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Book review

Author(s) Keckic, D.S. Mitinovic and J.D.
Title The Cauchy Method of Residues, Vol. II. Theory and Applications
Publisher Kluwer Academic Publishers
Year of publication 1993
Reviewed by S. Cobzas

The first volume of this book was published by D. Reidel Publishing Co. in 1984, being a revised, extended and updated translation of the authors' book "Cauchyjev racun ostataka sa primenama", published in Serbian by Naucna Knjiga, Belgrade 1978. The favourable reception by the mathematical community of the first volume encouraged the authors to write a second one, based this time on the second Serbian edition of the above mentioned book, published in 1991.

The books (volumes I and II) are the only ones in mathematical literature covering all known applications of residues theory as theory of equations, theory of numbers, matrix analysis, evaluation of real definite integrals, summation of finite and infinite series, expansions of functions into infinite series or products, ordinary and partial differential equations, mathematical and theoretical physics, calculus of finite differences and difference equations.

Chapter 1 has an introductory character, providing the reader with the basic notions and results of complex function theory, used in the rest of the book. It contains also some errata to the first volume signaled by the readers or discovered by the authors.

Chapters 2 - 6 are supplements to the corresponding chapters of Volume I.

Chapter 7 is devoted to the Master's Thesis of the Russian mathematician J.V. Sohocki, published in 1868, and containing many original contributions to complex function theory which usually have been ascribed to other mathematicians. For instance, Sohocki discovered 8 years before Weierstrass the theorem on the behaviour of an analytic function in a neighbourhood of an essential singularity. Also he applied the residue calculus to the study of Legendre polynomials.

The rest of the chapters (Chapters 8 - 13) are written by various authors and are based mainly on their own research work.

Chapter 8 is written by D.S. Dimitrovski, University of Skopje, and is concerned with principal and generalized values of improper integrals. Chapter 9, written by D.D. Tosic, from University of Beograd, is devoted to numerical evaluation of integrals.

In Chapter 10, M.S. Petrovic, university of Nis, using a disc arithmetic in the complex plane (an analogue of the interval arithmetic in R), develops a calculus of residues for functions having discs as arguments and values. This approach is justified by the use of computers in the calculation of integrals, when rounding errors and errors of initial data must be considered.

Chapter 11, Complex Polynomials Orthogonal on the Semicircle is written by G.V. Milovanovic, University of Nis, and Chapters 12, A Representation of Half Plane Meromorphic Function and 13, Calculus of Residues and Distributions, are written by D. Mitrovic, University of Zagreb. This last chapter contains an interesting generalization of the residue theorems to the distribution framework, obtained by the author.

The result is a valuable book which, together with Volume I, will find undoubtely a large audience, including researchers in complex analysis as well as physicists and engineers.

The book as a whole is suitable for post graduate courses but some part can be used also for undergraduate courses.