Operational Calculus and Related Topics
This book covers a reasonable portion of topics encountered in the theory and practice of operational calculi for functions of one real variable. It is intended and can be used both as an introductory text or a reference book for mathematicians, physicists and engineers. In the first of three chapters a vast collection of integral transforms is gathered. Operational rules, convolution, some special inverse formulas and selected important properties are presented for each of the transforms. The second chapter is dedicated to Mikusiński's operational calculus. A field of so-called operators is built out of a ring of functions. Most importantly, there is an operator p in this field that represents differentiation. Many algebraic and analytical properties and operational rules are investigated. The third chapter deals with generalized functions. After a brief exposition of the ‘functional approach’ (i.e. distributions as functionals on the space D(R)), a bigger part of this section presents ‘sequential approach’ (pioneered again by J. Mikusiński), which defines distributions as equivalence classes of certain sequences of smooth functions.
All three chapters are packed with many examples of concrete computations of transforms of selected functions. Careful attention is given to the application of exposed calculi to the solution of differential and integral equations, summation of series and evaluation of integrals. Most of the assertions are proved and the proofs are very detailed. The reader is expected to know the basics of calculus, Lebesgue integration, complex analysis and algebra. Functional analysis is intentionally avoided. As it happens, there are some typos and grammatical mishaps. Mostly they are insignificant. Unfortunately, in some cases (e.g. "Fourier transform of L1 function must not be L1") the very mathematical meaning is influenced.