integral transforms · b.a./b.sc.(mathematics)–3rd year–6th sem. e-content by dr.n.a.p ande...
Post on 24-Sep-2020
1 Views
Preview:
TRANSCRIPT
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
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
PAPER DETAILS
2
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
PAPER DETAILS
2
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
2
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
2
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
PAPER DETAILS
2
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 DETAILS
2
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.)
PAPER DETAILS
2
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 DETAILS
2
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
PAPER DETAILS
2
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
PAPER DETAILS
2
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
PAPER DETAILS
3
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
PAPER DETAILS
3
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.
PAPER DETAILS
3
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
PAPER DETAILS
4
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
PAPER DETAILS
4
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
PAPER DETAILS
4
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
PAPER DETAILS
5
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
PAPER DETAILS
5
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.
PAPER DETAILS
5
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: : : : ॥॥॥॥
Text Book & Scope
PAPER DETAILS
6
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: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
PAPER DETAILS
6
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: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Advanced Engineering
Mathematics
PAPER DETAILS
6
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: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Advanced Engineering
Mathematics
�Author : H. K. Dass
PAPER DETAILS
6
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: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Advanced Engineering
Mathematics
�Author : H. K. Dass
�Publisher : S. Chand & Company Ltd.
PAPER DETAILS
6
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: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Advanced Engineering
Mathematics
�Author : H. K. Dass
�Publisher : S. Chand & Company Ltd.
� ISSN : 81-219-0345-9
PAPER DETAILS
6
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: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�Title : Advanced Engineering
Mathematics
�Author : H. K. Dass
�Publisher : S. Chand & Company Ltd.
� ISSN : 81-219-0345-9
• Scope :
PAPER DETAILS
6
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: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�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
PAPER DETAILS
6
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: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�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
PAPER DETAILS
6
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: : : : ॥॥॥॥
Text Book & Scope • Recommended Text Book :
�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
PAPER DETAILS
6
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: : : : ॥॥॥॥
Reference Books
PAPER DETAILS
7
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: : : : ॥॥॥॥
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.
PAPER DETAILS
7
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-I
8
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 • 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
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 • 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
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
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 ====
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
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
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
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 >>>>
−−−−====
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 >>>>
−−−−====
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
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
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
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
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 >>>>
−−−−−−−−====
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 >>>>
−−−−−−−−−−−−====
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
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
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
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
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
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
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
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
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
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
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
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−−−−
====−−−−
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
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
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
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
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
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
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
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
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
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
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
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'
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
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
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
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
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
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
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
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
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
====
∫∫∫∫
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
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
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 −−−−====
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 −−−−====
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 )(
)(
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 )(
)(
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 )(
)(
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 )(
)(
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
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
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
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
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
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
====
−−−−∫∫∫∫
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
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
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.
UNIT-II
18
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
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
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
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
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
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
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
0 for ,cos22
1 >>>>====
++++−−−− sat
as
sL
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
0 for ,cos22
1 >>>>====
++++−−−− sat
as
sL
22
22
1 for ,sinh asatas
aL >>>>====
−−−−−−−−
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
0 for ,cos22
1 >>>>====
++++−−−− sat
as
sL
22
22
1 for ,sinh asatas
aL >>>>====
−−−−−−−−
22
22
1 for ,cosh asatas
sL >>>>====
−−−−
−−−−
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
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|
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|
(((( )))) 0 for ,sin22
1 >>>>====
++++−−−−−−−− sbte
bas
bL at
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|
(((( )))) 0 for ,sin22
1 >>>>====
++++−−−−−−−− sbte
bas
bL at
(((( )))) 0 for ,cos22
1 >>>>====
++++−−−−−−−−−−−− sbte
bas
asL at
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|
(((( )))) 0 for ,sin22
1 >>>>====
++++−−−−−−−− sbte
bas
bL at
(((( )))) 0 for ,cos22
1 >>>>====
++++−−−−−−−−−−−− sbte
bas
asL at
(((( ))))22
221 for ,sinh asbte
bas
bL at >>>>====
−−−−−−−−−−−−
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|
(((( )))) 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 >>>>====
−−−−−−−−−−−−−−−−
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|
(((( )))) 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|
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
21
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 :
UNIT-II
21
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)].
UNIT-II
21
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
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
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++++====−−−−
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 )()(
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
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
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 −−−−====
−−−−
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 −−−−====
−−−−
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 ====
∫∫∫∫∞∞∞∞
−−−−
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 ====
∫∫∫∫∞∞∞∞
−−−−
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 ====
∫∫∫∫∞∞∞∞
−−−−
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
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
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
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
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
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 )()()()(
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
29
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
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
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 :
There is no explicit formula for its inverse.
UNIT-III
29
[[[[ ]]]] (((( )))) ∫∫∫∫∞∞∞∞
−−−−========0
1 )()( dxxfxsFxfM s
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 :
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
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
UNIT-III
30
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
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
(((( ))))∫∫∫∫ ∫∫∫∫∞∞∞∞ ∞∞∞∞
∞∞∞∞−−−−
−−−−====0
cos)(1
)( dtduxtutfxfπ
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 :
2. Fourier Sine Integral :
UNIT-III
30
(((( ))))∫∫∫∫ ∫∫∫∫∞∞∞∞ ∞∞∞∞
∞∞∞∞−−−−
−−−−====0
cos)(1
)( dtduxtutfxfπ
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 :
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
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 :
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
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
UNIT-III
31
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 3. Fourier Cosine Integral :
UNIT-III
31
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 3. Fourier Cosine Integral :
If f(x) is an even function, then it has Fourier cosine integral given by
UNIT-III
31
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 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
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 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
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 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
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
32
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 :
UNIT-III
32
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
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
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)(
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)(
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====−−−− )(
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
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
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)(
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)(
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)(
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)(
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)(
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)(
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)(
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
top related