Theory of Functions

Overview

This is a one-volume edition of Parts I and II of the classic five-volume set The Theory of Functions prepared by renowned mathematician Konrad Knopp. Concise, easy to follow, yet complete and rigorous, the work includes full demonstrations and detailed proofs.
Part I stresses the general foundation of the theory of functions, providing the student with background for further books on a more advanced level.
Part II places major emphasis on ...

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Theory of Functions, Parts I and II

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Overview

This is a one-volume edition of Parts I and II of the classic five-volume set The Theory of Functions prepared by renowned mathematician Konrad Knopp. Concise, easy to follow, yet complete and rigorous, the work includes full demonstrations and detailed proofs.
Part I stresses the general foundation of the theory of functions, providing the student with background for further books on a more advanced level.
Part II places major emphasis on special functions and characteristic, important types of functions, selected from single-valued and multiple-valued classes.

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Product Details

  • ISBN-13: 9780486692197
  • Publisher: Dover Publications
  • Publication date: 8/12/1996
  • Series: Dover Books on Mathematics Series
  • Pages: 320
  • Product dimensions: 5.40 (w) x 8.00 (h) x 0.63 (d)

Table of Contents

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
Introduction

Section 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 Functions
 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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