Chapter 5 - Methods for Representing Analytic Functions by Continued Fractions

Summary

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 ν_n power of z, where ν_n 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 {R_n(z)} meromorphic at the origin. As a special case R_n(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 {R_n(z)} corresponds are given by Theorem 5.1. A method for obtaining a sequence {R_n(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 {R_n(z)} is equivalent to uniform bounded-ness, and that when a sequence {R_n(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].