Introduction To The Operational Calculus

1st Edition - January 1, 1967
Author: Lothar Berg
Language: English
eBook ISBN:
9 7 8 - 0 - 3 2 3 - 1 6 2 4 5 - 6

Introduction to the Operational Calculus is a translation of "Einfuhrung in die Operatorenrechnung, Second Edition." This book deals with Heaviside's interpretation, on the… Read more

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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.

Introduction 1. General Survey 2. The Heaviside Method 3. A Rigorous Approach 4. Derivation of an Integral Transformation 5. Numerical Evaluations 6. Improved ApproximationsChapter 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 RingsChapter 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. ApplicationsChapter 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 pChapter IV. Applications 19. Differential Equations with Constant Coefficients 20. Examples 21. Systems of Differential Equations 22. Degenerate Systems 23. Control Engineering 24. Integral EquationsChapter 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 OperatorChapter 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 TheoremChapter 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 EquationsChapter VIII. Asymptotic Properties 43. Definitions 44. Abelian Theorems 45. Further Types of Singularity 46. Tauberian Theorems 47. Problems of Stability 48. Euler's Summation FormulaChapter 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 IntegralChapter 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 CoefficientsAppendixAnswers to ExercisesReferences A. Text-Books and Monographs B. Original PapersFormulaeSubject Index