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Continued Fractions : Analytic Theory and Applications / William B. Jones, W.J. Thron.

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This is an exposition of the analytic theory of continued fractions in the complex domain with emphasis on applications and computational methods.

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Bibliographic Details
Online Access: Full Text (via Cambridge)
Other Authors: Jones, William B., Thron, W. J.
Format: Electronic eBook
Language:English
Published: Cambridge : Cambridge University Press, 1984.
Series:Encyclopedia of mathematics and its applications ; no. 11.
Subjects:
Continued fractions.
Table of Contents:
  • Cover
  • Half-title
  • Title
  • Copyright
  • Dedication
  • Contents
  • Editor's Statement
  • Section Editor's Foreword
  • Introduction by Peter Henrici
  • Preface
  • Symbols
  • Chapter 1 Introduction
  • 1.1 History
  • 1.1.1 Beginnings
  • 1.1.2 Number-Theoretic Results
  • 1.1.3 Analytic Theory
  • 1.2 Overview of Contents
  • Chapter 2 Elementary Properties of Continued Fractions
  • 2.1 Preliminaries
  • 2.1.1 Basic Definitions and Theorems
  • 2.1.2 Regular Continued Fractions
  • 2.1.3 Other Continued-Fraction Expansions ...
  • 2.1.4 Algorithms for Computing Approximants ...
  • 2.2 Sequences Generated by Linear Fractional Transformations
  • 2.3 Equivalence Transformations
  • 2.3.1 Equivalent Continued Fractions
  • 2.3.2 Euler's [1748] Connection between Continued Fractions and Infinite Series
  • 2.4 Contractions and Extensions
  • 2.4.1 Contraction of Continued Fractions
  • 2.4.2 Even Part of a Continued Fraction
  • 2.4.3 Odd Part of a Continued Fraction
  • 2.4.4 Extension of a Continued Fraction
  • Chapter 3 Periodic Continued Fractions
  • 3.1 Introduction.
  • 3.2 Convergence of Periodic Continued Fractions ...
  • 3.3 Dual Periodic Continued Fractions
  • Chapter 4 Convergence of Continued Fractions
  • 4.1 Introduction
  • 4.2 Element Regions, Value Regions, and Sequences of Nested Circular Disks
  • 4.3 Necessary Conditions for Convergence
  • 4.3.1 Stern-Stolz Theorem
  • 4.3.2 Necessary Conditions for Best Value Regions and Convergence Regions
  • 4.4 Sufficient Conditions for Convergence: Constant Elements
  • 4.4.1 Classical Results and Generalizations
  • 4.4.2 Parabolic Convergence Regions.
  • 4.4.3 Convergence Neighborhoods for K(an/1) ...
  • 4.4.4 Twin Convergence Regions
  • 4.4.5 Miscellaneous Convergence Criteria
  • 4.5 Sufficient Conditions for Convergence
  • Variable Elements .
  • 4.5.1 Introduction
  • Classification of Continued Fractions
  • 4.5.2 Regular C-fractions
  • 4.5.3 Positive Definite/-fractions
  • 4.5.4 General T-fractions
  • Chapter 5 Methods for Representing Analytic Functions by Continued Fractions
  • 5.1 Correspondence
  • 5.2 Three-Term Recurrence Relations.
  • 5.3 Minimal Solutions of Three-Term Recurrence Relations .
  • 5.4 Uniform Convergence
  • 5.5 Pade Table
  • 5.5.1 Pade Approximants
  • 5.5.2 Multiple-Point Pade Tables
  • Chapter 6 Representations of Analytic Functions by Continued Fractions
  • 6.1 Continued Fractions of Gauss
  • 6.1.1 Hypergeometrie Functions F(a, b
  • c
  • z) ...
  • 6.1.2 Confluent Hypergeometric Functions Ø(&
  • c
  • z) .
  • 6.1.3 Confluent Hypergeometric Functions? (c
  • z) . .
  • 6.1.4 Confluent Hypergeometric Functions O (a, b
  • z).

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