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Full 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).
Contents
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.
