ

∫∫∫∫ ∫∫∫∫∞∞∞∞ ∞∞∞∞

====0 0

sin)(sin2

)( utdttfuxduxfπ



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Various Integrals

UNIT-III

31



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Various Integrals 3. Fourier Cosine Integral :

UNIT-III

31



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


If f(x) is an even function, then it has Fourier cosine integral given by

UNIT-III

31



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



UNIT-III

31

∫∫∫∫ ∫∫∫∫∞∞∞∞ ∞∞∞∞

====0 0

cos)(cos2

)( utdttfuxduxfπ



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



4. Fourier Complex Integral :

UNIT-III

31

∫∫∫∫ ∫∫∫∫∞∞∞∞ ∞∞∞∞

====0 0

cos)(cos2

)( utdttfuxduxfπ



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



4. Fourier Complex Integral :

UNIT-III

31

∫∫∫∫ ∫∫∫∫∞∞∞∞ ∞∞∞∞

====0 0

cos)(cos2

)( utdttfuxduxfπ

∫∫∫∫ ∫∫∫∫∞∞∞∞

∞∞∞∞−−−−

∞∞∞∞

∞∞∞∞−−−−

−−−−==== dtetfduexf iutiux )(1

)(2π



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Properties of Fourier Transform

UNIT-III

32



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Properties of Fourier Transform • Linearity Property :

UNIT-III

32



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Properties of Fourier Transform • Linearity Property : F[af1(x) + bf2(x)] = aF[f1(x)] + bF[f2(x)].

UNIT-III

32



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• Change of Scale Property :

UNIT-III

32



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



UNIT-III

32

[[[[ ]]]]

====a

sF

aaxfF

1)(



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Shifting Property :

UNIT-III

32

[[[[ ]]]]

====a

sF

aaxfF

1)(



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Shifting Property :

UNIT-III

32

[[[[ ]]]]

====a

sF

aaxfF

1)(

[[[[ ]]]] (((( ))))sFeaxfF isa====−−−− )(



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Properties of Fourier Transform

UNIT-III

33



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Properties of Fourier Transform • Modulation Theorem :

UNIT-III

33



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

33

[[[[ ]]]] (((( )))) (((( ))))[[[[ ]]]]asFasFaxxfF −−−−++++++++====2

1cos)(



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• Fourier Transform of Multiplication by x :

UNIT-III

33

[[[[ ]]]] (((( )))) (((( ))))[[[[ ]]]]asFasFaxxfF −−−−++++++++====2

1cos)(



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



UNIT-III

33

[[[[ ]]]] (((( )))) (((( ))))sFds

dixfxF

n

nnn −−−−====)(

[[[[ ]]]] (((( )))) (((( ))))[[[[ ]]]]asFasFaxxfF −−−−++++++++====2

1cos)(



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Fourier Transform of Derivative :

UNIT-III

33

[[[[ ]]]] (((( )))) (((( ))))sFds

dixfxF

n

nnn −−−−====)(

[[[[ ]]]] (((( )))) (((( ))))[[[[ ]]]]asFasFaxxfF −−−−++++++++====2

1cos)(



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Fourier Transform of Derivative : F[f '(x)] = isF(s) if f(x) → 0 as x → ±∞.

UNIT-III

33

[[[[ ]]]] (((( )))) (((( ))))sFds

dixfxF

n

nnn −−−−====)(

[[[[ ]]]] (((( )))) (((( ))))[[[[ ]]]]asFasFaxxfF −−−−++++++++====2

1cos)(



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• Fourier Transform of Integral :

UNIT-III

33

[[[[ ]]]] (((( )))) (((( ))))sFds

dixfxF

n

nnn −−−−====)(

[[[[ ]]]] (((( )))) (((( ))))[[[[ ]]]]asFasFaxxfF −−−−++++++++====2

1cos)(



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• Fourier Transform of Integral :

UNIT-III

33

[[[[ ]]]] (((( )))) (((( ))))sFds

dixfxF

n

nnn −−−−====)(

(((( ))))(((( ))))is

sFdxxfF

x

a−−−−

====

∫∫∫∫ )(

[[[[ ]]]] (((( )))) (((( ))))[[[[ ]]]]asFasFaxxfF −−−−++++++++====2

1cos)(



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Integral Transforms INTEGRAL TRANSFORMS

34

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INTEGRAL TRANSFORMS · B.A./B.Sc.(Mathematics)–3rd Year–6th Sem. E-CONTENT BY DR.N.A.P ANDE ॥॥॥॥ °ीह ïर ° °ीीहह ï ïरर °ीह ïर: :: : ॥॥॥॥