Theory of Functions Part 1: Elements of the General Theory of Analytical Functions

180 Pages·1945·4.66 MB·English

by  Konrad Knopp

THEORY OF FUNCTIONS.
PART I. ELEMENTS OF THE GENERAL THEORY
OF ANALYTIC FUNCTIONS
by Konrad Knopp
This is the second volume in the five-volume set THE THEORY OF
FUNCTIONS prepared by Konrad Knopp, a mathematician of inter·
national renown. It may be used separately, or with other volumes
in the series, or with any other text on theory of functions. It is un
usual in its field in being concise, clear, easy to follow, yet complete
and rigorous. Demonstrations are full, and proofs are given in detail.
THEORY OF FUNCTIONS, PART I considers the general foundations of
theory of functions. It provides the student with background for fur
ther books on a more advanced level. Stress is upon general founda
tions rather than specific functions.
PARTIAL CONTENTS. I. FUNDAMENTAL CONCEPTS. Numbers, points,
functions of a complex variable, Cauchy-Riemann differential equa
tions. II. INTEGRAL THEOREMS. Integral of a continuous function.
Cauchy's integral theorem, integral formulas. Ill. SERIES AND THE
EXPANSION OF ANALYTIC FUNCTIONS IN SERIES. Series with variable
terms, convergence. Expansion of analytic functions in power series,
identity theorem for analytical functions. Coiiti-nJlation and complete
definition of analytic functions, monodromy theorem', multiple-valued
functions. Entire transcendental functions. IV. SINGULARITIES. laurent
expansion. Various types of singularities.
"Not only for the young student, but also for the student who knows
all about what is in it," MATHEMATICAL JOURNAL.
+
Bibliography. Index. 8 figures. vii 146pp. 5 3fs x 8.
S1 56 Paperbound $1.35
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First American Edition
CoPYRIGHT 1945
By DOVER PUBLICATIONS
THEORY
OF FUNCTIONS
By
DR. KONRAD K VOPP
Professor of Mathematics at the University of Tubingen
Translated by
FREDERICK BAGEMIHL, M.A.
Instructor in Mathematics at the University of Rochester
PART ONE
ELEMENTS OF THE GENERAL THEORY OF
ANALYTIC FUNCTIONS
NEw YoRK
DOVER PUBLICATIONS

PREFACE
This little }book follows rather closely the fifth
edition of Dr. Knopp's Funktionentheorie. Several
changes have been made in order to conform to common
English terminology and notation, and to render cer
tain passages more precise or rigorous than they are in
the German volume. The proofs of Lemmas 1 and 2
in § 4 were found to be incorrect, and proofs remedy
ing this defect were substituted. Typographical errors
have been corrected, the bibliography has undergone
some minor changes, and a few helpful references have
been added to the text.
The Translator

CONTENTS
SECTION I
Fundamental Concepts
CHAPTER 1. NuMBERs AND PoiNTs PAGE
§ 1. Prerequisites . . . . . . . . . . . . . 1
§ 2. The Plane and Sphere of Complex N um-
bers .............. . 2
§ 3. Point Sets and Sets of Numbers 5
§ 4. Paths, Regions, Continua . . . . . . . 13
CHAPTER 2. FUNCTIONS OF A COMPLEX VARIABLE
§ 5. The Concept of a Most General (Single
valued) Function of a Complex Variable 21
§ 6. Continuity and Differentiability 23
§ 7. The Cauchy-Riemann Differential Equa-
tions . . . . . . . . . . . . . . . . 28
SECTION II
Integral Theorem,:;
3.
CHAPTER THE INTEGRAL oF A CoNTINuous
FuNCTION
§ 8. Definition of the Definite Integral 32
§ 9. Existence Theorem for the Definite
Integral . 34
§ 10. Evaluation of Definite Integrals 38
§ 11. Elementary Integral Theorems . 44
v
vi CONTENTS
CHAPTER 4. CAUCHY's INTEGRAL THEOREM PAGE
§ 12. Formulation of the Theorem . . . . 47
§ 13. Proof of the Fundamental Theorem . 49
§ 14. Simple Consequences and Extensions 55
CHAPTER 5. CAUCHY'S INTEGRAL FORMULAS
§ 15. The Fundamental Formula . . . . . 61
§ 16. Integral Formulas for the Derivatives . 62
SECTION III
Series and the Expansion of Analytic Functions
in Series
CHAPTER 6. SERIES WITH VARIABLE TERMS
§ 17. Domain of Convergence . . . . . . 67
§ 18. Uniform Convergence . . . . . . . 71
§ 19. Uniformly Convergent Series of Analytic
Functions . . . . . . . . . . . . . . 73
CHAPTER 7. THE EXPANSION OF ANALYTIC
FuNcTIONS IN PowER SERIES
§ 20. Expansion and Identity Theorems for
Power Series . . . . . . . . . . . . . 79
§21. The Identity Theorem for Analytic Func-
tions ........... . 85
CHAPTER 8. ANALYTIC CoNTINUATION AND
CoMPLETE DEFINITION OF
ANALYTIC FuNCTIONS
§22. The Principle of Analytic Continuation 92
§ 23. The Elementary Functions ..... 96
§ 24. Continuation by Means of Power Series
and Complete Definition of Analytic
Functions ............ . 98
§25. The Monodromy Theorem . . . . . . 105
§ 26. Examples of Multiple-valued Functions . 107