An up-to-date exposition of the analytic theory of continued fractions has been long overdue. To remedy this is the intent of the present book. It deals with continued fractions in the complex domain, and places emphasis on applications and computational methods. All analytic functions have various expansions into continued fractions. Among those functions which have fairly simple expansions are many of the special functions of mathematical physics. Other applications deal with analytic continuation, location of zeros and singular points, stable polynomials, acceleration of convergence, summation of divergent series, asymptotic expansions, moment problems and birth-death processes.

The present volume is intended for mathematicians (pure and applied), theoretical physicists, chemists, and engineers. It is accessible to anyone familiar with the rudiments of complex analysis. We hope that it will be of interest to specialists in the theory of functions, approximation theory, and numerical analysis. Some of the material presented here has been developed for seminars given at the University of Colorado over a number of years. It also has been used in a seminar at the University of Trondheim.

The three most recent books on the analytic theory of continued fractions are those by Wall [1948], Perron [1957a] and Khovanskii [1963; the original Russian edition was published in 1956]. More recently Henrici [1977] has included an excellent chapter on continued fractions in the second volume of his treatise on Applied and Computational Complex Analysis. We owe much to the books of Perron and Wall, but since these books were written, many advances have been made in the subject. We have tried to incorporate the most significant of these in this volume.