New insights into mathematical problems and physical applications continue to arise from the study of power series representation of functions. The present work is devoted to an intensive description of the Pade approximant technique*. The value of this scheme of approximation in a wide variety of physical problems has been increasingly recognized in recent years. This two-part presentation is a fine example of the interplay between physics and mathematics, each stimulating the other to new concepts and techniques.
One could not imagine better qualified authors for a contemporary major set of volumes on Pade approximants. Baker and Graves-Morris are widely known for their original contributions both to the mathematics and serious physical applications. The result is a lucid and explicit treatment of the subject which does not compromise mathematical accuracy yet is easily accessible to the modern theorist.
We may mention that this work is an example of a healthy trend developing in recent years in which modern mathematical developments are increasingly providing the language in which the most advanced physical theories are expressed. In the present case the renaissance in statistical mechanics and field theory studies in recent years has required such developments as Wilson's renormalization group method and Pade approximants. We may also mention the serious studies of continuous groups and their representations inspired by efforts to unite the weak, electromagnetic, strong, and gravitational forces. These same theories seem to be best formulated as non-Abelian gauge field theories, whose content and consequences involve the concepts of differential geometry and topology.
Stated briefly, the Pade approximant represents a function by the ratio of two polynomials. The coefficients of the powers occurring in the polynomials are, however, determined by the coefficients in the Taylor series expansion of the function. Thus, given a power series expansion we can by the methods described in the text, make an optimal choice of the Foreword
Exploitation of this simple idea and its extensions has led to many insights and by now has become practically a major industry. I shall not spoil the story by revealing more of the plot.
Inspection of the table of contents reveals an intensive development of the mathematical texture inherent in the subject. Many excellent examples illustrate the concepts. Some recent results appear here for the first time in monograph form. Among these are included the developments of reliable algorithms (1.2.1, 2.4, 4.5, and II.1.1), Saff-Varga theorems on Pade ap- proximants to the exponential (1.5.7), the theory of convergence in capacity of Pommerenke (1.6.6), Canterbury (two variable) approximants (II. 1.4) and results from \<f>4 Euclidean field theory derived using Pade approximants. In addition the approach to Laguerre's method in (II.3.7) is new. The treatment of applications is well done and has sufficient depth to be useful to the research scientist. Part II, Chapter 2, describes the connection with integral equations and quantum mechanics. The connection with numerical analysis is made in Part II, Chapter 3. The authors close with a frontier topic, the application of Pade approximants to problems in quantum field theory. Finally, an extensive bibliography documents the subject and provides references to the treatment of further related problems.
This two-volume presentation is a fine example of a creative review because it weaves together the vital ideas of the subject of Pade approximants and sets the stage for vigorous new developments in theory and applications. It should fill this role for some time to come.