# Operational Calculus

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Operational Calculus, Volume II is a methodical presentation of operational calculus. An outline of the general theory of linear differential equations with constant coefficients is presented. Integral operational calculus and advanced topics in operational calculus, including locally integrable functions and convergence in the space of operators, are also discussed. Formulas and tables are included. Comprised of four sections, this volume begins with a discussion on the general theory of linear differential equations with constant coefficients, focusing on such topics as homogeneous and non-homogeneous equations and applications of operational calculus to partial differential equations. The section section deals with the integral of an operational function and its applications, along with integral transformations. A definition of operators in terms of abstract algebra is then presented. Operators as generalized functions, power series of operators, and Laplace transform are also discussed. Formulas of the operational calculus and tables of functions round out the book. This monograph will be useful to engineers, who regard the operational calculus merely as a tool in their work, and readers who are interested in proofs of theorems and mathematical problems.

## Table of Contents

Foreword to the First English Edition

Foreword to the Second English Edition

Supplements to Volume I

(A) Supplement to Part I, Chapter VI

(B) Supplement to Part III, Chapter VIII

Part IV —An Outline of the General Theory of Linear Differential Equations with Constant Coefficients

Chapter I. Homogeneous Equations

§ 1. Introductory Remarks

§ 2. Characteristic Equations

§ 3. On exponential Functions

§ 4. Logarithms

§ 5. Multiple Roots of the Characteristic Equation

§ 6. The General Solution

§ 7. Theorem on Uniqueness of Solution

§ 8. The Logarithmic Equation

§ 9. Linear Differential Expressions

§ 10. Operations on Linear Differential Expressions

§ 11. Characteristic Polynomials of Linear Differential Expressions

§ 12. Pure Equations

§ 13. Mixed Equations

§ 14. Adapting the Solution to Given Initial, Boundary and Other Conditions

Chapter II. Non-Homogeneous Equations

§ 15. The General Solution of a Non-Homogeneous Equation

§ 16. The Case where the Right Side is a Polynomial

§ 17. The Case where the Right Side is an Exponential Function

§ 18. The Case where the Right Side is a Product of a Polynomial and an Exponential Function

§ 19. The Case where the Right Side is a Linear Combination of Two Functions

§ 20. The Case where the Right Side is a Trigonometric Function

§ 21. Adapting the Solution to Additional Conditions

Chapter III. Applications to Partial Differential Equations

§ 22. Reducing Partial Operational Equations to Operational Equations

§ 23. Remarks on Additional Conditions

§ 24. An Incorrect Solution

§ 25. Explaining the Apparent Contradiction

§ 26. The Cauchy Conditions and the Question of their Being Equivalent to the General Conditions

§ 27. Solving Restrictive Equations

§ 28. The Question of the Equivalence of a Partial Equation and an Operational Equation

§ 29. Further Examples of Solving Partial Equations

§ 30. General Remarks on Solving Partial Equations by the Operational Method

§ 31. Mixed Problems

Part V—Integral Operational Calculus

Chapter I. The Integral of an Operational Function and its Applications

§ 1. Operational Functions of Class (ℋ)

§ 2. The Definition of the Integral

§ 3. Properties of the Integral

§ 4. Operational Functions of Two Variables

§ 5. Cutting Down a Function

§ 6. The Integral Form of a Certain Particular Solution of the Logarithmic Differential Equation

§ 7. Application to the Equation of a Vibrating String

§ 8. Application of Infinite Series and Definite Integrals

Chapter II. Integral Transformations

§ 9. The Laplace Transform

§ 10. The Laplace Transform as a Basis for the Operational Calculus

§ 11. A comparison of the Direct Method and the Method of Laplace Transform

§ 12. Other Related Methods

Part VI—Advanced Topics in the Operational Calculus

Chapter I. Definition of Operators in Terms of Abstract Algebra

§ 1. Commutative Ring

§ 2. Quotient Field

§ 3. Operators

§ 4. Rings with Divisors of Zero

§ 5. Periodic Operators

§ 6. The Fourier Series of a Periodic Operator

Chapter II. Locally Integrable Functions

§ 1. The Convolution of Integrable Functions

§ 2. Properties of this Convolution

§ 3. Locally Integrable Functions as Operators

§ 4. Functions of Class ℋ

§ 5. Absolutely Continuous Functions

§ 6. The Ring of Locally Integrable Functions

Chapter III. Operators as Generalized Functions

§ 1. Introduction

§ 2. The Class of Left-Bounded Functions on the Whole Real Line

§ 3. The Support Number of an Operator

§ 4. The Support of Continuous Functions

§ 5. Approximate Identities

§ 6. Convolution of Functions Defined on an Arbitrary Open Set

§ 7. Almost Uniform Convergence on Open Sets

§ 8. Regular Operators

§ 9. Examples of Regular Operators

§ 10. An Example of an Operator which is not Regular

§ 11. The Restriction of Operators to an Open Set

§ 12. A Characterization of Support

§ 13. Derivatives of an Operator

§ 14. The Proof of Theorem A

§ 15. The Proof of Theorem B

§ 16. Some Examples of Derivatives

§ 17. Regular Convergence of Operators

§ 18. A Comparison with Other Types of Generalized Functions

§ 19. Defining Sequences

§ 20. Generalized Functions on the Whole Real Line

Chapter IV. Convergence in the Space of Operators

§ 1. Introduction

§ 2. Approximation by Convolution

§ 3. Infinite Convolution Products

§ 4. Continuous Convolution Multiples

§ 5. Common Denominators for a Sequence of Operators

§ 6. The Diagonal Subsequence Property

§ 7. The Space of Operators which have Positive Support Numbers

§ 8. A Convergent Sequence

§ 9. Topological Convergence

§ 10. The Connection Between Sequential and Topological Convergence

§ 11. A Topological Convergence in the Space of Operators

§ 12. Functional on the Space ℱ0

§ 13. A Characterization of Operator Convergence Type I'

§ 14. Metrizable Topological Vector Spaces

§ 15. The Space ℱ0 as a Metric Space

§ 16. The Space of Operators as the Union of Metric Spaces

§ 17. Bounded Sets in the Space of Operators

§ 18. Precompact Sets in the Space of Operators

§ 19. Precompact Collections in ℓ

§ 20. Convolution Takes Bounded Sets into Precompact Sets

§ 21. Any Bounded Set of Operators is Precompact

§ 22. Periodic Operators

§ 23. Continuous Operator Functions

§ 24. Differentiable Operator Functions

§ 25. Integrals of Operator Functions

Chapter V. Power Series of Operators

§ 1. Power Series of Integrable Functions

§ 2. A More General Case

§ 3. Some Particular Power Series

§ 4. The Classes C{Mn}

§ 5. The Logarithmically Convex Hull of a Sequence

§ 6. Power Series in the Operators s

§ 7. The Support of Power Series in s

Chapter VI. Laplace Transform

§ 1. Fundamental Properties of the Laplace Transform

§ 2. Complex Inversion Formula

§ 3. Post's Inversion Formula

§ 4. Laplace Transform of Convolution

§ 5. Finite Laplace Transform

§ 6. The Laplace Transform of a Function which Vanishes on an Initial Interval

§ 7. The Theorem of Titchmarsh for ƒ = g

§ 8. The Full Theorem of Titchmarsh

§ 9. Convex Sets in Rk

§ 10. Functions with Compact Support in Rk

§ 11. A Theorem of Lions

§ 12. The Proof of the Theorem of Lions

§ 13. A Titchmarsh Type Theorem for Rk

§ 14. Operators in Several Variables

Chapter VII. A Class of Dirichlet Series

§ 1. Introduction

§ 2. Hirschman-Widder Functions

§ 3. A Sequence of Translated Hirschman-Widder Functions

§ 4. Entire Function Generated by a Given Sequence of Exponents

§ 5. Generalized Euler's Gamma Function

§ 6. Generalized Phragmén's Discontinuity Factor

§ 7. A Theorem on Bounded Moments

§ 8. Titchmarsh's Theorem on Convolution

Chapter VIII. The Exponential Function exp(-λsa)

§ 1. Introduction

§ 2. Case α < 0

§ 3. Case α = 0

§ 4. Case 0 < α < 1

§ 5. Expansion into Taylor's Series

§ 6. Function Uα(t)

§ 7. Real Formula of Uα(t)

§ 8. Properties of Uα(t) on the Real Axis

§ 9. The Case of α > 1 and real λ

§ 10. Real and Imaginary Part of an Operator

§ 11. The Case of α > 1 and a Complex, but not Imaginary λ

§ 12. The Case of α ≥ 1 and Imaginary λ

§ 13. The Case of α = 1

§ 14. Table of Existence

Part VII—Formulae and Tables

I. Special Functions

1. Gamma Function of Euler

2. Error Function

3. Bessel Functions

II. Formulae of the Operational Calculus

III. Electrotechnical Applications

1. Equation of the Circuit

2. Stationary Current

3. Table of Simple Quadripoles and their Matrices

IV. Tables of Functions

1. Gamma Function of Euler Γ(λ)

2. Error Function Erf λ

3. Bessel Function J0(λ)

4. Bessel Function J1(λ)

5. Functions J0(iλ) and -iJ1(iλ)

Appendix—Some Topics from Real Analysis

§ 1. Introduction

§ 2. Some Properties of Rk

§ 3. A Theorem of Carathéodory

§ 4. The Riemann-Lebesgue Lemma

§ 5. Metric and Normed Spaces

§ 6. The Hahn-Banach Theorem

§ 7. A Corollary to the Hahn-Banach Theorem

§ 8. The Birkhoff-Kakutani Theorem for Topological Vector Spaces

§ 9. The Arzelà-Ascoli Theorem

§ 10. Two Lemmas

§ 11. The Proof of the Denjoy-Carleman Theorem

Answers to Problems

Bibliography

Index

Other Titles in the Series

## Product details

- No. of pages: 264
- Language: English
- Copyright: © Pergamon 1987
- Published: January 1, 1987
- Imprint: Pergamon
- eBook ISBN: 9781483161457