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Operational Calculus and Related Topics [Kõva köide]

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  • Formaat: Hardback, 420 pages, kõrgus x laius: 254x178 mm, kaal: 612 g
  • Sari: Analytical Methods and Special Functions
  • Ilmumisaeg: 15-Aug-2006
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-10: 1584886498
  • ISBN-13: 9781584886495
Teised raamatud teemal:
Operational Calculus and Related TopicsSuurem pilt
  • Formaat: Hardback, 420 pages, kõrgus x laius: 254x178 mm, kaal: 612 g
  • Sari: Analytical Methods and Special Functions
  • Ilmumisaeg: 15-Aug-2006
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-10: 1584886498
  • ISBN-13: 9781584886495
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781584886495.html
Märksõnad:
Even though the theories of operational calculus and integral transforms are centuries old, these topics are constantly developing, due to their use in the fields of mathematics, physics, and electrical and radio engineering. Operational Calculus and Related Topics highlights the classical methods and applications as well as the recent advances in the field.

Combining the best features of a textbook and a monograph, this volume presents an introduction to operational calculus, integral transforms, and generalized functions, the backbones of pure and applied mathematics. The text examines both the analytical and algebraic aspects of operational calculus and includes a comprehensive survey of classical results while stressing new developments in the field. Among the historical methods considered are Oliver Heavisides algebraic operational calculus and Paul Diracs delta function. Other discussions deal with the conditions for the existence of integral transforms, Jan Mikusiskis theory of convolution quotients, operator functions, and the sequential approach to the theory of generalized functions.

Benefits

· Discusses theory and applications of integral transforms

· Gives inversion, complex-inversion, and Diracs delta distribution formulas, among others

· Offers a short survey of actual results of finite integral transforms, in particular convolution theorems

Because Operational Calculus and Related Topics provides examples and illustrates the applications to various disciplines, it is an ideal reference for mathematicians, physicists, scientists, engineers, and students.
Preface xi
List of Symbols xv
1 Integral Transforms 1(162)
1.1 Introduction to Operational Calculus
1(4)
1.2 Integral Transforms — Introductory Remarks
5(3)
1.3 The Fourier Transform
8(19)
1.3.1 Definition and Basic Properties
8(3)
1.3.2 Examples
11(2)
1.3.3 Operational Properties
13(4)
1.3.4 The Inversion Formula
17(4)
1.3.5 Applications
21(6)
1.4 The Laplace Transform
27(28)
1.4.1 Definition and Basic Properties
27(4)
1.4.2 Examples
31(2)
1.4.3 Operational Properties
33(4)
1.4.4 The Complex Inversion Formula
37(3)
1.4.5 Inversion Methods
40(4)
1.4.6 Asymptotic Behavior
44(3)
1.4.7 Remarks on the Bilateral Laplace Transform
47(2)
1.4.8 Applications
49(6)
1.5 The Mellin Transform
55(12)
1.5.1 Definition and Basic Properties
55(3)
1.5.2 Operational Properties
58(4)
1.5.3 The Complex Inversion Formula
62(1)
1.5.4 Applications
63(4)
1.6 The Stieltjes Transform
67(11)
1.6.1 Definition and Basic Properties
67(3)
1.6.2 Operational Properties
70(3)
1.6.3 Asymptotics
73(2)
1.6.4 Inversion and Application
75(3)
1.7 The Hilbert Transform
78(6)
1.7.1 Definition and Basic Properties
78(3)
1.7.2 Operational Properties
81(2)
1.7.3 Applications
83(1)
1.8 Bessel Transforms
84(23)
1.8.1 The Hankel Transform
85(8)
1.8.2 The Meijer (K-) Transform
93(7)
1.8.3 The Kontorovich—Lebedev Transform
100(6)
1.8.4 Application
106(1)
1.9 The Mehler—Fock Transform
107(8)
1.10 Finite Integral Transforms
115(48)
1.10.1 Introduction
115(1)
1.10.2 The Chebyshev Transform
116(6)
1.10.3 The Legendre Transform
122(9)
1.10.4 The Gegenbauer Transform
131(6)
1.10.5 The Jacobi Transform
137(7)
1.10.6 The Laguerre Transform
144(9)
1.10.7 The Hermite Transform
153(10)
2 Operational Calculus 163(108)
2.1 Introduction
163(4)
2.2 Titchmarsh's Theorem
167(13)
2.3 Operators
180(39)
2.3.1 Ring of Functions
180(5)
2.3.2 The Field of Operators
185(5)
2.3.3 Finite Parts of Divergent Integrals
190(11)
2.3.4 Rational Operators
201(4)
2.3.5 Laplace Transformable Operators
205(8)
2.3.6 Examples
213(4)
2.3.7 Periodic Functions
217(2)
2.4 Bases of the Operator Analysis
219(17)
2.4.1 Sequences and Series of Operators
219(7)
2.4.2 Operator Functions
226(3)
2.4.3 The Derivative of an Operator Function
229(1)
2.4.4 Properties of the Continuous Derivative of an Operator Function
229(3)
2.4.5 The Integral of an Operator Function
232(4)
2.5 Operators Reducible to Functions
236(11)
2.5.1 Regular Operators
236(3)
2.5.2 The Realization of Some Operators
239(3)
2.5.3 Efros Transforms
242(5)
2.6 Application of Operational Calculus
247(24)
2.6.1 Ordinary Differential Equations
247(11)
2.6.2 Partial Differential Equations
258(13)
3 Generalized Functions 271(118)
3.1 Introduction
271(1)
3.2 Generalized Functions — Functional Approach
272(15)
3.2.1 Introduction
272(2)
3.2.2 Distributions of One Variable
274(5)
3.2.3 Distributional Convergence
279(1)
3.2.4 Algebraic Operations on Distributions
280(7)
3.3 Generalized Functions — Sequential Approach
287(24)
3.3.1 The Identification Principle
287(2)
3.3.2 Fundamental Sequences
289(8)
3.3.3 Definition of Distributions
297(3)
3.3.4 Operations with Distributions
300(3)
3.3.5 Regular Operations
303(8)
3.4 Delta Sequences
311(21)
3.4.1 Definition and Properties
311(6)
3.4.2 Distributions as a Generalization of Continuous Functions
317(3)
3.4.3 Distributions as a Generalization of Locally Integrable Functions
320(2)
3.4.4 Remarks about Distributional Derivatives
322(3)
3.4.5 Functions with Poles
325(1)
3.4.6 Applications
326(6)
3.5 Convergent Sequences
332(15)
3.5.1 Sequences of Distributions
332(7)
3.5.2 Convergence and Regular Operations
339(2)
3.5.3 Distributionally Convergent Sequences of Smooth Functions
341(3)
3.5.4 Convolution of Distribution with a Smooth Function of Bounded Support
344(2)
3.5.5 Applications
346(1)
3.6 Local Properties
347(8)
3.6.1 Inner Product of Two Functions
347(3)
3.6.2 Distributions of Finite Order
350(2)
3.6.3 The Value of a Distribution at a Point
352(3)
3.6.4 The Value of a Distribution at Infinity
355(1)
3.6.5 Support of a Distribution
355(1)
3.7 Irregular Operations
355(26)
3.7.1 Definition
355(2)
3.7.2 The Integral of Distributions
357(12)
3.7.3 Convolution of Distributions
369(6)
3.7.4 Multiplication of Distributions
375(4)
3.7.5 Applications
379(2)
3.8 Hilbert Transform and Multiplication Forms
381(8)
3.8.1 Definition of the Hilbert Transform
381(2)
3.8.2 Applications and Examples
383(6)
References 389(12)
Index 401


Prudnikov, A. P.; Skórnik, K.A.