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.

Mathematics derives much of its vitality from the fact that it has several faces, each face having its own sharply distinguished features. One face, which might be called the dialectic face of mathematics, is the face of a scholar, or even of a philosopher. It is the face which tells us whether theorems are true or false, and whether mathematical objects with specified properties do or do not exist. Dialectic mathematics is an intellectual game, played according to rules about which there is a high degree of consensus, and where progress can be measured sharply in terms of the generality of a result that has been achieved.

There is another, entirely different face of mathematics, which I like to call its algorithmic face. This is the face of an engineer. The algorithmic mathematician tells us how to construct the beautiful things of whose existence we are assured by the dialectic mathematician. The rules of the game of algorithmic mathematics, and in particular the significance attached to results in algorithmic mathematics, depend on the equipment that is available to carry out the required constructions.

Dialectic mathematics has experienced a continuous growth at least since the time of C. F. Gauss. Algorithmic mathematics, on the other hand, has stagnated from Euler's time until very recently, because no really new computing equipment came into existence; even the manually operated desk calculator did not significantly increase the speed of computation. It has been brought to light by H. H. Goldstine that Gauss in essence invented the fast-Fourier-transform algorithm.

This chapter deals with the representation of analytic functions by continued fractions. Two main approaches are considered. In the first approach a formal continued-fraction expansion is obtained by requiring that the Laurent expansion of the nth approximant agree term by term with a given Laurent series L up to the νn power of z, where νn tends to infinity with n. Continued fractions defined in this manner are said to correspond to the series L [or to the function ƒ(z) of which L is a Laurent expansion]. A general theory of correspondence is developed in Sections 5.1, 5.2, and 5.4 for sequences of functions {Rn(z)} meromorphic at the origin. As a special case Rn(z) can be the nth approximant of a continued fraction. A norm is introduced for the field of all formal Laurent series, such that convergence with respect to the norm is equivalent to correspondence. Necessary and sufficient conditions for the existence of a Laurent series L to which a given sequence {Rn(z)} corresponds are given by Theorem 5.1. A method for obtaining a sequence {Rn(z)} (or continued fraction) corresponding to a given Laurent series L is provided by Theorems 5.2 and 5.3, in which three-term recurrence relations play an important role. As a consequence of the property of correspondence, it is shown (Section 5.4) that, with suitable restrictions, uniform convergence of a sequence {Rn(z)} is equivalent to uniform bounded-ness, and that when a sequence {Rn(z)} converges uniformly, its limit ƒ(z) is a function whose Laurent expansion is L. Although the basic ideas of correspondence go back to Gauss [1813], the general theory described here is based on [Jones and Thron, 1975, 1979].

Introduction

Multiplicative integration was initiated by Volterra [33] in 1887 as a method of solving systems of linear differential equations. The early history of the subject bears the impress of this initiation. Schlesinger, who took up the further development of the theory and became its chief proponent, stressed the link with differential equations in his 1908 lectures on the subject [26]. But the historical growth of mathematics only roughly displays its authentic design as a logical edifice. From a logical or architectural standpoint multiplicative integration belongs to the area of Lie groups and Lie algebras. This was perceived by Birkhoff [2] in the late 1930s, and has been borne out by subsequent work. Accordingly, our first task here will be to lay bare the intrinsic Lie group-theoretic aspect of the subject. For this it is very convenient, as again Birkhoff indicated [2, Sec. 4], to consider the kinematics of fluid flows. It is with this that we shall begin.

Our exposition will not be rigorous in every detail. We shall not, for instance, define what “smooth” means in each case. Nor shall we tarry over questions concerning the existence or interchange of limits. We shall assume that the reader has the empathy to see that our assertions can be made correct by the imposition of reasonable restraints and that rigorization is feasible.

Introduction

Ergodic theory has its origins in the attempt to explain the macroscopic characteristics of physical systems, in particular thermodynamical systems of gases, in terms of the behavior of the microscopic structure of the system. This problem, along with other physical problems, led to what has been known as the “many body problem” in mathematics, and mathematical research on this problem led to what is now known as ergodic theory.

For example, if a gas is considered to be a large collection of molecules (thought of as point masses) which are in motion, such characteristics of the gas as its pressure, temperature, and volume should be determined by the dynamic behavior of these molecules, i.e., their positions and momenta as functions of time.

If such a system is conservative, i.e., its total energy does not change with time, then the Hamiltonian theory of dynamical systems can be used to obtain transformations on manifolds in a high dimensional Euclidean space called phase space. These manifolds in phase space consist of those states of the gas (generalized coordinates of position and momentum of each molecule) with the same total energy. The transformations preserve the measure induced by Lebesgue measure on the manifold; their construction was sketched in Section 2.8.

It is impossible to determine the microscopic state of a gas at an instant of time, so that even if the Hamilton equations of motion could be solved to find the transformation explicitly, it would not be possible to predict exactly the future states of the gas.

Introduction

The task of statistical mechanics is to derive macroscopic properties of matter from the laws governing the microscopic actions and interactions of individual particles. The systems that are considered in statistical mechanics are those that consist of a large number (on the order of 1027 particles, say, for the molecules in one liter of air) of subsystems (the molecules). To specify such a system on a microscopic level would require the coordinates of a point in 6N-dimensional space, where N is the number of subsystems (or particles) of the system. Recall that we considered these systems in another context in Section 2.8 and the introductions to Chapters 4 and 5.

A macroscopic description of such a system can be given in terms of relatively few quantities such as energy, volume, specific heat, etc., which are called thermodynamic variables, or functions. The entropy of a system is one such thermodynamic variable. Thermodynamics is a study of the relationships that exist between the various thermodynamic variables, and this subject, from a mathematical perspective, can be completely axiomatized [28]. In particular, the equilibrium states of a system can be described in terms of relatively few thermodynamic variables.

We shall not discuss the known relationships between the entropy of a system in an equilibrium state and the other thermodynamic variables of the state. The reader interested in this topic may consult [28] or [154].

An editor's preface to a mathematics book does not have a clearly defined role in contemporary usage. If some past precedents were followed, I would merely remark that the present work by John Dollard and Charles Friedman is a completely self-contained treatment of the product integral on a simple and elementary basis. As such, I believe it to be unique as far as this topic is concerned. The applications that are presented fall mainly within the domain of ordinary differential equations. Some amplifications of the generality of the theme with applications to a wider circle of mathematical topics are described in the accompanying Appendix by P. R. Masani.

For the benefit of some readers at least, I shall go beyond this conventional restriction of the editor's function. Though in the last analysis, mathematical topics must be treated in full technical detail and with logical completeness (as they are indeed treated in the body of the present work), it is often useful to preface such a detailed development with a more discursive and less technical discussion.

What is the product integral? As the text tells us, it is an analytic process or class of processes first put forward by Volterra in the last decades of the nineteenth century for the study of various questions relating to the theory of ordinary differential equations. As Professor Masani reminds us in his Appendix, it was extensively developed by Schlesinger in the early part of the twentieth century, particularly in connection with differential equations in the complex domain.

We indicate here, briefly, the content of the chapters. Chapter 1 is an elementary introduction to the product integral of a continuous matrix-valued function and its properties. (The generalization to “Lebesgue product integrals” is sketched in Section 8.) This chapter should be accessible to readers with quite minimal mathematical background. It is prerequisite for understanding of the other chapters. Chapter 2 deals with contour product integrals; the development is parallel to that of the theory of ordinary contour integrals in complex variable theory. Contour product integrals are not used in later chapters. In Chapter 3 we present a theory of product integration in a much more general setting. More mathematical background is required here, for example, familiarity with functional analysis in Banach space and some integration theory. Included are results on the equation of evolution (1) with unbounded A(x). The other chapters (except Chapter 4, Section 5) are independent of this chapter. Chapter 4 presents applications of product integration to the theory of differential equations; some new results concerning solutions of the Schrodinger equation with rather singular potentials are included. In Chapter 5 we discuss product integration of (matrix-valued) measures. Some familiarity with measure theory is assumed here, but nothing very sophisticated is required. Chapter 6 contains a discussion of work on product integration by various other authors and some remarks on generalizations of the theory. Following Chapter 6 is an appendix on matrix theory containing elementary definitions and results and a few special results which may not be familiar to all readers.