Theory of Functions, Parts I and II (eBook)

Konrad Knopp (Autor)

eBook Download: EPUB

2013
Dover Publications (Verlag)
9780486318707 (ISBN)

Infos

Handy one-volume edition. Part I considers general foundations of theory of functions; Part II stresses special and characteristic functions. Proofs given in detail. Introduction. Bibliographies.

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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.

Inhalt

PART I: ELEMENTS OF THE GENERAL THEORY OF ANALYTIC FUNCTIONSSection I. Fundamental ConceptsChapter 1. Numbers and Points 1. Prerequisites 2. The Plane and Sphere of Complex Numbers 3. Point Sets and Sets of Numbers 4. Paths, Regions, ContinuaChapter 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 EquationsSection II. Integral TheoremsChapter 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 TheoremsChapter 4. Cauchy's Integral Theorem 12. Formulation of the Theorem 13. Proof of the Fundamental Theorem 14. Simple Consequences and ExtensionsChapter 5. Cauchy's Integral Formulas 15. The Fundamental Formula 16. Integral Formulas for the DerivativesSection III. Series and the Expansion of Analytic Functions in SeriesChapter 6. Series with Variable Terms 17. Domain of Convergence 18. Uniform Convergence 19. Uniformly Convergent Series of Analytic FunctionsChapter 7. The Expansion of Analytic Functions in Power Series 20. Expansion and Identity Theorems for Power Series 21. The Identity Theorem for Analytic FunctionsChapter 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 FunctionsChapter 9. Entire Transcendental Functions 27. Definitions 28. Behavior for Large | z |Section IV. SingularitiesChapter 10. The Laurent Expansion 29. The Expansion 30. Remarks and ExamplesChapter 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 THEORYIntroductionSection I. Single-valued FunctionsChapter 1. Entire Functions 1. Weierstrass's Factor-theorem 2. Proof of Weierstrass's Factor-theorem 3. Examples of Weierstrass's Factor-theoremChapter 2. Meromorphic Functions 4. Mittag-Leffler's Theorem 5. Proof of Mittag-Leffler’s Theorem 6. Examples of Mittag-Leffler's TheoremChapter 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 FunctionsChapter 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 FunctionChapter 6. The Analytic Configuration 16. The Monogenic Analytic Function 17. The Riemann Surface 18. The Analytic ConfigurationBibliography, Index

Produkt-Details

• Erscheint lt. Verlag: 24.7.2013
• Reihe/Serie: Dover Books on Mathematics
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• ISBN-13: 9780486318707 / 9780486318707