# Operational Calculus

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Pure and Applied Mathematics, Volume 109: Operational Calculus, Second Edition. Volume I presents the foundations of operational calculus and its applications to physics and engineering. This book introduces the operators algebraically as a kind of fractions. Organized into three parts, this volume begins with an overview of the concept as well as the characteristics of a convolution of continuous functions. This text then examines the transitivity, associativity, and distributivity of convolution with regard to addition. Other parts consider the methods of solving other difference equations, particularly in the field of electrical engineering, in which the variable runs over integer values only. This book discusses as well the solution of differential equations under given initial conditions. The final part deals with the characteristic properties of a derivative and provides the definition of algebraic derivative to any operators. This book is a valuable resource for physicists, electrical engineers, mathematicians, and research workers.

## Table of Contents

Foreword

Foreword to the Second English Edition

Part I - Operational Algebra

Chapter I. The Concept and Properties of a Convolution of Continuous Functions

§ 1. Definition of Convolution

§ 2. Class 𝓒

§ 3. Commutativity of Convolution

§ 4. Associativity of Convolution

§ 5. Addition and Convolution as Fundamental Operations in the Operational Calculus

§ 6. Function and Value of a Function

§ 7. Symbolics

§ 8. Integral Operator

Chapter II. Theorem of Titchmarsh

§ 9. Formulation of the Theorem and General Remarks

§ 10. Theorem of Phragmén

§ 11. Theorem on Moments

§ 12. Proof of Titschmarsh's Theorem in the Case f = g

§ 13. General Proof

Chapter III. Operators

§ 14. An Operation Inverse to Convolution

§ 15. Operators

§ 16. Operations on Operators

§ 17. Numerical Operators

§ 18. Remarks on Terminology

§ 19. Product of a Number and a Function

§ 20. Numbers 0 and 1

§ 21. Differential Operator

§ 22. Powers of Operator 8

§ 23. Polynomials of Operator 8

§ 24. Connections of the Operator 8 with the Exponential Function

§ 25. Connections of the Operator 8 with Trigonometric Functions

§ 26. Rational Operators

§ 27. Some Properties of Operators. Moments

Chapter IV. Ordinary Differential Equations with Constant Coefficients

§ 28. General Method and Examples

Chapter V. Theory of Electric Circuit

§ 29. Remarks on the Application of the Operational Calculus to Physical and Technical Problems

§ 30. The Electric Circuit

§ 31. The Short-Circuit Current

§ 32. Impedance

§ 33. Sine Wave Currents

§ 34. Kirchhoff's Laws

§ 35. Wheatstone Bridge

§ 36. Anderson Bridge

§ 37. General Remarks on Forming Equations for Network Currents

§ 38. Impedance and the Short-Circuit Current of Compound Two-Terminal Networks

§ 39. The Case of a Sinusoidal Electromotive Force

§ 40. The Electromotive Impulse and its Application to the Measurement of Impedance

§ 41. Inductive Couplings

§ 42. Four-Terminal Networks

§ 43. Connecting Four-Terminal Networks

§ 44. Connection of Three Four-Terminal Networks

§ 45. Four-Terminal Networks with Short-Circuited Terminals

§ 46. Four-Terminal Networks with Free Terminals

§ 47. Four-Terminal Network with Terminals Short-Circuited by a Two-Terminal Network

§ 48. Chain of Four-Terminal Networks

§ 49. Transformers

§ 50. Cathode-Ray Tube as a Four-Terminal Network

§ 51. Four Terminal Network with Determinant 1

§ 52. Inverted Four-Terminal Networks and Symmetrical Four-Terminal Networks

Chapter VI. General Eolutione of Differential Equations and Boundary Problems

§ 53. General Solution. Transfer Functions

§ 54. Boundary Problems

§ 55. Solution of Differential Equations Under Given Initial Conditions At the Point to ≠ 0

Chapter VII. Discontinuous Functions

§ 56. Functions of Class 𝓚

§ 57. Operations on Functions of Class 𝓚

§ 58. Euler's Gamma Integral

§ 59. Non-Integer Powers of the Operators 𝑙 and 𝒔 — α

§ 60. Functions Having a Derivative of Class 𝓚

§ 61. Differential Equations with a Discontinuous Right Side

§ 62. Jump Function and Translation Operator

§ 63. Derivatives of Certain Discontinuous Functions

§ 64. Approximation of the Translation Operator by Functions

§ 65. Various Interpretations of the Translation Operator

§ 66. Functions Defined on the Whole Axis t

Chapter VIII. Applications to the Statics of Beams

§ 67. Kinds of Load

§ 68. Transverse Force and Bending Moment

§ 69. Conditions of Balance

§ 70. Deflection of Beams

§ 71. Support Beams

§ 72. Statically Indeterminate Cases

Part II - Sequences and Series of Operators

Chapter I. Sequences of Operators

§ 1. Uniform Convergence

§ 2. Limit of a Sequence of Operators

§ 3. Properties of the Limit of a Sequence of Operators

Chapter II. Series of Translation Operators

§ 4. Series with Numerical Coefficients

§ 5. More General Series of Translation Operators

§ 6. Operator 1/(1-β𝒉λ)𝒌

§ 7. Electrical Systems Excited Periodically

Chapter III. Difference Equations

§ 8. Examples of Difference Equations

§ 9. Operators 1/(1-β𝒉λ)𝒌

§ 10. Difference Equations of Arbitrary Order

§ 11. Reduction of the Complex Form of a Solution to the Real Form

§ 12. General Remarks

Chapter IV. Power Series

§ 13. Power Series of a Numerical Variable with Operational Coefficients

§ 14. Multiplication of Power Series

§ 15. Power Series of Operators with Numerical Coefficients

§ 16. Powers with Arbitrary Real Exponents

§ 17. A Problem in Electrical Engineering

§ 18. The Bessel Function Jo

§ 19. A More General Problem in Electrical Engineering

§ 20. Bessel Functions Jn with an Arbitrary Natural n

Part III - The Operational Differential Calculus

Chapter I. Operational Functions and their Derivatives

§ 1. Operational Functions

§ 2. Continuity of the Operational Function

§ 3. Continuous Derivative of An Operational Function

§ 4. Properties of the Continuous Derivative

§ 5. Continuous Derivatives of Higher Order

§ 6. The Continuous Derivative in An Infinite Interval

§ 7. General Definition of the Derivative

Chapter II. Exponential Functions

§ 8. Differential Equation χ'(λ) = wx(λ)

§ 9. Generalized Exponential Functions

§ 10. Derivatives of Power Series

§ 11. Function Expλ(s—√82+α2) and Cognate Functions

Chapter III. Differential Equation χ'(λ) = wx(λ)

§ 12. A Uniqueness Theorem

§ 13. Continuation of Solutions

Chapter IV. Vibrations of a String

§ 14. Operational Equation of the Vibrating String

§ 15. The Shape of a Vibrating String

§ 16. More General Boundary Conditions

§ 17. Uniqueness of Solution

§ 18. An Infinitely Long String

§ 19. String Vibrations in the Case of Certain Particular Initial Positions

§ 20. String Vibrations in Case of an Arbitrarily Fixed Initial Position

§ 21. String Vibrations with Given Initial Velocity

§ 22. Other Interpretations

Chapter V. The Equation of Heat

§ 23. The Parabolic Exponential Function

§ 24. Some Analytic Properties of the Parabolic Exponential Function

§ 25. The Temperature of a Heat-Conducting Bar

§ 26. Expanding the Solution in an Infinite Series

§ 27. Inequalities and the Module

§ 28. An Infinitely Long Bar

§ 29. A Bar Without an Outgoing Flow of Heat

§ 30. Trigonometric Series

§ 31. Changes of Temperature in a Bar with a Given Initial Temperature

§ 32. Checking the Correctness of the Solution

§ 33. Certain Particular Cases

§ 34. A Bar Insulated at One End

§ 35. Heat Inflow Regulated at One End

§ 36. A Heat-Conducting Ring

§ 37. Operation Tα and its Application

§ 38. Non-Insulated Heat Conductor

Chapter VI. The Equation of Telegraphy

§ 39. The General Form of the Equation of Telegraphy

§ 40. Conductance Without Loss

§ 41. Conductance Without Deformation

§ 42. The Thomson Cable

§ 43. A Cable Without Self-Induction

§ 44. A Cable Without Leak-Conductance

§ 45. The Case Where all the Four Conductance Parameters are Positive

Chapter VII. Applications to Chromatography

§ 46. Perfectly Stirred Vessel

§ 47. A Vessel Containing Two Phases

§ 48. A Theoretical Modification of the Problem

§ 49. Perfectly Stirred Vessels in Sequence

§ 50. A Limiting Case

§ 51. Chromatographical Effect

§ 52. Plug Flow Model

§ 53. Diffusion

§ 54. Plug Flow with Axial Dispersion

§ 55. A Mathematical Proof

§ 56. Case with the Absorption of a Solute

Chapter VIII. The Algebraic Derivative

§ 57. Definition and Properties

§ 58. Powers of the Operator 1/(82+β2)

Answers to Problems

Bibliography

Index

Other Titles in the Series

## Product details

- No. of pages: 322
- Language: English
- Copyright: © Pergamon 1983
- Published: January 1, 1983
- Imprint: Pergamon
- eBook ISBN: 9781483278933