PREFACE xiii In §§6-9 we discuss asymptotic properties of certain kinds of generic asymp- totics, such as those occurring in optics in the vicinity of a simple caustic. The simple caustic case is discussed in §6, and used to obtain results of Ludwig on uniform asymptotic expansions. To obtain analogous results for more complicat- ed types of caustics we need canonical form theorems for the associated phase functions. These canonical forms are discussed in §§7 and 8. Finally in §9 we indicate some generalizations of the results of §6. The results of §§6-9 are joint research of Guillemin and Schaeffer, the results being obtained in 1972. Since then, two articles have appeared covering similar grounds, one by Duistermaat and one by Arnol'd. We feel that the point of view developed here, and the results obtained are sufficiently different from the above-mentioned articles to warrent publication in their original form. For the reader who prefers to deal directly with the mathematical theory, unencumbered by historical allusions or physical applications we recommend that he begin with Chapter VI, and refer back to those sections of Chapters IV and V as they become necessary. As we indicated in the above chapter by chapter analysis, much of the present book represents joint work with others. Many of the results of Chapter IV were obtained jointly with Kostant Chapter V represents joint work of Blattner, Kostant, and Sternberg, The second half of Chapter VII is joint work of Guillemin and Schaeffer and many comments by Schaeffer were extremely helpful in the development of Chapter VII. As many of the results are presented here for the first time, we are grateful to Bob Blattner, Bert Kostant and Dave Schaeffer for allowing the material to be published as part of this book. The book itself is based on an intensive joint Harvard-MIT course in the fall of 1973, together with seminars over the past few years. We wish to thank John Guckenheimer and Marty Golubitsky for their contribution to the course and seminar, and to Molly Scheffe for taking careful notes on which a lot of the current manuscript is based. Above all, we wish to thank Mary McQuillin for her assistance in seeing this book through its many editorial stages. Both authors were supported in part by the National Science Foundation and the second author by the John Simon Guggenheim Memorial Foundation to whom we wish to express our thanks.
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