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 JordanForm. - 12. 3. General Holomorphic Block Diagonalization. - 12. 4. Holomorphic Transformation of Matrices into Arnold's Form. - References. Language: English