
Introduction To The Operational Calculus
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Introduction to the Operational Calculus is a translation of "Einfuhrung in die Operatorenrechnung, Second Edition." This book deals with Heaviside's interpretation, on the Laplace integral, and on Jan Mikusinki's fundamental work "Operational Calculus." Throughout the book, basic algebraic concepts appear as aids to understanding some relevant points of the subject. An important field for research in analysis is asymptotic properties. This text also discusses examples to show the potentialities in applying operational calculus that run beyond ordinary differential equations with constant coefficients. In using operational calculus to solve more complicated problems than those of ordinary differential equations with constant coefficients, the concept of convergence assumes a significant role in the field of operators. This book also extends the Laplace transformation and applies it to non-transformable functions. This text also present three methods in which operational calculus can be modified and become useful in solving specific ranges of problems. These methods pertain to the finite Laplace transformation, to partial differential equations, and to the Volterra integral equations and ordinary differential equations with variable coefficients. This book can prove valuable for mathematicians, students, and professor of calculus and advanced mathematics.
Table of Contents
Introduction
1. General Survey
2. The Heaviside Method
3. A Rigorous Approach
4. Derivation of an Integral Transformation
5. Numerical Evaluations
6. Improved Approximations
Chapter I. Algebraic Foundations
1. Rings and Domains of Integrity
2. Fields
3. Polynomial Rings
4. Rational Functions
5. Isomorphisms and Extensions
6. Ideals and Residue Class Rings
Chapter II. Functions of a Discrete Variable
7. The Function Ring
8. Quotient Fields
9. Linear Difference Equations
10. Passage to the Limit
11. The Operator q as Complex Variable
12. Applications
Chapter III. Functions of a Continuous Variable
13. The Duhamel Product
14. Function Powers of t
15. Comparison Between Function and Value Products
16. Titchmarsh's Theorem
17. The Field of Operators
18. Rational Operators in p
Chapter IV. Applications
19. Differential Equations with Constant Coefficients
20. Examples
21. Systems of Differential Equations
22. Degenerate Systems
23. Control Engineering
24. Integral Equations
Chapter V. Convergent Sequences of Operators
25. The Concept of Convergence
26. Infinite Series
27. Discontinuous Functions
28. The Displacement Operator
29. Step Functions
30. The Delta Operator
Chapter VI. The Laplace Transformation
31. The Operator p as Complex Variable
32. Properties
33. Examples
34. Inverse Transformations
35. The Complex Inversion Formula
36. Fourier's Integral Theorem
Chapter VII. Applications
37. The Method of Residues
38. Series Expansions
39. Differential Equations with Polynomial Coefficients
40. Partial Differential Equations
41. Difference Equations in the Image Domain
42. Integral Equations
Chapter VIII. Asymptotic Properties
43. Definitions
44. Abelian Theorems
45. Further Types of Singularity
46. Tauberian Theorems
47. Problems of Stability
48. Euler's Summation Formula
Chapter IX. Generalizations
49. Asymptotic Series
50. Asymptotic Integrals
51. The General Operational Calculus
52. Partial Differential Equations
53. Differential-Difference Equations
54. The Finite Part of an Integral
Chapter X. Further Operational Methods
55. The Finite Laplace Transformation
56. Boundary Value Problems
57. Functions of Two Variables
58. Partial Differential Equations
59. Groups
60. Differential Equations with Variable Coefficients
Appendix
Answers to Exercises
References
A. Text-Books and Monographs
B. Original Papers
Formulae
Subject Index
Product details
- No. of pages: 304
- Language: English
- Copyright: © North Holland 1967
- Published: January 1, 1967
- Imprint: North Holland
- eBook ISBN: 9780323162456
About the Author
Lothar Berg
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