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One service mathematics has rendered the l moil ..., Ii j'avait su
comment en revenir, je n'y serais point aUe.' human race. It has
put common sense back Jules Verne where it belongs, on the topmost
shelf next to the dusty canister labelled 'discarded non- The
series is divergent; therefore we may be sense'. Eric T. Bell able
to do something with it. O. Heaviside Mathematics is a tool for
thought. A highly necessary tool in a world where both feedback and
non linearities abound. Similarly, all kinds of parts of
mathematics serve as tools for other parts and for other sciences.
Applying a simple rewriting rule to the quote on the right above
one finds such statements as: 'One service topology has rendered
mathematical physics .. .'; 'One service logic has rendered com
puter science .. .'; 'One service category theory has rendered
mathematics .. .'. All arguably true. And all statements obtainable
this way form part of the raison d'(ftre of this series."
This handbook covers a wealth of topics from number theory,
special attention being given to estimates and inequalities. As a
rule, the most important results are presented, together with their
refinements, extensions or generalisations. These may be applied to
other aspects of number theory, or to a wide range of mathematical
disciplines. Cross-references provide new insight into fundamental
research.
Audience: This is an indispensable reference work for
specialists in number theory and other mathematicians who need
access to some of these results in their own fields of
research.
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."
This volume is a sequel to the much-appreciated The Cauchy Method
of Residues published in 1984 (also by Kluwer under the D.Reidel
imprint). Volume 1 surveyed the main results published in the
period 1814--1982. The present volume contains various results
which were omitted from the first volume, some results mentioned
briefly in Volume 1 and discussed here in greater detail, and new
results published since 1982. It also contains short expositions,
by various authors, dealing with new and interesting aspects of the
theory and applications of residues. This volume will be of
interest to researchers and graduate students in complex analysis,
and also physicists and engineers whose work involves the
application of complex functions.
One service mathematic;., has Jcndcml the 'Et moi, .. si j'avait su
comment CD revcnir, human race. It has put COIDDlOJI SCIISC back je
n'y scrais point allC.' whc: rc it belongs, on the topmost shell
next Jules Verne to the dusty canister labc1lcd 'dilcardcd nOD- The
series is divergent; tbcre(on: we may be sense'. Eric T. Bcll able
to do something with it o. Hcavisidc Mathematics is a tool for
thought. A highly necessary tooll in a world where both feedbaclt
and non linearities abound. Similarly, all kinds of parts of
mathematics serve as tools for other paJts and for other sciences.
Applying a simple rewriting rule to the quote on the right above
one finds such statements as: 'One service topology has rendered
mathematical physics .. .'; 'One service logic has rendered com
puter science .. .'; 'One service category theory has rendered
mathematics .. .'. All arguably true. And all statements obtainable
this way form part of the raison d'etre of this series."
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.
One service mathematics has rendered the l moil ..., Ii j'avait su
comment en revenir, je n'y serais point aUe.' human race. It has
put common sense back Jules Verne where it belongs, on the topmost
shelf next to the dusty canister labelled 'discarded non- The
series is divergent; therefore we may be sense'. Eric T. Bell able
to do something with it. O. Heaviside Mathematics is a tool for
thought. A highly necessary tool in a world where both feedback and
non linearities abound. Similarly, all kinds of parts of
mathematics serve as tools for other parts and for other sciences.
Applying a simple rewriting rule to the quote on the right above
one finds such statements as: 'One service topology has rendered
mathematical physics .. .'; 'One service logic has rendered com
puter science .. .'; 'One service category theory has rendered
mathematics .. .'. All arguably true. And all statements obtainable
this way form part of the raison d'(ftre of this series."
The Theory of Inequalities began its development from the time when
C. F. GACSS, A. L. CATCHY and P. L. CEBYSEY, to mention only the
most important, laid the theoretical foundation for approximative
meth ods. Around the end of the 19th and the beginning of the 20th
century, numerous inequalities were proyed, some of which became
classic, while most remained as isolated and unconnected results.
It is almost generally acknowledged that the classic work "Inequali
ties" by G. H. HARDY, J. E. LITTLEWOOD and G. POLYA, which appeared
in 1934, transformed the field of inequalities from a collection of
isolated formulas into a systematic discipline. The modern Theory
of Inequalities, as well as the continuing and growing interest in
this field, undoubtedly stem from this work. The second English
edition of this book, published in 1952, was unchanged except for
three appendices, totalling 10 pages, added at the end of the book.
Today inequalities playa significant role in all fields of
mathematics, and they present a very active and attractive field of
research. J. DIEUDONNE, in his book "Calcullnfinitesimal" (Paris
1968), attri buted special significance to inequalities, adopting
the method of exposi tion characterized by "majorer, minorer,
approcher." Since 1934 a multitude of papers devoted to
inequalities have been published: in some of them new inequalities
were discovered, in others classical inequalities, vere sharpened
or extended, various inequalities, vere linked by finding their
common source, while some other papers gave a large number of
miscellaneous applications."
One service mathematic;., has Jcndcml the 'Et moi, .. si j'avait su
comment CD revcnir, human race. It has put COIDDlOJI SCIISC back je
n'y scrais point allC.' whc: rc it belongs, on the topmost shell
next Jules Verne to the dusty canister labc1lcd 'dilcardcd nOD- The
series is divergent; tbcre(on: we may be sense'. Eric T. Bcll able
to do something with it o. Hcavisidc Mathematics is a tool for
thought. A highly necessary tooll in a world where both feedbaclt
and non linearities abound. Similarly, all kinds of parts of
mathematics serve as tools for other paJts and for other sciences.
Applying a simple rewriting rule to the quote on the right above
one finds such statements as: 'One service topology has rendered
mathematical physics .. .'; 'One service logic has rendered com
puter science .. .'; 'One service category theory has rendered
mathematics .. .'. All arguably true. And all statements obtainable
this way form part of the raison d'etre of this series."
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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