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Growing specialization and diversification have brought a host of
monographs and textbooks on increasingly specialized topics.
However, the "tree" of knowledge of mathematics and related fields
does not' grow only by putting forth new branches. It also happens,
quite often in fact, that branches which were thought to be
completely disparate are suddenly seen to be related. Further, the
kind and level of sophistication of mathematics applied in various
sciences has changed drastically in recent years: measure theory is
used (non-trivially) in regional and theoretical economics;
algebraic geometry interacts with physics; the Minkowsky lemma,
coding theory arid the struc ture of water meet one another in
packing and covering theory; quantum fields, crystal defects and
mathematical programming profit from homotopy theory; lie algebras
are relevant to filtering; and prediction and electrical
engineering can use Stein spaces. And in addition to this there are
such new emerging subdisciplines as "completely integrable
systems," "chaos, synergetics and large-5cale order," which are
almost impossible to fit into the existing classification schemes.
They draw upon widely different sections of mathematics. This
program, Mathematics and Its Applications, is devoted to such (new)
interrelations as exampla gratia: - a central concept which plays
an important role in several different mathe matical and/or
scientific specialized areas; - new applications of the results and
ideas from one area of scientific en deavor into another; -
influences which the results, problems and concepts of one field of
enquiry have and have had on the development of another."
Volume 1, i. e. the monograph The Cauchy Method of Residues -
Theory and Applications published by D. Reidel Publishing Company
in 1984 is the only book that covers all known applications of the
calculus of residues. They range from the 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 and products, ordinary and partial
differential equations, mathematical and theoretical physics, to
the calculus of finite differences and difference equations. The
appearance of Volume 1 was acknowledged by the mathematical
community. Favourable reviews and many private communications
encouraged the authors to continue their work, the result being the
present book, Volume 2, a sequel to Volume 1. We mention that
Volume 1 is a revised, extended and updated translation of the book
Cauchyjev raeun ostataka sa primenama published in Serbian by Nau
na knjiga, Belgrade in 1978, whereas the greater part of Volume 2
is based upon the second Serbian edition of the mentioned book from
1991. Chapter 1 is introductory while Chapters 2 - 6 are
supplements to the corresponding chapters of Volume 1. They mainly
contain results missed during the preparation of Volume 1 and also
some new results published after 1982. Besides, certain topics
which were only briefly mentioned in Volume 1 are treated here in
more detail.
Growing specialization and diversification have brought a host of
monographs and textbooks on increasingly specialized topics.
However, the "tree" of knowledge of mathematics and related fields
does not' grow only by putting forth new branches. It also happens,
quite often in fact, that branches which were thought to be
completely disparate are suddenly seen to be related. Further, the
kind and level of sophistication of mathematics applied in various
sciences has changed drastically in recent years: measure theory is
used (non-trivially) in regional and theoretical economics;
algebraic geometry interacts with physics; the Minkowsky lemma,
coding theory arid the struc ture of water meet one another in
packing and covering theory; quantum fields, crystal defects and
mathematical programming profit from homotopy theory; lie algebras
are relevant to filtering; and prediction and electrical
engineering can use Stein spaces. And in addition to this there are
such new emerging subdisciplines as "completely integrable
systems," "chaos, synergetics and large-5cale order," which are
almost impossible to fit into the existing classification schemes.
They draw upon widely different sections of mathematics. This
program, Mathematics and Its Applications, is devoted to such (new)
interrelations as exampla gratia: - a central concept which plays
an important role in several different mathe matical and/or
scientific specialized areas; - new applications of the results and
ideas from one area of scientific en deavor into another; -
influences which the results, problems and concepts of one field of
enquiry have and have had on the development of another."
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