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THE purpose of this book is to prescnt a straightforward
introduction to complex numbers and their properties. Complex
numbers, like other kinds of numbers, are essen tially objects with
which to perform calculations a: cording to certain rules, and when
this principle is borne in mind, the nature of complex numbers is
no more mysterious than that of the more familiar types of numbers.
This formal approach has recently been recommended in a Reportt
prepared for the Mathematical Association. We believe that it has
distinct advantages in teaching and that it is more in line with
modern algebraical ideas than the alternative geometrical or
kinematical definitions of v -1 that used to be proposed. On the
other hand, an elementary textbook is clearly not the place to
enter into a full discussion of such questions as logical
consistency, which would have to be included in a rigorous
axiomatic treatment. However, the steps that had to be omitted
(with due warning) can easily be filled in by the methods of
abstract algebra, which do not conflict with the 'naive' attitude
adopted here. I should like to thank my friend and colleague Dr. J.
A. Green for a number of valuable suggestions, especially in
connection with the chapter on convergence, which is a sequel to
his volume Sequences and Series in this Library."
The aim of this book is to give an elementary treatment of multiple
integrals. The notions of integrals extended over a curve, a plane
region, a surface and a solid are introduced in tum, and methods
for evaluating these integrals are presented in detail. Especial
reference is made to the results required in Physics and other
mathematical sciences, in which multiple integrals are an
indispensable tool. A full theoretical discussion of this topic
would involve deep problems of analysis and topology, which are
outside the scope of this volume, and concessions had to be made in
respect of completeness without, it is hoped, impairing precision
and a reasonable standard of rigour. As in the author's Integral
Calculus (in this series), the main existence theorems are first
explained informally and then stated exactly, but not proved.
Topological difficulties are circumvented by imposing some what
stringent, though no unrealistic, restrictions on the regions of
integration. Numerous examples are worked out in the text, and each
chapter is followed by a set of exercises. My thanks are due to my
colleague Dr. S. Swierczkowski, who read the manuscript and made
valuable suggestions. w. LEDERMANN The University of Sussex,
Brighton."
THE purpose of this book is to present a straightforward
introduction to complex numbers and their properties. Complex
numbers, like other kinds of numbers, are essen tially objects with
which to perform calculations according to certain rules, and when
this principle is borne in mind, the nature of complex numbers is
no more mysterious than that of the more familiar types of numbers.
This formal approach has recently been recommended in a Reportt
prepared for the Mathematical Association. We believe that it has
distinct advantages in teaching and that it is more in line with
modern algebraical ideas than the alternative geometrical or
kinematical definitions of ..; - 1 that used to be proposed. On the
other hand, an elementary textbook is clearly not the place to
enter into a full discussion of such questions as logical
consistency, which would have to be included in a rigorous
axiomatic treatment. However, the steps that had to be omitted
(with due warning) can easily be filled in by the methods of
abstract algebra, which do not conflict with the 'naive' attitude
adopted here. I should like to thank my friend and colleague Dr. J.
A. Green for a number of valuable suggestions, especially in
connection with the chapter on convergence, which is a sequel to
his volume Sequences and Series in this Library."
Die elementaren Operationen in der Arithmetik bestehen darin, daB
man zwei ZaWen a und b in Ubereinstimmung mit einigen
wohldefinierten Regeln verkniipft und so eine neue eindeutig
bestimmte zaW c erMlt. Nehmen wir zum Beispiel als
Verkniipfungsregel die Multiplikation, so schreiben wir c = ab.
Wenn a und b gegeben sind, dann kann die zaW c in jedem Fall
gefunden werden. Es ist bekannt, daB die Multiplikation von zwei
oder mehreren Zahlen gewissen for- malen Regeln gehorcht, welche
fur aile Produkte gelten, unabhiingig yom spezieilen nume- rischen
Wert: (Ll) ab = ba; Kommutativgesetz (1. 2) (ab)c = a(bc)
Assoziativgesetz (1. 3) la=al=a Die letzte Gleichung hat die
Einftihrung eines spezieilen Elementes, des Einselementes, zur
Folge. Das zweite Gesetz lautet ausftihrlicher: wenn wir ab = s und
bc = t setzen, dann gilt immer sc = at. In der axiomatischen
Behandlung der Arithmetik ist es iiblich, zuerst die Axiome oder
Postulate etwa solche wie (1. 1), (1. 2) und (1. 3) festzulegen,
sowie auch gewisse andere Ver- fahrensregeln beziiglich der
Addition oder der Multiplikation einzuftihren, und man leitet davon
dann die logischen Folgerungen abo Es ist dabei am Anfang
unwesentlich, ob die Symbole a, b, . . . ZaWen, wie wir sie im
iiblichen Sinne verstehen darstellen, oder etwa an- dere
mathematische Gr6Ben, ja man verzichtet oft auf eine konkrete
Interpretation. Es sind auch zaWreiche axiomatische Systeme im
logischen Sinne m6glich, jedoch sind diese nicht alle in gleicher
Weise interessant oder wichtig.
Professor Walter Ledermann is one of the great algebraists of the
twentieth century. His memoirs begin with life in pre-war Germany,
the murder of several members of his family, and of the joy he
found in mathematics and music. As the story of his remarkable life
unfolds, we are entranced by tales of Scotland during the war and
of academic life in Manchester and Sussex. His memoirs contain
numerous entertaining, and often hilarious anecdotes of his
encounters with famous mathematicians and physicists, such as Issai
Schur, Heinz Hopf, Max Plank, Erwin Schroedinger, Edmund Whittaker,
Alec Aitkin, Max Born and Alan Turing.
To an algebraist the theory of group characters presents one of
those fascinating situations, where the structure of an abstract
system is elucidated by a unique set of numbers inherent in the
system. But the subject also has a practical aspect, since group
characters have gained importance in several branches of science,
in which considerations of symmetry play a decisive part. This is
an introductory text, suitable for final-year undergraduates or
postgraduate students. The only prerequisites are a standard
knowledge of linear algebra and a modest acquaintance with group
theory. Especial care has been taken to explain how group
characters are computed. The character tables of most of the
familiar accessible groups are either constructed in the text or
included amongst the exercise, all of which are supplied with
solutions. The chapter on permutation groups contains a detailed
account of the characters of the symmetric group based on the
generating function of Frobenius and on the Schur functions. The
exposition has been made self-sufficient by the inclusion of
auxiliary material on skew-symmetric polynomials, determinants and
symmetric functions.
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