Operational Calculus

¿Foreword to the First English EditionForeword to the Second English EditionSupplements to Volume I (A) Supplement to Part I, Chapter VI (B) Supplement to Part III, Chapter VIIIPart 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 ProblemsPart 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 MethodsPart 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.