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Zbl 0532.44003
Mikusiński, Jan
Operational calculus. Transl. from the Polish by Janina Sḿolska. 2nd ed. Vol. I. (English)
[B] International Series in Pure and Applied Mathematics, Vol. 109. Oxford etc.: Pergamon Press; Warszawa: PWN - Polish Scientific Publishers. 321 p. {\$} 27.00; \sterling 12.00 (1983).

The present edition of this book is the sixth one. The first (in Polish) appeared in 1953 (for a review see Zbl 0078.113; English translation (1959; Zbl 0088.330)). Now, it is devided in two volumes. We have to speak about Volume I. Six editions in 30 years tell how this book is popular for both engineers and mathematicians. Based on an abstract mathematical theory, elaborated with minimal mathematical tools, written outstandingly logical and clear the book is accepted by a very wide users. The additional value is that it includes many applications of a practical interest. The basic idea is to enlarge the ring C of continuous functions defined in the interval $[0,\infty)$, with operations: sum and convolution, to a field. This is the field M of Mikusinski's operators. So, we have the set of operators with a rich algebraic structure. The topological structure is based on a convergence class which is not topological. This convergence class has excited many researches till these days. It is a matter of fact that this operational calculus is based on numerical functions in one dimension which restricts the application of it. This first volume includes foundations of the operational calculus and its applications to physics and engineering. (The theoretical treatments are left for the second volume). It contains the first three parts: Operational algebra, sequences and series of operators and operational differential calculus. In the third part there is a completely new chapter (with 20 pages): Applications to chromatography, written in collaboration with J. Rogut. As in all applications in this book, the mathematical operations follow the physical meaning and the practical interest, but one can find also the mathematical generalisation of the treated problem. Between corrections and exchanges we will mention only the introduction of the transfer function in the analysis of the solution of a differential equation.
[B.Stanković]

MSC 2000:: *44A40 Calculus of Mikusinski, etc.
34A25 Analytical theory of ODE
44-01 Textbooks (integral transforms)
00A06 Mathematics for non-mathematicians

Keywords: Mikusinski's operators; algebraic structure; topological structure; convergence class; operational calculus; foundations; Operational algebra; sequences and series of operators; operational differential calculus; chromatography

Citations: Zbl 0078.113; Zbl 0088.330

Cited in: Zbl 1069.47034 Zbl 0753.44001 Zbl 0643.44005 Zbl 0542.44001

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