Peter A. Carruthers General Editor, Section on Mathematics of PhysicsPreface

These two volumes are intended to serve as a basic text on one approach to the problem of assigning a value to a power series. We have attempted to present the basic results and methods in as transparent a form as possible, in line with the general objectives of the Encyclopedia. The general topic of Pade Approximants, which is, among other things, a highly practical method of definition and of construction of the value of a power series, seems to have begun independently at least twice. Pade's claim for credit is based on his thesis (1892), in which he developed the approximants and organized them in a table. He paid particular attention to the exponential function. He was presumably unaware of the prior work of Jacobi (1846), who gave the determinantal representation in his paper on the simplification of Cauchy's solution to the problem of rational interpolation. Also, Pade's work was preceded by that of Frobenius (1881), who derived identities between the neighboring rational fractions of Jacobi. It is interesting to note that Anderson seems to have stumbled upon some Pade Approximants for the logarithmic function in 1740. A photograph of H. Pade is to be found in The Pade Approximant Method and Its Application to Mechanics, edited by H. Cabannes. A copy of his autographed thesis is to be found in the Cornell University Library.

This work has been distilled from an extensive literature, and The Essentials of Pade Approximants, written by one of us, has been an essential reference. We use the abbreviation EPA for this book, and refer to it often for a different or fuller treatment of some of the more advanced topics. While each book is entirely self-contained, our notation is normally compat­ible with EPA, and to a large extent the books complement each other. An important exception is that the Pade table in EPA is reflected through its main diagonal in our present notation. The proceedings of the Canterbury Summer School and International Conference, edited by the other of us, contain diverse contributions which initiated in print the multidisciplinary view of the subject—a view we hope we have transmitted herein. The many publications which have contributed substantially to our text are listed in the bibliography. We are grateful to our numerous colleagues at Brook- haven, Canterbury, Cornell, Los Alamos, and Saclay in freely discussing so many topics which have made possible the breadth of our treatment. Especially, we thank Roy Chisholm, John Gammel, and Daniel Bessis for many conversations, and the C.E.A. at Saclay, where part of this book was written, for hospitality.Our hardest task in writing this book was to choose a presentation which is both correct and readily comprehensible. A fully precise system based on rigorous analysis and set-theoretic language would have ensured total ob­scurity of the more practical techniques. Conversely, omission of all the conditions under which the theorems hold good would be absurdly mislead­ing. We have chosen a level of presentation suitable for the topic in hand. For example, the connectivity of sets is mentioned where it is important, and otherwise it is omitted. The meaning of the order notation is clear in context. Both applications in physics and techniques recently developed are treated in a practical fashion.

Equations are referenced by a default option. Equation (1.6.5.3) is Equa­tion (5.3) of Part I, Chapter 6; the Part and Chapter are dropped by default if they are the same as the source of the reference.

Finally, a spirit of evangelism may be detected in the text. When a review of rational approximation in 1963 can claim that Pade approximants cannot approximate on the entire range (0, oo) and be believed, a revision of view is overdue.

George A. Baker, Jr. Peter Graves-Morris