integral transforms · b.a./b.sc.(mathematics)–3rd year–6th sem. e-content by dr.n.a.p ande...
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![Page 1: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/1.jpg)
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 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: : : : ॥॥॥॥
Paper Details
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![Page 3: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/3.jpg)
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: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth Marathwada University, Nanded, India
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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: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth Marathwada University, Nanded, India
• Course : B.A./B.Sc.
PAPER DETAILS
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![Page 5: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/5.jpg)
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: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth Marathwada University, Nanded, India
• Course : B.A./B.Sc. • Subject : Mathematics
PAPER DETAILS
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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: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth Marathwada University, Nanded, India
• Course : B.A./B.Sc. • Subject : Mathematics • Year : 3rd
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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: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth Marathwada University, Nanded, India
• Course : B.A./B.Sc. • Subject : Mathematics • Year : 3rd • Semester : 6th
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![Page 8: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/8.jpg)
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: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth Marathwada University, Nanded, India
• Course : B.A./B.Sc. • Subject : Mathematics • Year : 3rd • Semester : 6th • Paper No. : 12(B.A.) / 17(B.Sc.)
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![Page 9: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/9.jpg)
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: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth Marathwada University, Nanded, India
• 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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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: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth Marathwada University, Nanded, India
• 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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![Page 11: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/11.jpg)
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: : : : ॥॥॥॥
Paper Details • University : Swami Ramanand Teerth Marathwada University, Nanded, India
• 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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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: : : : ॥॥॥॥
Syllabus
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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: : : : ॥॥॥॥
Syllabus • Unit-I : Laplace Transforms
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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: : : : ॥॥॥॥
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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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: : : : ॥॥॥॥
Syllabus
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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: : : : ॥॥॥॥
Syllabus • Unit-II : Inverse Laplace Transforms
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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: : : : ॥॥॥॥
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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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: : : : ॥॥॥॥
Syllabus
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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: : : : ॥॥॥॥
Syllabus • Unit-III : : Fourier Transforms
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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: : : : ॥॥॥॥
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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�Publisher : S. Chand & Company Ltd.
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• Scope :
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�Title : Advanced Engineering
Mathematics
�Author : H. K. Dass
�Publisher : S. Chand & Company Ltd.
� ISSN : 81-219-0345-9
• Scope : �Unit–I : 13.1 to 13.15, and 13.17 to 13.19
�Unit–II : 13.20 to 13.31
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�Title : Advanced Engineering
Mathematics
�Author : H. K. Dass
�Publisher : S. Chand & Company Ltd.
� ISSN : 81-219-0345-9
• Scope : �Unit–I : 13.1 to 13.15, and 13.17 to 13.19
�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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Introduction
UNIT-I
8
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Introduction • Laplace transforms help in solving the differential equations with boundary conditions without finding the general solution and the values of the arbitrary constants.
UNIT-I
8
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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
UNIT-I
8
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
−−−−========0
)()( dttfesFtfL st
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Important Formulae •
UNIT-I
9
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Important Formulae •
UNIT-I
9
(((( ))))s
L1
1 ====
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Important Formulae •
UNIT-I
9
(((( ))))s
L1
1 ====
(((( )))) asas
eL at >>>>−−−−
==== for ,1
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Important Formulae •
UNIT-I
9
(((( ))))s
L1
1 ====
(((( )))) asas
eL at >>>>−−−−
==== for ,1
(((( )))) 0 for ,sin22
>>>>++++
==== sas
aatL
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Important Formulae •
UNIT-I
9
(((( ))))s
L1
1 ====
(((( )))) asas
eL at >>>>−−−−
==== for ,1
(((( )))) 0 for ,sin22
>>>>++++
==== sas
aatL
(((( )))) 0 for ,cos22
>>>>++++
==== sas
satL
![Page 41: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/41.jpg)
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: : : : ॥॥॥॥
Important Formulae •
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 >>>>
−−−−====
![Page 42: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/42.jpg)
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: : : : ॥॥॥॥
Important Formulae •
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 >>>>
−−−−====
![Page 43: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/43.jpg)
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: : : : ॥॥॥॥
Important Formulae •
UNIT-I
10
![Page 44: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/44.jpg)
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: : : : ॥॥॥॥
Important Formulae •
UNIT-I
10
(((( ))))1
1|++++++++====
n
n
s
ntL
![Page 45: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/45.jpg)
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: : : : ॥॥॥॥
Important Formulae •
UNIT-I
10
(((( ))))1
1|++++++++====
n
n
s
ntL
(((( )))) (((( )))) 0 for ,sin22 >>>>
++++−−−−==== s
bas
bbteL at
![Page 46: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/46.jpg)
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: : : : ॥॥॥॥
Important Formulae •
UNIT-I
10
(((( ))))1
1|++++++++====
n
n
s
ntL
(((( )))) (((( )))) 0 for ,sin22 >>>>
++++−−−−==== s
bas
bbteL at
(((( )))) (((( )))) 0 for ,cos22 >>>>
++++−−−−−−−−==== s
bas
asbteL at
![Page 47: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/47.jpg)
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: : : : ॥॥॥॥
Important Formulae •
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 >>>>
−−−−−−−−====
![Page 48: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/48.jpg)
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: : : : ॥॥॥॥
Important Formulae •
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 >>>>
−−−−−−−−−−−−====
![Page 49: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/49.jpg)
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: : : : ॥॥॥॥
Important Formulae •
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
![Page 50: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/50.jpg)
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: : : : ॥॥॥॥
Special Functions
UNIT-I
11
![Page 51: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/51.jpg)
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: : : : ॥॥॥॥
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
![Page 52: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/52.jpg)
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: : : : ॥॥॥॥
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
(((( ))))
≥≥≥≥<<<<
====−−−−at
atatu
if ,1
if ,0
![Page 53: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/53.jpg)
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: : : : ॥॥॥॥
Special Functions • Unit Step Function : If a is any constant, the unit step function, denoted by u(t − a) is given by
• The graph of unit step function is
UNIT-I
11
(((( ))))
≥≥≥≥<<<<
====−−−−at
atatu
if ,1
if ,0
![Page 54: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/54.jpg)
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: : : : ॥॥॥॥
Special Functions • Unit Step Function : If a is any constant, the unit step function, denoted by u(t − a) is given by
• The graph of unit step function is
UNIT-I
11
(((( ))))
≥≥≥≥<<<<
====−−−−at
atatu
if ,1
if ,0
![Page 55: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/55.jpg)
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: : : : ॥॥॥॥
Special Functions
UNIT-I
12
![Page 56: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/56.jpg)
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: : : : ॥॥॥॥
Special Functions • Functions with conditional definitions can be represented by using unit step function :
UNIT-I
12
![Page 57: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/57.jpg)
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: : : : ॥॥॥॥
Special Functions • Functions with conditional definitions can be represented by using unit step function :
• If
UNIT-I
12
(((( )))) <<<<<<<<
====otherwise ,0
if , btaEtf
![Page 58: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/58.jpg)
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: : : : ॥॥॥॥
Special Functions • Functions with conditional definitions can be represented by using unit step function :
• If
• then, f(t) = E(u(t − a) − u(t − a))
UNIT-I
12
(((( )))) <<<<<<<<
====otherwise ,0
if , btaEtf
![Page 59: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/59.jpg)
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: : : : ॥॥॥॥
Special Functions • Functions with conditional definitions can be represented by using unit step function :
• 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
![Page 60: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/60.jpg)
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: : : : ॥॥॥॥
Special Functions • Functions with conditional definitions can be represented by using unit step function :
• 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−−−−
====−−−−
![Page 61: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/61.jpg)
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: : : : ॥॥॥॥
Special Functions
UNIT-I
13
![Page 62: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/62.jpg)
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: : : : ॥॥॥॥
Special Functions • For a large force acting for a short time, the product of force and time is called impulse.
UNIT-I
13
![Page 63: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/63.jpg)
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: : : : ॥॥॥॥
Special Functions • For a large force acting for a short time, the product of force and time is called impulse.
• Unit Impulse Function : The limiting value :
UNIT-I
13
![Page 64: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/64.jpg)
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: : : : ॥॥॥॥
Special Functions • For a large force acting for a short time, the product of force and time is called impulse.
• Unit Impulse Function : The limiting value :
UNIT-I
13
(((( ))))
→→→→++++<<<<<<<<====−−−−
otherwise ,0
0 as if ,1
εεεδ
ataat
![Page 65: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/65.jpg)
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: : : : ॥॥॥॥
Special Functions • For a large force acting for a short time, the product of force and time is called impulse.
• Unit Impulse Function : The limiting value :
• The hard core value :
UNIT-I
13
(((( ))))
→→→→++++<<<<<<<<====−−−−
otherwise ,0
0 as if ,1
εεεδ
ataat
![Page 66: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/66.jpg)
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: : : : ॥॥॥॥
Special Functions • For a large force acting for a short time, the product of force and time is called impulse.
• Unit Impulse Function : The limiting value :
• The hard core value :
UNIT-I
13
(((( ))))
→→→→++++<<<<<<<<====−−−−
otherwise ,0
0 as if ,1
εεεδ
ataat
(((( )))) ====∞∞∞∞
====−−−−otherwise ,0
if , atatδ
![Page 67: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/67.jpg)
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: : : : ॥॥॥॥
Special Functions • For a large force acting for a short time, the product of force and time is called impulse.
• Unit Impulse Function : The limiting value :
• The hard core value :
• 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
![Page 68: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/68.jpg)
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: : : : ॥॥॥॥
Special Functions
UNIT-I
14
![Page 69: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/69.jpg)
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: : : : ॥॥॥॥
Special Functions • The area under the graph of unit impulse function is 1, i.e.,
UNIT-I
14
![Page 70: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/70.jpg)
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: : : : ॥॥॥॥
Special Functions • The area under the graph of unit impulse function is 1, i.e.,
UNIT-I
14
(((( )))) 1====−−−−∫∫∫∫∞∞∞∞
0
δ dtat
![Page 71: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/71.jpg)
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: : : : ॥॥॥॥
Special Functions • The area under the graph of unit impulse function is 1, i.e.,
UNIT-I
14
(((( )))) 1====−−−−∫∫∫∫∞∞∞∞
0
δ dtat
(((( )))) )()( afdtattf ====−−−−∫∫∫∫∞∞∞∞
0
δ
![Page 72: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/72.jpg)
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: : : : ॥॥॥॥
Special Functions • The area under the graph of unit impulse function is 1, i.e.,
UNIT-I
14
(((( )))) 1====−−−−∫∫∫∫∞∞∞∞
0
δ dtat
(((( )))) )()( afdtattf ====−−−−∫∫∫∫∞∞∞∞
0
δ
(((( )))) )(')( afdtattf −−−−====−−−−∫∫∫∫∞∞∞∞
0
δ'
![Page 73: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/73.jpg)
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: : : : ॥॥॥॥
Special Functions • The area under the graph of unit impulse function is 1, i.e.,
UNIT-I
14
(((( )))) 1====−−−−∫∫∫∫∞∞∞∞
0
δ dtat
(((( )))) )()( afdtattf ====−−−−∫∫∫∫∞∞∞∞
0
δ
(((( )))) )(')( afdtattf −−−−====−−−−∫∫∫∫∞∞∞∞
0
δ'
(((( ))))(((( )))) aseatL −−−−====−−−−δ
![Page 74: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/74.jpg)
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: : : : ॥॥॥॥
Properties of Laplace Transforms
UNIT-I
15
![Page 75: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/75.jpg)
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: : : : ॥॥॥॥
Properties of Laplace Transforms • Linearity Property :
UNIT-I
15
![Page 76: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/76.jpg)
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: : : : ॥॥॥॥
Properties of Laplace Transforms • Linearity Property :
�L[af1(t) + bf2(t)] = aL[f1(t)] + bL[f2(t)].
UNIT-I
15
![Page 77: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/77.jpg)
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: : : : ॥॥॥॥
Properties of Laplace Transforms • Linearity Property :
�L[af1(t) + bf2(t)] = aL[f1(t)] + bL[f2(t)].
• First Shifting Theorem :
UNIT-I
15
![Page 78: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/78.jpg)
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: : : : ॥॥॥॥
Properties of Laplace Transforms • Linearity Property :
�L[af1(t) + bf2(t)] = aL[f1(t)] + bL[f2(t)].
• First Shifting Theorem : �L[eatf(t)] = F(s − a)
UNIT-I
15
![Page 79: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/79.jpg)
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: : : : ॥॥॥॥
Properties of Laplace Transforms • Linearity Property :
�L[af1(t) + bf2(t)] = aL[f1(t)] + bL[f2(t)].
• First Shifting Theorem : �L[eatf(t)] = F(s − a)
• Laplace Transform of Derivative :
UNIT-I
15
![Page 80: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/80.jpg)
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: : : : ॥॥॥॥
Properties of Laplace Transforms • Linearity Property :
�L[af1(t) + bf2(t)] = aL[f1(t)] + bL[f2(t)].
• First Shifting Theorem : �L[eatf(t)] = F(s − a)
• Laplace Transform of Derivative : �L[f(n)(t)] = snF(s) - sn-1f(0) - sn-2f’(0)- ⋯⋯⋯⋯ -f(n-1)(0).
UNIT-I
15
![Page 81: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/81.jpg)
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: : : : ॥॥॥॥
Properties of Laplace Transforms • Linearity Property :
�L[af1(t) + bf2(t)] = aL[f1(t)] + bL[f2(t)].
• First Shifting Theorem : �L[eatf(t)] = F(s − a)
• 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
====
∫∫∫∫
![Page 82: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/82.jpg)
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: : : : ॥॥॥॥
Properties of Laplace Transforms
UNIT-I
16
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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: : : : ॥॥॥॥
Properties of Laplace Transforms • Laplace Transform of Multiplication by t :
UNIT-I
16
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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: : : : ॥॥॥॥
Properties of Laplace Transforms • Laplace Transform of Multiplication by t :
UNIT-I
16
[[[[ ]]]] )()1()( sFds
dtftL
n
nnn −−−−====
![Page 85: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/85.jpg)
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: : : : ॥॥॥॥
Properties of Laplace Transforms • Laplace Transform of Multiplication by t :
• Laplace Transform of Division by t :
UNIT-I
16
[[[[ ]]]] )()1()( sFds
dtftL
n
nnn −−−−====
![Page 86: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/86.jpg)
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: : : : ॥॥॥॥
Properties of Laplace Transforms • Laplace Transform of Multiplication by t :
• Laplace Transform of Division by t :
UNIT-I
16
[[[[ ]]]] )()1()( sFds
dtftL
n
nnn −−−−====
∫∫∫∫∞∞∞∞
====
s
dssFt
tfL )(
)(
![Page 87: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/87.jpg)
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: : : : ॥॥॥॥
Properties of Laplace Transforms • Laplace Transform of Multiplication by t :
• Laplace Transform of Division by t :
• Second Shifting Theorem :
UNIT-I
16
[[[[ ]]]] )()1()( sFds
dtftL
n
nnn −−−−====
∫∫∫∫∞∞∞∞
====
s
dssFt
tfL )(
)(
![Page 88: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/88.jpg)
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: : : : ॥॥॥॥
Properties of Laplace Transforms • Laplace Transform of Multiplication by t :
• Laplace Transform of Division by t :
• Second Shifting Theorem : L[u(t − a)f(t − a)] = e−asF(s)
UNIT-I
16
[[[[ ]]]] )()1()( sFds
dtftL
n
nnn −−−−====
∫∫∫∫∞∞∞∞
====
s
dssFt
tfL )(
)(
![Page 89: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/89.jpg)
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: : : : ॥॥॥॥
Properties of Laplace Transforms • Laplace Transform of Multiplication by t :
• Laplace Transform of Division by t :
• 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 )(
)(
![Page 90: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/90.jpg)
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: : : : ॥॥॥॥
Properties of Laplace Transforms
UNIT-I
17
![Page 91: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/91.jpg)
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: : : : ॥॥॥॥
Properties of Laplace Transforms • Laplace Transform of Periodic Function with Period T :
UNIT-I
17
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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: : : : ॥॥॥॥
Properties of Laplace Transforms • Laplace Transform of Periodic Function with Period T :
UNIT-I
17
sT
T
st
e
dttfe
tfL −−−−
−−−−
−−−−====∫∫∫∫
1
)(
)]([ 0
![Page 93: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/93.jpg)
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: : : : ॥॥॥॥
Properties of Laplace Transforms • Laplace Transform of Periodic Function with Period T :
• Convolution Theorem :
UNIT-I
17
sT
T
st
e
dttfe
tfL −−−−
−−−−
−−−−====∫∫∫∫
1
)(
)]([ 0
![Page 94: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/94.jpg)
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: : : : ॥॥॥॥
Properties of Laplace Transforms • Laplace Transform of Periodic Function with Period T :
• 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
![Page 95: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/95.jpg)
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: : : : ॥॥॥॥
Properties of Laplace Transforms • Laplace Transform of Periodic Function with Period T :
• 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
)()()()( 21
0
21 sFsFdtxtfxfLt
====
−−−−∫∫∫∫
![Page 96: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/96.jpg)
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: : : : ॥॥॥॥
Introduction
UNIT-II
18
![Page 97: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/97.jpg)
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: : : : ॥॥॥॥
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
![Page 98: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/98.jpg)
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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: : : : ॥॥॥॥
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).
• We use indirect form of inverse Laplace transform.
UNIT-II
18
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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: : : : ॥॥॥॥
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).
• We use indirect form of inverse Laplace transform.
• That is
UNIT-II
18
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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: : : : ॥॥॥॥
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).
• We use indirect form of inverse Laplace transform.
• That is • L[f(t)] = F(s) ⇔ L−1[F(s)] = f(t).
UNIT-II
18
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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: : : : ॥॥॥॥
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Important Formulae •
UNIT-II
19
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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: : : : ॥॥॥॥
Important Formulae •
UNIT-II
19
111 ====
−−−−
sL
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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: : : : ॥॥॥॥
Important Formulae •
UNIT-II
19
111 ====
−−−−
sL
aseas
L at >>>>====
−−−−−−−− for ,
11
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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: : : : ॥॥॥॥
Important Formulae •
UNIT-II
19
111 ====
−−−−
sL
aseas
L at >>>>====
−−−−−−−− for ,
11
0 for ,sin22
1 >>>>====
++++−−−− sat
as
aL
![Page 105: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/105.jpg)
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: : : : ॥॥॥॥
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Important Formulae •
UNIT-II
19
111 ====
−−−−
sL
aseas
L at >>>>====
−−−−−−−− for ,
11
0 for ,sin22
1 >>>>====
++++−−−− sat
as
aL
0 for ,cos22
1 >>>>====
++++−−−− sat
as
sL
![Page 106: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/106.jpg)
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: : : : ॥॥॥॥
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Important Formulae •
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 >>>>====
−−−−−−−−
![Page 107: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/107.jpg)
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: : : : ॥॥॥॥
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Important Formulae •
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 >>>>====
−−−−
−−−−
![Page 108: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/108.jpg)
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: : : : ॥॥॥॥
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Important Formulae •
UNIT-II
20
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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: : : : ॥॥॥॥
Important Formulae •
UNIT-II
20
n
nt
s
nL ====
++++++++
−−−−1
1 1|
![Page 110: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/110.jpg)
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: : : : ॥॥॥॥
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Important Formulae •
UNIT-II
20
n
nt
s
nL ====
++++++++
−−−−1
1 1|
(((( )))) 0 for ,sin22
1 >>>>====
++++−−−−−−−− sbte
bas
bL at
![Page 111: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/111.jpg)
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: : : : ॥॥॥॥
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Important Formulae •
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
![Page 112: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/112.jpg)
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: : : : ॥॥॥॥
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Important Formulae •
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 >>>>====
−−−−−−−−−−−−
![Page 113: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/113.jpg)
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: : : : ॥॥॥॥
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Important Formulae •
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 >>>>====
−−−−−−−−−−−−−−−−
![Page 114: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/114.jpg)
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: : : : ॥॥॥॥
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Important Formulae •
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|
![Page 115: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/115.jpg)
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: : : : ॥॥॥॥
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inverse Laplace Transform Properties
UNIT-II
21
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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: : : : ॥॥॥॥
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inverse Laplace Transform Properties • Linearity Property :
UNIT-II
21
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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: : : : ॥॥॥॥
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Inverse Laplace Transform Properties • Linearity Property :
�L−1[aF1(s) + bF2(s)] = aL−1[F1(s)] + bL
−1 [F2(s)].
UNIT-II
21
![Page 118: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/118.jpg)
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: : : : ॥॥॥॥
Inverse Laplace Transform Properties • Linearity Property :
�L−1[aF1(s) + bF2(s)] = aL−1[F1(s)] + bL
−1 [F2(s)].
• First Shifting Theorem :
UNIT-II
21
![Page 119: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/119.jpg)
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: : : : ॥॥॥॥
Inverse Laplace Transform Properties • Linearity Property :
�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
![Page 120: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/120.jpg)
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: : : : ॥॥॥॥
Inverse Laplace Transform Properties • Linearity Property :
�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)
• Inverse Laplace Transform of multiplication by s :
UNIT-II
21
)()0()()]([1 tftfdt
dssFL δ++++====−−−−
![Page 121: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/121.jpg)
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: : : : ॥॥॥॥
Inverse Laplace Transform Properties • Linearity Property :
�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)
• 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 )()(
![Page 122: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/122.jpg)
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: : : : ॥॥॥॥
Inverse Laplace Transform Properties
UNIT-II
22
![Page 123: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/123.jpg)
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: : : : ॥॥॥॥
Inverse Laplace Transform Properties • Inverse Laplace Transform of Derivative :
UNIT-II
22
![Page 124: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/124.jpg)
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: : : : ॥॥॥॥
Inverse Laplace Transform Properties • Inverse Laplace Transform of Derivative :
UNIT-II
22
)()(1 ttfsFds
dL −−−−====
−−−−
![Page 125: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/125.jpg)
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: : : : ॥॥॥॥
Inverse Laplace Transform Properties • Inverse Laplace Transform of Derivative :
• Inverse Laplace Transform of Integral :
UNIT-II
22
)()(1 ttfsFds
dL −−−−====
−−−−
![Page 126: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/126.jpg)
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: : : : ॥॥॥॥
Inverse Laplace Transform Properties • Inverse Laplace Transform of Derivative :
• Inverse Laplace Transform of Integral :
UNIT-II
22
)()(1 ttfsFds
dL −−−−====
−−−−
t
tfdssFL
s
)()(1 ====
∫∫∫∫∞∞∞∞
−−−−
![Page 127: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/127.jpg)
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: : : : ॥॥॥॥
Inverse Laplace Transform Properties • Inverse Laplace Transform of Derivative :
• Inverse Laplace Transform of Integral :
• Second Shifting Theorem :
UNIT-II
22
)()(1 ttfsFds
dL −−−−====
−−−−
t
tfdssFL
s
)()(1 ====
∫∫∫∫∞∞∞∞
−−−−
![Page 128: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/128.jpg)
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: : : : ॥॥॥॥
Inverse Laplace Transform Properties • Inverse Laplace Transform of Derivative :
• Inverse Laplace Transform of Integral :
• Second Shifting Theorem : L−1[e−asF(s)] = f(t − a)u(t − a).
UNIT-II
22
)()(1 ttfsFds
dL −−−−====
−−−−
t
tfdssFL
s
)()(1 ====
∫∫∫∫∞∞∞∞
−−−−
![Page 129: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/129.jpg)
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: : : : ॥॥॥॥
Inverse Laplace Transform Properties
UNIT-II
23
![Page 130: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/130.jpg)
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: : : : ॥॥॥॥
Inverse Laplace Transform Properties • If f(t) is periodic with period T and L[f(t)] = F(s), then :
UNIT-II
23
![Page 131: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/131.jpg)
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: : : : ॥॥॥॥
Inverse Laplace Transform Properties • If f(t) is periodic with period T and L[f(t)] = F(s), then :
UNIT-II
23
(((( )))) (((( ))))sFedttfeL sT
T
st −−−−−−−−−−−− −−−−====
∫∫∫∫ 1)(0
1
![Page 132: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/132.jpg)
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: : : : ॥॥॥॥
Inverse Laplace Transform Properties • If f(t) is periodic with period T and L[f(t)] = F(s), then :
• Convolution Theorem :
UNIT-II
23
(((( )))) (((( ))))sFedttfeL sT
T
st −−−−−−−−−−−− −−−−====
∫∫∫∫ 1)(0
1
![Page 133: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/133.jpg)
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: : : : ॥॥॥॥
Inverse Laplace Transform Properties • If f(t) is periodic with period T and L[f(t)] = F(s), then :
• Convolution Theorem : If L[f1(t)] = F1(s) and L[f2(t)] = F2(s), then
UNIT-II
23
(((( )))) (((( ))))sFedttfeL sT
T
st −−−−−−−−−−−− −−−−====
∫∫∫∫ 1)(0
1
![Page 134: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/134.jpg)
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: : : : ॥॥॥॥
Inverse Laplace Transform Properties • If f(t) is periodic with period T and L[f(t)] = F(s), then :
• Convolution Theorem : If L[f1(t)] = F1(s) and L[f2(t)] = F2(s), then
UNIT-II
23
(((( )))) (((( ))))sFedttfeL sT
T
st −−−−−−−−−−−− −−−−====
∫∫∫∫ 1)(0
1
[[[[ ]]]] ∫∫∫∫ −−−−====−−−−t
dtxtfxfsFsFL0
21211 )()()()(
![Page 135: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/135.jpg)
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: : : : ॥॥॥॥
Solution of Differential Equations
UNIT-II
24
![Page 136: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/136.jpg)
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: : : : ॥॥॥॥
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
![Page 137: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/137.jpg)
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: : : : ॥॥॥॥
Solution of Differential Equations • Differential equations with boundary/initial conditions can be solved by coupled use of Laplace and inverse Laplace transforms
• Systems of simultaneous differential equations with similar conditions can also be solved by these techniques.
UNIT-II
24
![Page 138: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/138.jpg)
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: : : : ॥॥॥॥
Solution of Differential Equations • Differential equations with boundary/initial conditions can be solved by coupled use of Laplace and inverse Laplace transforms
• Systems of simultaneous differential equations with similar conditions can also be solved by these techniques.
• Steps in obtaining solutions :
UNIT-II
24
![Page 139: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/139.jpg)
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: : : : ॥॥॥॥
Solution of Differential Equations • Differential equations with boundary/initial conditions can be solved by coupled use of Laplace and inverse Laplace transforms
• Systems of simultaneous differential equations with similar conditions can also be solved by these techniques.
• Steps in obtaining solutions : �Take Laplace transform on both sides of equation(s).
UNIT-II
24
![Page 140: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/140.jpg)
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: : : : ॥॥॥॥
Solution of Differential Equations • Differential equations with boundary/initial conditions can be solved by coupled use of Laplace and inverse Laplace transforms
• Systems of simultaneous differential equations with similar conditions can also be solved by these techniques.
• Steps in obtaining solutions : �Take Laplace transform on both sides of equation(s).
�Substitute boundary or initial values.
UNIT-II
24
![Page 141: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/141.jpg)
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: : : : ॥॥॥॥
Solution of Differential Equations • Differential equations with boundary/initial conditions can be solved by coupled use of Laplace and inverse Laplace transforms
• Systems of simultaneous differential equations with similar conditions can also be solved by these techniques.
• Steps in obtaining solutions : �Take Laplace transform on both sides of equation(s).
�Substitute boundary or initial values.
�Rearrange the terms.
UNIT-II
24
![Page 142: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/142.jpg)
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: : : : ॥॥॥॥
Solution of Differential Equations • Differential equations with boundary/initial conditions can be solved by coupled use of Laplace and inverse Laplace transforms
• Systems of simultaneous differential equations with similar conditions can also be solved by these techniques.
• Steps in obtaining solutions : �Take Laplace transform on both sides of equation(s).
�Substitute boundary or initial values.
�Rearrange the terms.
�Take inverse Laplace transform to get solution(s).
UNIT-II
24
![Page 143: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/143.jpg)
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: : : : ॥॥॥॥
Introduction
UNIT-III
25
![Page 144: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/144.jpg)
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: : : : ॥॥॥॥
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
![Page 145: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/145.jpg)
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: : : : ॥॥॥॥
Introduction • Integral transforms help in solving the differential equations with boundary conditions without finding the general solution and the values of the arbitrary constants.
• There are various integral transforms.
UNIT-III
25
![Page 146: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/146.jpg)
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: : : : ॥॥॥॥
Introduction • Integral transforms help in solving the differential equations with boundary conditions without finding the general solution and the values of the arbitrary constants.
• 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
![Page 147: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/147.jpg)
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: : : : ॥॥॥॥
Introduction • Integral transforms help in solving the differential equations with boundary conditions without finding the general solution and the values of the arbitrary constants.
• 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
![Page 148: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/148.jpg)
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: : : : ॥॥॥॥
Various Integral Transforms
UNIT-III
26
![Page 149: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/149.jpg)
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: : : : ॥॥॥॥
Various Integral Transforms 1. Laplace Transform :
UNIT-III
26
![Page 150: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/150.jpg)
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: : : : ॥॥॥॥
Various Integral Transforms 1. Laplace Transform :
UNIT-III
26
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
−−−−========0
)()( dxxfesFxfL sx
![Page 151: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/151.jpg)
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: : : : ॥॥॥॥
Various Integral Transforms 1. Laplace Transform :
There is no explicit formula for its inverse.
UNIT-III
26
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
−−−−========0
)()( dxxfesFxfL sx
![Page 152: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/152.jpg)
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: : : : ॥॥॥॥
Various Integral Transforms 1. Laplace Transform :
There is no explicit formula for its inverse.
2. Fourier Complex Transform :
UNIT-III
26
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
−−−−========0
)()( dxxfesFxfL sx
![Page 153: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/153.jpg)
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: : : : ॥॥॥॥
Various Integral Transforms 1. Laplace Transform :
There is no explicit formula for its inverse.
2. Fourier Complex Transform :
UNIT-III
26
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
−−−−========0
)()( dxxfesFxfL sx
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
∞∞∞∞−−−−
======== dxxfesFxfF isx )(2
1)(
π
![Page 154: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/154.jpg)
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: : : : ॥॥॥॥
Various Integral Transforms 1. Laplace Transform :
There is no explicit formula for its inverse.
2. Fourier Complex Transform :
Inverse Fourier complex transform is given by
UNIT-III
26
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
−−−−========0
)()( dxxfesFxfL sx
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
∞∞∞∞−−−−
======== dxxfesFxfF isx )(2
1)(
π
![Page 155: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/155.jpg)
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: : : : ॥॥॥॥
Various Integral Transforms 1. Laplace Transform :
There is no explicit formula for its inverse.
2. Fourier Complex Transform :
Inverse Fourier complex transform is given by
UNIT-III
26
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
−−−−========0
)()( dxxfesFxfL sx
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
∞∞∞∞−−−−
======== dxxfesFxfF isx )(2
1)(
π
∫∫∫∫∞∞∞∞
∞∞∞∞−−−−
−−−−==== dssFexf isx )(2
1)(
π
![Page 156: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/156.jpg)
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: : : : ॥॥॥॥
Various Integral Transforms
UNIT-III
27
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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: : : : ॥॥॥॥
Various Integral Transforms 3. Fourier Sine Transform :
UNIT-III
27
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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: : : : ॥॥॥॥
Various Integral Transforms 3. Fourier Sine Transform :
UNIT-III
27
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
========0
sin)(2
)( sxdxxfsFxfF ssπ
![Page 159: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/159.jpg)
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: : : : ॥॥॥॥
Various Integral Transforms 3. Fourier Sine Transform :
Inversion :
UNIT-III
27
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
========0
sin)(2
)( sxdxxfsFxfF ssπ
![Page 160: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/160.jpg)
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: : : : ॥॥॥॥
Various Integral Transforms 3. Fourier Sine Transform :
Inversion :
UNIT-III
27
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
========0
sin)(2
)( sxdxxfsFxfF ssπ
∫∫∫∫∞∞∞∞
====0
sin)(2
)( sxdssFxf sπ
![Page 161: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/161.jpg)
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: : : : ॥॥॥॥
Various Integral Transforms 3. Fourier Sine Transform :
Inversion :
4. Fourier Cosine Transform :
UNIT-III
27
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
========0
sin)(2
)( sxdxxfsFxfF ssπ
∫∫∫∫∞∞∞∞
====0
sin)(2
)( sxdssFxf sπ
![Page 162: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/162.jpg)
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: : : : ॥॥॥॥
Various Integral Transforms 3. Fourier Sine Transform :
Inversion :
4. Fourier Cosine Transform :
UNIT-III
27
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
========0
sin)(2
)( sxdxxfsFxfF ssπ
∫∫∫∫∞∞∞∞
====0
sin)(2
)( sxdssFxf sπ
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
========0
cos)(2
)( sxdxxfsFxfF ccπ
![Page 163: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/163.jpg)
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: : : : ॥॥॥॥
Various Integral Transforms 3. Fourier Sine Transform :
Inversion :
4. Fourier Cosine Transform :
Inversion :
UNIT-III
27
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
========0
sin)(2
)( sxdxxfsFxfF ssπ
∫∫∫∫∞∞∞∞
====0
sin)(2
)( sxdssFxf sπ
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
========0
cos)(2
)( sxdxxfsFxfF ccπ
![Page 164: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/164.jpg)
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: : : : ॥॥॥॥
Various Integral Transforms 3. Fourier Sine Transform :
Inversion :
4. Fourier Cosine Transform :
Inversion :
UNIT-III
27
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
========0
sin)(2
)( sxdxxfsFxfF ssπ
∫∫∫∫∞∞∞∞
====0
sin)(2
)( sxdssFxf sπ
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
========0
cos)(2
)( sxdxxfsFxfF ccπ
∫∫∫∫∞∞∞∞
====0
cos)(2
)( sxdssFxf cπ
![Page 165: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/165.jpg)
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: : : : ॥॥॥॥
Various Integral Transforms
UNIT-III
28
![Page 166: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/166.jpg)
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: : : : ॥॥॥॥
Various Integral Transforms 5. Hankel Transform :
UNIT-III
28
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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: : : : ॥॥॥॥
Various Integral Transforms 5. Hankel Transform :
UNIT-III
28
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
========0
)()()( dxxfsxxJsFxfH n
![Page 168: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/168.jpg)
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: : : : ॥॥॥॥
Various Integral Transforms 5. Hankel Transform :
Inversion :
UNIT-III
28
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
========0
)()()( dxxfsxxJsFxfH n
![Page 169: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/169.jpg)
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: : : : ॥॥॥॥
Various Integral Transforms 5. Hankel Transform :
Inversion :
UNIT-III
28
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
========0
)()()( dxxfsxxJsFxfH n
∫∫∫∫∞∞∞∞
====0
)()()( dssFsxsJxf n
![Page 170: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/170.jpg)
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: : : : ॥॥॥॥
Various Integral Transforms 5. Hankel Transform :
Inversion :
6. Hilbert Transform :
UNIT-III
28
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
========0
)()()( dxxfsxxJsFxfH n
∫∫∫∫∞∞∞∞
====0
)()()( dssFsxsJxf n
![Page 171: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/171.jpg)
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: : : : ॥॥॥॥
Various Integral Transforms 5. Hankel Transform :
Inversion :
6. Hilbert Transform :
UNIT-III
28
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
========0
)()()( dxxfsxxJsFxfH n
∫∫∫∫∞∞∞∞
====0
)()()( dssFsxsJxf n
(((( )))) ∫∫∫∫∞∞∞∞
∞∞∞∞−−−− −−−−==== dx
xs
xfsF
)(1
π
![Page 172: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/172.jpg)
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: : : : ॥॥॥॥
Various Integral Transforms 5. Hankel Transform :
Inversion :
6. Hilbert Transform :
Inversion :
UNIT-III
28
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
========0
)()()( dxxfsxxJsFxfH n
∫∫∫∫∞∞∞∞
====0
)()()( dssFsxsJxf n
(((( )))) ∫∫∫∫∞∞∞∞
∞∞∞∞−−−− −−−−==== dx
xs
xfsF
)(1
π
![Page 173: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/173.jpg)
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: : : : ॥॥॥॥
Various Integral Transforms 5. Hankel Transform :
Inversion :
6. Hilbert Transform :
Inversion :
UNIT-III
28
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
========0
)()()( dxxfsxxJsFxfH n
∫∫∫∫∞∞∞∞
====0
)()()( dssFsxsJxf n
(((( )))) ∫∫∫∫∞∞∞∞
∞∞∞∞−−−− −−−−==== dx
xs
xfsF
)(1
π
∫∫∫∫∞∞∞∞
∞∞∞∞−−−− −−−−−−−−==== ds
xs
sFxf
)(1)(
π
![Page 174: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/174.jpg)
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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: : : : ॥॥॥॥
Various Integral Transforms
UNIT-III
29
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Various Integral Transforms 7. Mellin Transform :
UNIT-III
29
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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: : : : ॥॥॥॥
Various Integral Transforms 7. Mellin Transform :
UNIT-III
29
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
−−−−========0
1 )()( dxxfxsFxfM s
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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: : : : ॥॥॥॥
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Various Integral Transforms 7. Mellin Transform :
There is no explicit formula for its inverse.
UNIT-III
29
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
−−−−========0
1 )()( dxxfxsFxfM s
![Page 178: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/178.jpg)
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: : : : ॥॥॥॥
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Various Integral Transforms 7. Mellin Transform :
There is no explicit formula for its inverse.
• 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
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Various Integrals
UNIT-III
30
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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: : : : ॥॥॥॥
Various Integrals 1. Fourier Integral Theorem :
UNIT-III
30
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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: : : : ॥॥॥॥
Various Integrals 1. Fourier Integral Theorem :
UNIT-III
30
(((( ))))∫∫∫∫ ∫∫∫∫∞∞∞∞ ∞∞∞∞
∞∞∞∞−−−−
−−−−====0
cos)(1
)( dtduxtutfxfπ
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Various Integrals 1. Fourier Integral Theorem :
2. Fourier Sine Integral :
UNIT-III
30
(((( ))))∫∫∫∫ ∫∫∫∫∞∞∞∞ ∞∞∞∞
∞∞∞∞−−−−
−−−−====0
cos)(1
)( dtduxtutfxfπ
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Various Integrals 1. Fourier Integral Theorem :
2. Fourier Sine Integral :
If f(x) is an odd function, then it has Fourier sine integral given by
UNIT-III
30
(((( ))))∫∫∫∫ ∫∫∫∫∞∞∞∞ ∞∞∞∞
∞∞∞∞−−−−
−−−−====0
cos)(1
)( dtduxtutfxfπ
![Page 184: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/184.jpg)
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Various Integrals 1. Fourier Integral Theorem :
2. Fourier Sine Integral :
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
![Page 185: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/185.jpg)
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Various Integrals
UNIT-III
31
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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: : : : ॥॥॥॥
Various Integrals 3. Fourier Cosine Integral :
UNIT-III
31
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Various Integrals 3. Fourier Cosine Integral :
If f(x) is an even function, then it has Fourier cosine integral given by
UNIT-III
31
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Various Integrals 3. Fourier Cosine Integral :
If f(x) is an even function, then it has Fourier cosine integral given by
UNIT-III
31
∫∫∫∫ ∫∫∫∫∞∞∞∞ ∞∞∞∞
====0 0
cos)(cos2
)( utdttfuxduxfπ
![Page 189: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/189.jpg)
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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: : : : ॥॥॥॥
Various Integrals 3. Fourier Cosine Integral :
If f(x) is an even function, then it has Fourier cosine integral given by
4. Fourier Complex Integral :
UNIT-III
31
∫∫∫∫ ∫∫∫∫∞∞∞∞ ∞∞∞∞
====0 0
cos)(cos2
)( utdttfuxduxfπ
![Page 190: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/190.jpg)
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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: : : : ॥॥॥॥
Various Integrals 3. Fourier Cosine Integral :
If f(x) is an even function, then it has Fourier cosine integral given by
4. Fourier Complex Integral :
UNIT-III
31
∫∫∫∫ ∫∫∫∫∞∞∞∞ ∞∞∞∞
====0 0
cos)(cos2
)( utdttfuxduxfπ
∫∫∫∫ ∫∫∫∫∞∞∞∞
∞∞∞∞−−−−
∞∞∞∞
∞∞∞∞−−−−
−−−−==== dtetfduexf iutiux )(1
)(2π
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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: : : : ॥॥॥॥
Properties of Fourier Transform
UNIT-III
32
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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥
Properties of Fourier Transform • Linearity Property :
UNIT-III
32
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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: : : : ॥॥॥॥
Properties of Fourier Transform • Linearity Property : F[af1(x) + bf2(x)] = aF[f1(x)] + bF[f2(x)].
UNIT-III
32
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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: : : : ॥॥॥॥
Properties of Fourier Transform • Linearity Property : F[af1(x) + bf2(x)] = aF[f1(x)] + bF[f2(x)].
• Change of Scale Property :
UNIT-III
32
![Page 195: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/195.jpg)
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: : : : ॥॥॥॥
Properties of Fourier Transform • Linearity Property : F[af1(x) + bf2(x)] = aF[f1(x)] + bF[f2(x)].
• Change of Scale Property :
UNIT-III
32
[[[[ ]]]]
====a
sF
aaxfF
1)(
![Page 196: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/196.jpg)
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: : : : ॥॥॥॥
Properties of Fourier Transform • Linearity Property : F[af1(x) + bf2(x)] = aF[f1(x)] + bF[f2(x)].
• Change of Scale Property :
• Shifting Property :
UNIT-III
32
[[[[ ]]]]
====a
sF
aaxfF
1)(
![Page 197: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/197.jpg)
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: : : : ॥॥॥॥
Properties of Fourier Transform • Linearity Property : F[af1(x) + bf2(x)] = aF[f1(x)] + bF[f2(x)].
• Change of Scale Property :
• Shifting Property :
UNIT-III
32
[[[[ ]]]]
====a
sF
aaxfF
1)(
[[[[ ]]]] (((( ))))sFeaxfF isa====−−−− )(
![Page 198: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/198.jpg)
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: : : : ॥॥॥॥
Properties of Fourier Transform
UNIT-III
33
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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: : : : ॥॥॥॥
Properties of Fourier Transform • Modulation Theorem :
UNIT-III
33
![Page 200: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/200.jpg)
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: : : : ॥॥॥॥
Properties of Fourier Transform • Modulation Theorem :
UNIT-III
33
[[[[ ]]]] (((( )))) (((( ))))[[[[ ]]]]asFasFaxxfF −−−−++++++++====2
1cos)(
![Page 201: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/201.jpg)
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: : : : ॥॥॥॥
Properties of Fourier Transform • Modulation Theorem :
• Fourier Transform of Multiplication by x :
UNIT-III
33
[[[[ ]]]] (((( )))) (((( ))))[[[[ ]]]]asFasFaxxfF −−−−++++++++====2
1cos)(
![Page 202: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/202.jpg)
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: : : : ॥॥॥॥
Properties of Fourier Transform • Modulation Theorem :
• Fourier Transform of Multiplication by x :
UNIT-III
33
[[[[ ]]]] (((( )))) (((( ))))sFds
dixfxF
n
nnn −−−−====)(
[[[[ ]]]] (((( )))) (((( ))))[[[[ ]]]]asFasFaxxfF −−−−++++++++====2
1cos)(
![Page 203: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/203.jpg)
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: : : : ॥॥॥॥
Properties of Fourier Transform • Modulation Theorem :
• Fourier Transform of Multiplication by x :
• Fourier Transform of Derivative :
UNIT-III
33
[[[[ ]]]] (((( )))) (((( ))))sFds
dixfxF
n
nnn −−−−====)(
[[[[ ]]]] (((( )))) (((( ))))[[[[ ]]]]asFasFaxxfF −−−−++++++++====2
1cos)(
![Page 204: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/204.jpg)
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: : : : ॥॥॥॥
Properties of Fourier Transform • Modulation Theorem :
• Fourier Transform of Multiplication by x :
• 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)(
![Page 205: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/205.jpg)
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: : : : ॥॥॥॥
Properties of Fourier Transform • Modulation Theorem :
• Fourier Transform of Multiplication by x :
• Fourier Transform of Derivative : F[f '(x)] = isF(s) if f(x) → 0 as x → ±∞.
• Fourier Transform of Integral :
UNIT-III
33
[[[[ ]]]] (((( )))) (((( ))))sFds
dixfxF
n
nnn −−−−====)(
[[[[ ]]]] (((( )))) (((( ))))[[[[ ]]]]asFasFaxxfF −−−−++++++++====2
1cos)(
![Page 206: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/206.jpg)
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: : : : ॥॥॥॥
Properties of Fourier Transform • Modulation Theorem :
• Fourier Transform of Multiplication by x :
• Fourier Transform of Derivative : F[f '(x)] = isF(s) if f(x) → 0 as x → ±∞.
• Fourier Transform of Integral :
UNIT-III
33
[[[[ ]]]] (((( )))) (((( ))))sFds
dixfxF
n
nnn −−−−====)(
(((( ))))(((( ))))is
sFdxxfF
x
a−−−−
====
∫∫∫∫ )(
[[[[ ]]]] (((( )))) (((( ))))[[[[ ]]]]asFasFaxxfF −−−−++++++++====2
1cos)(
![Page 207: 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: :: : ॥॥॥॥](https://reader031.vdocuments.mx/reader031/viewer/2022011918/600a5a5b4b198a4c7736a6d7/html5/thumbnails/207.jpg)
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 INTEGRAL TRANSFORMS
34