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Book review

Author(s) Mitrinovic, Dragoslav S.
Keckic, Jovan B.
Title THE CAUCHY METHOD OF RESIDUES, vol. 2 Theory and Applications
Publisher Kluwer Academic Publishers
Year of publication 1993
Reviewed by S.Rogosin

This book is highly following to East European pedagogica1 traditions. Here are given not only results on the subject but also are described the simple methods of their obtaining. It makes possible to use this book even by people not familiar with high level modern mathematics. As a result it is extremely useful also for teachers delivered courses in Complex Analysis for graduate and postgraduate students. Besides along the book are scattered a great number of results which can be interesting for the specialists in Analysis.

The history of this volume's appearance is quite long. In 1978 in Serbian was published the book "Cauchyiev racun ostatka sa primenama" (by Naucna Knjiga, Belgrade). Its revized and extended English translation appeared in 1984 under the title "The Cauchy Method Of Residues Decry and Application" (D.Reidel Publishing Company). The latter was reprinted then by Kluwer Academic Publishers as the first volume for present book. At last the extended and enlarged edition of the Serbian variant was published in Belgrade in 1991. The reviewed volume is based upon the latter edition and contains all the results not presented in the previous editions (so neither in the volume 1) as well as new formulas and approaches appeared around the mentioned method in mathematical journals and books after 1982.

There are also some interesting additions made specially for this publication. First of all it is included as the chapter 7 discussion on the contents and ideas of Master's Dissertation of J.V. Sokhocki. It was practically first text in Russian literature on Cauchy method of residues not published yet since 1868. It has a great historical interest. The dissertation contains many original contributions which have always been ascribed to other mathematicians. Besides in the book are included a number of modern original results on theme. They are such topics as generalized values of improper integral (by D.S. Dimitrovski), numerical evaluations of definite integrals (by D.D. Tosic), inclusive calculus of residues (by M.S. Petkovic), polynomials orthogonal on a semicircle (by G.V. Milovanovic), residue theorem for distributions (by D. Mitrovic).

The main part of the reviewed book is highly sequel to the main part of the volume 1 (even the titles of the chapters are practically the same as those in vol. 1). On the discussion are the questions connected with the old Cauchy residue theorem and the based on it method of residues. This method is of nature of analytic functions. Its attractiveness is in possibility to represent "infinite-type" things (as sum of infinite series, value of definite integrals etc.) through the "finite-type" ones (a sum of residues). It seems very simple but is still highly applicable because of the possibility of its involving in many mathematical problems.

The authors could find the style of material's introduction which show the readers the advantages of the proposing method as understandable as possible. Nevertheless the book remains the deep, full of ideas mathematical text. All this makes it very useful for wide audience from the students in mathematics, physics and engineering until the specialists in different branches of human knowledge. It can be interesting as a brilliant example of a textbook in mathematics. Its contents is also interesting from the point of vies of History of Mathematics.