The theory of continued fractions has been defined by a small handful of books. This is one of them. The focus of Wall's book is on the study of continued fractions in the theory of analytic functions, rather than on arithmetical aspects. There are extended discussions of orthogonal polynomials, power series, infinite matrices and quadratic forms in infinitely many variables, definite integrals, the moment problem and the summation of divergent series.'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...' - from the Preface.
Part I: Convergence Theory: The continued fraction as a product of linear fractional transformations Convergence theorems Convergence of continued fractions whose partial denominators are equal to unity Introduction to the theory of positive definite continued fractions Some general convergence theorems Stieltjes type continued fractions Extensions of the parabola theorem The value region problem Part II: Function Theory: J-fraction expansions for rational functions Theory of equations J-fraction expansions for power series Matrix theory of continued fractions Continued fractions and definite integrals The moment problem for a finite interval Bounded analytic functions Hausdorff summability The moment problem for an infinite interval The continued fraction of Gauss Stieltjes summability The Pade table Bibliography Index.