He took an active part in secret education of secondary school pupils and students in Kraków. Because of the teaching activity he was arrested twice by the occupying Nazi force. He also took part in Professor Tadeusz Ważewski's underground seminar together with a group of Kraków mathematicians. The participants of Ważewski's seminar, in 1943, were the first persons to come in contact with a new theory which is now very well known in the world of mathematics as the Mikusiński operational calculus. The objects of the theory, operators, provide a common generalization of numbers and locally integrable functions on the positive half-line. The author first called them "hypernumbers" and gave this title to a paper containing the main ideas of the theory.

This paper is quite unusual. It is not a new paper. It has been published already (in Polish), but the size of the first edition and the circumstances of its publication were out of the ordinary. The edition was limited to seven copies, and printing was made by the author himself by using X-ray films. The matter will become clear when we add that the paper appeared in Kraków, in the year 1944, during the Nazi occupation of Poland. It also explains why the author conceals his identity under the initials J.G.-M., and why only initials appear in the acknowledgement at the end of the author's introduction. The time has come to solve this riddle. The letters J.G.-M. stand for Jan G.-Mikusiński (G, being the first letter of Geniusz, the author's patronymic, which he later abandoned). This name is well known nowadays in the world of mathematics. It belongs to an eminent Polish mathematician ... The paper 'Hypernumbers' represents the first version of the Mikusiński operational calculus (improved afterwards by the use of the Titchmarsh theorem) and contains already the main ideas of this theory. The date of the edition of the paper is 1944, but the results were already presented by the author in 1943, during one of the secret wartime seminars conducted by Professor Tadeusz Ważewski in Kraków.

In the 1950's, Jan Mikusiński proposed a new approach to develop an operational calculus for the operator of differentiation (see J Mikusiński, Operational Calculus (Pergamon Press, New York, 1959)). This algebraic approach was based on the interpretation of the Laplace convolution as a multiplication in the ring of the continuous functions on the real half-axis. The Mikusiński operational calculus was successfully used in ordinary differential equations, integral equations, partial differential equations and in the theory of special functions. It is worth mentioning that the Mikusiński scheme was extended by several mathematicians to develop operational calculi for differential operators with variable coefficients ...

Mikusiński's book is a remarkable, and pioneering, venture. It presents a modern and abstract mathematical theory, in a completely elementary manner, to students of physics and engineering. Although the essence of the author's theory is the extension of a ring to a field, and the introduction of a suitable topology in this field, the terms ring, field, topology do not occur in the book. The exposition is precise but never abstract. The analogy with the extension of the number concept from integers to rationals to reals is pointed out. No knowledge of algebra or topology, and no training in abstract thinking, is required. The author assumes that the reader is familiar with the notions of limit, continuity, convergence, but not necessarily with uniform convergence (which is explained in the book together with the principal theorems relating to it) or Euler's integral of the second kind and the gamma function (which are also explained). By and large, freshman and sophomore calculus are not quite sufficient for reading this book, a good course in advanced calculus is more than sufficient.

In several papers the author has published a theory containing a direct justification of the Heaviside Calculus as opposed to the various well known indirect methods using functional transforms. The purpose of the book under review is to present this theory and its applications both to engineers primarily interested in the use of efficient computational procedures and to readers desiring to understand why these procedures work. To reach such a heterogeneous readership, the author uses the text-book approach and leads the reader gently and with great skill from a completely elementary level to rather abstract concepts.

His arrival to Katowice was an event of great importance for the mathematicians working there. Shortly afterwards a group of young mathematicians from the Higher Pedagogical School in Katowice got in touch with him. The Professor delivered for them a series of lectures on operational calculus. After a year, the lectures turned to a regular seminar, first held at the Higher Pedagogical School, then at Mikusiński's home. The participants of this seminar ... remember the unique atmosphere of these meetings, discussions at a blackboard hanging in the garage or on the terrace, the rooms replacing a seminar room. The informal scientific contacts of Professor Mikusiński with Silesian mathematicians became formal only in 1966 when the Mathematical Laboratory (now the Branch of the Mathematical Institute of the Polish Academy of Sciences) was established.

The topics of the seminar were closely connected with the mathematical interests of Professor Jan Mikusiński and included the operational calculus, generalized functions, convergence structures and integration theory. In 1966 the members of the seminar organized in Katowice an international conference on generalized functions, in which about seventy mathematicians took part, among them those as eminent as Sergei Sobolev, Laurent Schwartz, Jean Dieudonné, George Temple, Hikosaburo Komatsu, Thomas K Boehme and others. Members of the seminar were invited to many mathematical centres all over the world. In the period of the duration of the seminar (1960-1987) its members published several books (more than 20 editions altogether) and about 250 scientific papers. Eight members of the seminar received their Ph.D.'s, two of them were habilitated and then two were promoted to the position of full professor.

Based on the lifetime work of leading teacher and researcher Jan Mikusiński, this classroom-tested book provides a thorough grounding in mathematical analysis, calculus and mathematical proofing. It introduces natural numbers through a new mathematical approach; replaces the Riemann integral with the more general Lebesgue integral; and rigorously develops the real number system from four simple axioms of natural numbers. Additional features include a wider range of problems than other texts - including simple and routine as well as problems requiring more in depth creativity, answers to common questions, a new approach to the concept of equivalence relation which simplifies the construction of real numbers, and a large number of computational applications.

In spite of bad conditions of his health, he was a very active researcher and worked to the last days of his life.

We would like to thank [Professor Jan Mikusiński] for all the theories and theorems with which he has enriched Polish and world mathematics and for his constant pursuit of beauty, simplicity and elegance in mathematics.