# Analytic Theory of Continued Fractions by H S Wall

Hubert Stanley Wall published 'Analytic Theory of Continued Fractions' in 1948. We give below the Table of Contents and an extract from the Preface:

**Table of Contents**

Part I: Convergence Theory

1. The continued fraction as a product of linear fractional transformations

2. Convergence theorems

3. Convergence of continued fractions whose partial denominators are equal to unity

4. Introduction to the theory of positive definite continued fractions

5. Some general convergence theorems

6. Stieltjes type continued fractions

7. Extensions of the parabola theorem

8. The value region problem

2. Convergence theorems

3. Convergence of continued fractions whose partial denominators are equal to unity

4. Introduction to the theory of positive definite continued fractions

5. Some general convergence theorems

6. Stieltjes type continued fractions

7. Extensions of the parabola theorem

8. The value region problem

Part II: Function Theory

9. J-fraction expansions for rational functions

10. Theory of equations

11. J-fraction expansions for power series

12. Matrix theory of continued fractions

13. Continued fractions and definite integrals

14. The moment problem for a finite interval

15. Bounded analytic functions

16. Hausdorff summability

17. The moment problem for an infinite interval

18. The continued fraction of Gauss

19. Stieltjes summability

20. The PadÃ© table

Bibliography

Index

10. Theory of equations

11. J-fraction expansions for power series

12. Matrix theory of continued fractions

13. Continued fractions and definite integrals

14. The moment problem for a finite interval

15. Bounded analytic functions

16. Hausdorff summability

17. The moment problem for an infinite interval

18. The continued fraction of Gauss

19. Stieltjes summability

20. The PadÃ© table

Bibliography

Index

**Preface**

In writing this book, I have tried to keep in mind the student of rather modest mathematical preparation, presupposing only a first course in function theory. Thus, I have included such things as a proof of Schwarz's inequality, theorems on uniformly bounded families of analytic functions, properties of Stieltjes integrals, and an introduction to the matrix calculus. I have presupposed a knowledge of the elementary properties of linear fractional transformations in the complex plane.

It has not been my intention to write a complete treatise on the subject of continued fractions, covering all the literature, but rather to present a unified theory correlating certain parts and applications of the subject within a larger analytic structure. I have not touched upon the arithmetic theory, and have, for the most part, refrained from developing formulas of a more general character than are actually used in the proofs. Neither have I made any attempt to compile a complete bibliography.

Certain parts of the book have been developed in courses. For instance, parts of Chapter X were used in a course in the theory of equations, and most of Part I was covered in a course in the theory of continued fractions. Some of the material of Chapters XII and XV was developed in seminar courses.

This approach to the theory of continued fractions is mainly the result of researches carried on during the past decade by my students and colleagues and myself. ...

It has not been my intention to write a complete treatise on the subject of continued fractions, covering all the literature, but rather to present a unified theory correlating certain parts and applications of the subject within a larger analytic structure. I have not touched upon the arithmetic theory, and have, for the most part, refrained from developing formulas of a more general character than are actually used in the proofs. Neither have I made any attempt to compile a complete bibliography.

Certain parts of the book have been developed in courses. For instance, parts of Chapter X were used in a course in the theory of equations, and most of Part I was covered in a course in the theory of continued fractions. Some of the material of Chapters XII and XV was developed in seminar courses.

This approach to the theory of continued fractions is mainly the result of researches carried on during the past decade by my students and colleagues and myself. ...

H W S

January, 1948

The University of Texas

Last Updated April 2015