I Historical Introduction.- 1.1. Early Asymptotic Theory Without Turning Points.- 1.2. Total Reflection and Turning Points.- 1.3. Hydrodynamic Stability and Turning Points.- 1.4. The So-Called WKB Method.- 1.5. The Contribution of R. E. Langer.- 1.6. Remarks on Recent Trends.- II Formal Solutions.- 2.1. Introduction.- 2.2. The Jordan Form of Holomorphic Functions.- 2.3. A Formal Block Diagonalization.- 2.4. Parameter Shearing: Its Nature and Purpose.- 2.5. Simplification by a Theorem of Arnold.- 2.6. Parameter Shearing: Its Application.- 2.7. Parameter Shearing: The Exceptional Case.- 2.8. Formal Solution of the Differential Equation.- 2.9. Some Comments and Warnings.- III Solutions Away From Turning Points.- 3.1. Asymptotic Power Series: Definition of Turning Points.- 3.2. A Method for Proving the Analytic Validity of Formal.- Solutions: Preliminaries.- 3.3. A General Theorem on the Analytic Validity of Formal.- Solutions.- 3.4. A Local Asymptotic Validity Theorem.- 3.5. Remarks on Points That Are Not Asymptotically Simple.- IV Asymptotic Transformations of Differential Equations.- 4.1. Asymptotic Equivalence.- 4.2. Formal Invariants.- 4.3. Formal Circuit Relations with Respect to the Parameter.- V Uniform Transformations at Turning Points: Formal Theory.- 5.1. Preparatory Simplifications.- 5.2. A Method for Formal Simplification in Neighborhoods of a Turning Point.- 5.3. The Case h 1.- 5.4. The General Theory for n = 2.- VI Uniform Transformations at Turning Points: Analytic Theory.- 6.1. Preliminary General Results.- 6.2. Differential Equations Reducible to Airy's Equation.- 6.3. Differential Equations Reducible to Weber's Equation.- 6.4. Uniform Transformations in a Full Neighborhood of.- a Turning Point.- 6.5. Complete Reduction to Airy's Equation.- 6.6. Reduction to Weber's Equation in Wider Sectors.- 6.7. Reduction to Weber's Equation in a Full Disk.- VII Extensions of the Regions of Validity of the Asymptotic Solutions.- 7.1. Introduction.- 7.2. Regions of Asymptotic Validity Bounded by Separation Curves: The Problem.- 7.3. Solutions Asymptotically Known in Sectors Bounded by.- Separation Curves.- 7.4. Singularities of Formal Solutions at a Turning Point.- 7.5. Asymptotic Expansions in Growing Domains.- 7.6. Asymptotic Solutions in Expanding Regions: A General Theorem.- 7.7. Asymptotic Solutions in Expanding Regions: A Local Theorem.- VIII Connection Problems.- 8.1. Introduction.- 8.2. Stretching and Parameter Shearing.- 8.3. Calculation of the Restraint Index.- 8.4. Inner and Outer Solutions for a Particular nth-Order System.- 8.5. Calculation of a Central Connection Matrix.- 8.6. Connection Formulas Calculated Through Uniform Simplification.- IX Fedoryuk's Global Theory of Second-Order Equations.- 9.1. Global Formal Solutions of ?2u"=a(x)u2u" = a(x)u.- 9.2. Separation Curves for ?2u"=a(x)u2u" = a(x)u.- 9.3. A Global Asymptotic Existence Theorem for ?2u"=a(x)u2u" = a(x)u.- X Doubly Asymptotic Expansions.- 10.1. Introduction.- 10.2. Formal Solutions for Large Values of the.- Independent Variable.- 10.3. Asymptotic Solutions for Large Values of the.- Independent Variable.- 10.4. Some Properties of Doubly Asymptotic Solutions.- 10.5. Central Connection Problems in Unbounded Regions.- XI A Singularly Perturbed Turning Point Problem.- 11.1. The Problem.- 11.2. A Simple Example.- 11.3. The General Case: Formal Part.- 11.4. The General Case: Analytic Part.- XII Appendix: Some Linear Algebra for Holomorphic Matrices.- 12.1. Vectors and Matrices of Holomorphic Functions.- 12.2. Reduction to Jordan Form.- 12.3. General Holomorphic Block Diagonalization.- 12.4. Holomorphic Transformation of Matrices into Arnold's Form.- References.
My book "Asymptotic Expansions for Ordinary Differential Equations" published in 1965 is out of print. In the almost 20 years since then, the subject has grown so much in breadth and in depth that an account of the present state of knowledge of all the topics discussed there could not be fitted into one volume without resorting to an excessively terse style of writing. Instead of undertaking such a task, I have concentrated, in this exposi tion, on the aspects of the asymptotic theory with which I have been particularly concerned during those 20 years, which is the nature and structure of turning points. As in Chapter VIII of my previous book, only linear analytic differential equations are considered, but the inclusion of important new ideas and results, as well as the development of the neces sary background material have made this an exposition of book length. The formal theory of linear analytic differential equations without a parameter near singularities with respect to the independent variable has, in recent years, been greatly deepened by bringing to it methods of modern algebra and topology. It is very probable that many of these ideas could also be applied to the problems concerning singularities with respect to a parameter, and I hope that this will be done in the near future. It is less likely, however, that the analytic, as opposed to the formal, aspects of turning point theory will greatly benefit from such an algebraization.
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