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Publisher:

Dover Publications

Publication Date:

1996

Number of Pages:

320

Format:

Paperback

Price:

14.95

ISBN:

978-0486692197

Category:

Textbook

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by , on ]

Allen Stenger

10/26/2015

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.

PART I: ELEMENTS OF THE GENERAL THEORY OF ANALYTIC FUNCTIONS | |||||||

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 | |||||||

PART II: APPLICATIONS AND CONTINUATION OF THE GENERAL THEORY | |||||||

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(z – a1)(z – a2) . . . (z – ak) |
|||||||

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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