Operational Calculus - Gregers Krabbe

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Résumé :
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).

Sommaire:
1.- ?0. Operators.- ? 1. Perfect Operators.- 2.- ?2. The Basic Facts.- ? 3. Elementary Applications.- ? 4. Partial Fraction Decomposition.- 3.- ?5. Further Applications.- ? 6. Calculus of Operators.- ?7. Vectors.- ?8 Non-integrable Functions.- 4.- ? 9. Partial Differential Equations.- ? 10. Diffusion Problems.- 5.- ? 11. Series of Operators.- ? 12. A Functional Calculus for D.- ? 13. Non-linear Equations.- ? 14. Differential Equations with Polynomial Coefficients.- ? 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.