Introduction To The Operational Calculus

Introduction To The Operational Calculus

1st Edition - January 1, 1967

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  • Author: Lothar Berg
  • eBook ISBN: 9780323162456

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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


    Answers to Exercises


    A. Text-Books and Monographs

    B. Original Papers


    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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