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Theory of Functions, Parts I and II

Dover Publications
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This is an extremely traditional and straightforward introductory course in complex variables. Its big strength is that it is so clear that everything seems obvious; there is no trickiness or “brilliancy” (D. J. Newman’s term) that you have to think of to produce the proof. Its big weakness is that there are no exercises.

Dover published a five-volume series of little books by Konrad Knopp in 1945–1952 under the series title “Theory of Functions”. The books were originally published in German starting in 1918 and then revised many times. The present book is a repackaging of the two main volumes into one volume. The two problem books in the series have also been packaged into one, Problem Book in the Theory of Functions (Dover, 2000). The remaining volume, Elements of the Theory of Functions (Dover, 1952) is out of print; it is introductory to the rest of the series and deals with complex numbers and convergence and elementary functions.

The present book only covers the essentials and so does not go very deep. The first half of the book concentrates on developing the basics of analytic functions: definition, Cauchy integral theorem, representation as uniformly convergent power series, and classification of singularities. The second half concentrates on developing the properties of particular classes of functions; for example, what can we say about a function if we know it is entire (analytic everywhere except at infinity)? Unusually for introductory books there is quite a lot about multi-valued functions and Riemann surfaces.

By modern standards this is not a textbook, because there are no exercises, and historically it has been used more as a review than as a textbook. The companion problem book in the series is not keyed exactly to this book, but is organized in the same way and can easily be used in conjunction with it. However, the problem book includes complete solutions, so this combination would not be chosen as a text today.

Fashions change in mathematics, just like everything else, and although what’s in the first half of the book is still the core of any introductory course today, we usually would treat the topics of the second half much more lightly, if at all. Modern books also integrate the exercises very tightly with the narrative. A good modern introductory book that covers these core items well is Bak & Newman’s Complex Analysis.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

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Section I. Fundamental Concepts
Chapter 1. Numbers and Points
1. Prerequisites
2. The Plane and Sphere of Complex Numbers
3. Point Sets and Sets of Numbers
4. Paths, Regions, Continua
Chapter 2. Functions of a Complex Variable
5. The Concept of a Most General (Single-valued) Function of a Complex Variable
6. Continuity and Differentiability
7. The Cauchy-Riemann Differential Equations
Section II. Integral Theorems
Chapter 3. The Integral of a Continuous Function
8. Definition of the Definite Integral
9. Existence Theorem for the Definite Integral
10. Evaluation of Definite Integrals
11. Elementary Integral Theorems
Chapter 4. Cauchy's Integral Theorem
12. Formulation of the Theorem
13. Proof of the Fundamental Theorem
14. Simple Consequences and Extensions
Chapter 5. Cauchy's Integral Formulas
15. The Fundamental Formula
16. Integral Formulas for the Derivatives
Section III. Series and the Expansion of Analytic Functions in Series
Chapter 6. Series with Variable Terms
17. Domain of Convergence
18. Uniform Convergence
19. Uniformly Convergent Series of Analytic Functions
Chapter 7. The Expansion of Analytic Functions in Power Series
20. Expansion and Identity Theorems for Power Series
21. The Identity Theorem for Analytic Functions
Chapter 8. Analytic Continuation and Complete Definition of Analytic Functions
22. The Principle of Analytic Continuation
23. The Elementary Functions
24. Continuation by Means of Power Series and Complete Definition of Analytic Functions
25. The Monodromy Theorem
26. Examples of Multiple-valued Functions
Chapter 9. Entire Transcendental Functions
27. Definitions
28. Behavior for Large | z |
Section IV. Singularities
Chapter 10. The Laurent Expansion
29. The Expansion
30. Remarks and Examples
Chapter 11. The Various types of Singularities
31. Essential and Non-essential Singularities or Poles
32. Behavior of Analytic Functions at Infinity
33. The Residue Theorem
34. Inverses of Analytic Functions
35. Rational Functions
Bibliography; Index
IntroductionSection I. Single-valued Functions
Chapter 1. Entire Functions
1. Weierstrass's Factor-theorem
2. Proof of Weierstrass's Factor-theorem
3. Examples of Weierstrass's Factor-theorem
Chapter 2. Meromorphic Func
4. Mittag-Leffler's Theorem
5. Proof of Mittag-Leffler’s Theorem
6. Examples of Mittag-Leffler's Theorem
Chapter 3. Periodic Functions
7. The Periods of Analytic Functions
8. Simply Periodic Functions
9. Doubly Periodic Functions; in Particular, Elliptic Functions
Section II. Multiple-valued Functions
Chapter 4. Root and Logarithm
10. Prefatory Remarks Concerning Multiple-valued Functions and Riemann Surfaces
11. The Riemann Surfaces for p(root)z and log z
12. The Riemann Surfaces for the Functions w = root(za1)(za2) . . . (zak)
Chapter 5. Algebraic Functions
13. Statement of the Problem
14. The Analytic Character of the Roots in the Small
15. The Algebraic Function
Chapter 6. The Analytic Configuration
16. The Monogenic Analytic Function
17. The Riemann Surface
18. The Analytic Configuration
Bibliography, Index
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