# Theory of Functions, Parts I and II

Publisher:
Dover Publications
Number of Pages:
320
Price:
14.95
ISBN:
978-0486692197
Friday, July 24, 2009
Reviewable:
Yes
Include In BLL Rating:
Yes
Publication Date:
1996
Format:
Paperback
Audience:
Category:
Textbook
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07/24/2009
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 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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