# 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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