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E-raamat: Introduction to Integral Transforms

  • Formaat: 428 pages
  • Ilmumisaeg: 13-Feb-2018
  • Kirjastus: CRC Press
  • Keel: eng
  • ISBN-13: 9780429994340
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Introduction to Integral TransformsSuurem pilt
  • Formaat: 428 pages
  • Ilmumisaeg: 13-Feb-2018
  • Kirjastus: CRC Press
  • Keel: eng
  • ISBN-13: 9780429994340
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An Introduction to Integral Transforms is meant for students pursuing graduate and post graduate studies in Science and Engineering. It contains discussions on almost all transforms for normal users of the subject. The content of the book is explained from a rudimentary stand point to an advanced level for convenience of its readers. Pre-requisite for understanding the subject matter of the book is some knowledge on the complex variable techniques.

Please note: Taylor & Francis does not sell or distribute the Hardback in India, Pakistan, Nepal, Bhutan, Bangladesh and Sri Lanka.

Arvustused

"This is a nice introductory book devoted to basic integral transforms of single-variable functions. It can be recommended to graduate and advanced undergraduate students, as well as to engineers and researchers in dierent areas of analysis and applications.

The book consists of fourteen chapters and covers the following topics: the Fou-rier, Laplace, and Mellin transforms; the Hilbert and Stieltjes transforms; the Hankel, Legendre, Mehler-Fock, Kantorovich-Lebedev transforms; the Jacobi, Gegenbauer, La-guerre and Hermite transforms; the Z-transform. It also includes the nite Fourier, Hankel, Mellin, and Laplace transforms.

The theory is accompanied with numerous examples and applications of integral transforms to diverse problems of mathematical physics."

- B. S. Rubin - Mathematical Reviews Clippings - April 2019

1 Fourier Transform 1(78)
1.1 Introduction
1(1)
1.2 Classes of functions
2(1)
1.3 Fourier Series and Fourier Integral Formula
2(4)
1.4 Fourier Transforms
6(2)
1.4.1 Fourier sine and cosine Transforms
7(1)
1.5 Linearity property of Fourier Transforms
8(1)
1.6 Change of Scale property
9(1)
1.7 The Modulation theorem
10(1)
1.8 Evaluation of integrals by means of inversion theorems
11(2)
1.9 Fourier Transform of some particular functions
13(7)
1.10 Convolution or Faltung of two integrable functions
20(1)
1.11 Convolution or Falting or Faltung Theorem for FT
21(2)
1.12 Parseval's relations for Fourier Transforms
23(3)
1.13 Fourier Transform of the derivative of a function
26(4)
1.14 Fourier Transform of some more useful functions
30(6)
1.15 Fourier Transforms of Rational Functions
36(1)
1.16 Other important examples concerning derivative of FT
37(10)
1.17 The solution of Integral Equations of Convolution Type
47(6)
1.18 Fourier Transform of Functions of several variables
53(2)
1.19 Application of Fourier Transform to Boundary Value Problems
55(24)
2 Finite Fourier Transform 79(23)
2.1 Introduction
79(1)
2.2 Finite Fourier cosine and sine Transforms
79(2)
2.3 Relation between finite Fourier Transform of the derivatives of a function
81(1)
2.4 Faltung or convolution theorems for finite Fourier Trans form
82(3)
2.5 Multiple Finite Fourier Transform
85(1)
2.6 Double Transforms of partial derivatives of functions
86(1)
2.7 Application of finite Fourier Transforms to boundary value problems
87(15)
3 The Laplace Transform 102(39)
3.1 Introduction
102(1)
3.2 Definitions
103(1)
3.3 Sufficient conditions for existence of Laplace Transform
103(1)
3.4 Linearity property of Laplace Transform
104(1)
3.5 Laplace transforms of some elementary functions
105(2)
3.6 First shift theorem
107(1)
3.7 Second shift theorem
107(1)
3.8 The change of scale property
107(1)
3.9 Examples
108(2)
3.10 Laplace Transform of derivatives of a function
110(2)
3.11 Laplace Transform of Integral of a function
112(1)
3.12 Laplace Transform of tn f (t)
113(1)
3.13 Laplace Transform of f (t)/t
114(1)
3.14 Laplace Transform of a periodic function
115(1)
3.15 The initial-value theorem and the final-value theorem of Laplace Transform
116(1)
3.16 Examples
117(4)
3.17 Laplace Transform of some special functions
121(10)
3.18 The Convolution of two functions
131(1)
3.19 Applications
132(9)
4 The Inverse Laplace Transform And Application 141(79)
4.1 Introduction
141(2)
4.2 Calculation of Laplace inversion of some elementary functions
143(2)
4.3 Method of expansion into partial fractions of the ratio of two polynomials
145(8)
4.4 The general evaluation technique of inverse Laplace transform
153(5)
4.5 Inversion Formula from a different stand point: The Tricomi's method
158(3)
4.6 The Double Laplace Transform
161(5)
4.7 The iterative Laplace transform
166(1)
4.8 The Bilateral Laplace Transform
166(2)
4.9 Application of Laplace Transforms
168(52)
5 Hilbert and Stieltjes Transforms 220(18)
5.1 Introduction
220(1)
5.2 Definition of Hilbert Transform
220(1)
5.3 Some Important properties of Hilbert Transforms
221(4)
5.4 Relation between Hilbert Transform and Fourier Transform
225(1)
5.5 Finite Hilbert Transform
226(1)
5.6 One-sided Hilbert Transform
227(1)
5.7 Asymptotic Expansions of one-sided Hilbert Transform
228(2)
5.8 The Stieltjes Transform
230(1)
5.9 Some Deductions
231(1)
5.10 The Inverse Stieltjes Transform
232(2)
5.11 Relation between Hilbert Transform and Stieltjes Transform
234(4)
6 Hankel Transforms 238(22)
6.1 Introduction
238(1)
6.2 The Hankel Transform
238(1)
6.3 Elementary properties
238(4)
6.4 Inversion formula for Hankel Transform
242(2)
6.5 The Parseval Relation for Hankel Transforms
244(1)
6.6 Illustrative Examples:
245(15)
7 Finite Hankel Transforms 260(17)
7.1 Introduction
260(1)
7.2 Expansion of some functions in series involving cylinder functions : Fourier-Bessel Series
260(2)
7.3 The Finite Hankel Transform
262(1)
7.4 Illustrative Examples
263(2)
7.5 Finite Hankel Transform of order n in 0 < or = to x < or = to 1 of the derivatrive of a function
265(1)
7.6 Finite Hankel Transform over 0 < or = to x < or = to 1 of order n of d2f/dx2 + 1/x, when p is the root of Jn(p) = 0
266(1)
7.7 Finite Hankel Transform of f"(x) + 1/xf(x) - n2/x2 f(x), where p is the root of Jn(p) = 0 in 0 < or = to x < or = to 1
266(1)
7.8 Other forms of finite Hankel Transforms
267(1)
7.9 Illustrations
268(1)
7.10 Application of finite Hankel Transforms
269(8)
8 The Mellin Transform 277(25)
8.1 Introduction
277(1)
8.2 Definition of Mellin Transform
278(3)
8.3 Mellin Transform of derivative of a function
281(2)
8.4 Mellin Transform of Integral of a function
283(2)
8.5 Mellin Inversion theorem
285(1)
8.6 Convolution theorem of Mellin Transform
286(1)
8.7 Illustrative solved Examples
287(5)
8.8 Solution of Integral equations
292(1)
8.9 Application to Summation of Series
293(2)
8.10 The Generalised Mellin Transform
295(2)
8.11 Convolution of generalised Mellin Transform
297(1)
8.12 Finite Mellin Transform
297(5)
9 Finite Laplace Transforms 302(15)
9.1 Introduction
302(1)
9.2 Definition of Finite Laplace Transform
302(2)
9.3 Finite Laplace Transform of elementary functions
304(3)
9.4 Operational Properties
307(4)
9.5 The Initial Value and the Final Value Theorem
311(1)
9.6 Applications
312(5)
10 Legendre Transforms 317(11)
10.1 Introduction
317(1)
10.2 Definition of Legendre Transform
317(1)
10.3 Elementary properties of Legendre Transforms
318(5)
10.4 Operational Properties of Legendre Transforms
323(2)
10.5 Application to Boundary Value Problems
325(3)
11 The Kontorovich-Lebedev Transform 328(7)
11.1 Introduction
328(1)
11.2 Definition of Kontorovich-Lebedev Transform
328(1)
11.3 Parseval Relation for Kontorovich-Lebedev Transforms
329(1)
11.4 Illustrative Examples
330(2)
11.5 Boundary Value Problem in a wedge of finite thickness
332(3)
12 The Mehler-Fock Transform 335(16)
12.1 Introduction
335(1)
12.2 Fock's Theorem (with weaker restriction)
335(2)
12.3 Mehler-Fock Transform of zero order and its properties
337(2)
12.4 Parseval type relation
339(2)
12.5 Mehler-Fock Transform of order m
341(1)
12.6 Application to Boundary Value Problems
342(6)
12.6.1 First Example
342(2)
12.6.2 Second Example
344(1)
12.6.3 Third Example
345(2)
12.6.4 Fourth Example
347(1)
12.7 Application of Mehler-Fock Transform for solving dual integral equation
348(3)
13 Jacobi, Gegenbauer, Laguerre and Hermite Transforms 351(21)
13.1 Introduction
351(1)
13.2 Definition of Jacobi Transform
351(4)
13.3 The Gegenbauer Transform
355(1)
13.4 Convolution Theorem
356(1)
13.5 Application of the Transforms
357(2)
13.6 The Laguerre Transform
359(2)
13.7 Operational properties
361(3)
13.8 Hermite Transform
364(2)
13.9 Operational Properties
366(1)
13.10 Hermite Transform of derivative of a function
367(5)
14 The Z-Transform 372(18)
14.1 Introduction
372(1)
14.2 Z - Transform : Definition
372(4)
14.3 Some Operational Properties of Z-Transform
376(7)
14.4 Application of Z-Transforms
383(7)
Appendix 390(15)
Bibliography 405(2)
Index 407
Baidyanath Patra is retired Professor of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, West Bengal, India. With his forty years teaching and research experience the author has published several research papers in national and international journals of repute and also published few books for under graduate, post graduate and research students.
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