Operational Calculus
Published on 1. January 1970
Book
Hardback
366 pages
978-3-540-04896-1 (ISBN)
Description
Since the publication of an article by G. DOETSCH in 1927 it has been known that the Laplace transform procedure is a reliable sub- stitute for HEAVISIDE'S operational calculus*. However, the Laplace transform procedure is unsatisfactory from several viewpoints (some of these will be mentioned in this preface); the most obvious defect: the procedure cannot be applied to functions of rapid growth (such as the 2 function t ~ exp (t )). In 1949 JAN MIKUSINSKI indicated how the un- necessary restrictions required by the Laplace transform can be avoided by a direct approach, thereby gaining in notational as well as conceptual simplicity; this approach is carefully described in MIKUSINSKI'S textbook "Operational Calculus" [M 1J. . The aims of the present book are the same as MIKUSINSKI'S [M 1J: a direct approach requiring no un-necessary restrictions. The present operational calculus is essentially equivalent to the "calcul symbolique" of distributions having left-bounded support (see 6.52 below and pp. 171 to 180 of the textbook "Theorie des distributions" by LAURENT SCHWARTZ).
More details
Language
English
Place of publication
Berlin
Germany
Publishing group
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Target group
Professional and scholarly
Illustrations
12 black & white illustrations, biography
Weight
900 gr
ISBN-13
978-3-540-04896-1 (9783540048961)
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Content
1.- 0. Operators.- Test-functions and Operators.- Convolution.- Entering Functions.- 1. Perfect Operators.- Algebra of Perfect Operators.- The Operator of an Entering Function.- Translation Property.- K-functions.- Numbers and Operators.- Ratios of Perfect Operators.- 2.- 2. The Basic Facts.- Translation Property.- Numbers and Operators.- Functions Having no Jumps on [0, ?).- Various Consequences.- Three Integral Equations.- Differential Equations: Procedure.- Higher Derivatives.- Continuity of the Convolution Integral.- 3. Elementary Applications.- Table of Formulas.- Illustrative Examples.- 4. Partial Fraction Decomposition.- Decompositions Obtained by Division.- Heaviside's Expansion.- Decomposition into Two Parts.- Repeated Linear Factors.- Heaviside Procedure.- Repeated Quadratic and Linear Factors.- Illustrative Examples.- 3.- 5. Further Applications.- Series of Translates.- Periodic Functions.- Advantages of Operational Calculus.- Use of Laplace Transform Tables.- Difference Equations.- Bibliographical Comments.- Notes and Further Comments.- 6. Calculus of Operators.- Convergence.- Value of an Operator at a Point.- Integral of an Operator.- Function-operators.- Impulse and Dipole.- Distributions and Operators.- Two Examples.- Multiplication by t.- Static Deflection of Beams.- 7. Vectors.- Electric Circuits.- The General Case.- 8 Non-integrable Functions.- Convolution of Operators.- Finite Part of a Divergent Integral.- Indexed Functions.- Derivatives on the Open Interval (0, ?).- Duhamel's Formula.- 4.- 9. Partial Differential Equations.- Differentiation with Respect to x.- Vibrating String.- A Diffusion Problem.- Wave Problems.- The Infinite Interval.- Classical Solutions.- The Finite Interval.- 10. Diffusion Problems.- The Infinite Interval.- Insulated Rod.- The Finite Interval.- Comparison with the Laplace Transformation.- 5.- 11. Series of Operators.- Series of Impulses.- Analytic Functions of h/D.- A Functional Calculus for h/D.- Applications.- Difference-differential Equations.- 12. A Functional Calculus for D.- Applications.- 13. Non-linear Equations.- Applications.- 14. Differential Equations with Polynomial Coefficients.- Table of Formulas.- Illustrative Examples.- 15. Theorems.- Three Basic Theorems.- A Theorem for 6.- A Theorem for 9.- A Theorem for 11.- Glossary of Terminology and Notations.- Terminology.- Notations.- Summary of Results and Table of Formulas.- Elementary Formulas.- Periodic Functions.- Bibliographical Comments.- Subject and Author Index.