Author | Wasow, Wolfgang. author |
---|---|
Title | Linear Turning Point Theory [electronic resource] / by Wolfgang Wasow |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1985 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-1090-0 |
Descript | IX, 246 p. online resource |
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 Airyโs Equation -- 6.3. Differential Equations Reducible to Weberโs Equation -- 6.4. Uniform Transformations in a Full Neighborhood of -- a Turning Point -- 6.5. Complete Reduction to Airyโs Equation -- 6.6. Reduction to Weberโs Equation in Wider Sectors -- 6.7. Reduction to Weberโs 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 Fedoryukโs Global Theory of Second-Order Equations -- 9.1. Global Formal Solutions of ?2uโ=a(x)u2uโ = a(x)u -- 9.2. Separation Curves for ?2uโ=a(x)u2uโ = a(x)u -- 9.3. A Global Asymptotic Existence Theorem for ?2uโ=a(x)u2uโ = 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 Arnoldโs Form -- References