Asymptotic Analysis: Linear Ordinary Differential Equations

In this book we present the main results on the asymptotic theory of ordinary linear differential equations and systems where there is a small parameter in the higher derivatives. We are concerned with the behaviour of solutions with respect to the parameter and for large values of the independent variable. The literature on this question is considerable and widely dispersed, but the methods of proofs are sufficiently similar for this material to be put together as a reference book. We have restricted ourselves to homogeneous equations. The asymptotic behaviour of an inhomogeneous equation can be obtained from the asymptotic behaviour of the corresponding fundamental system of solutions by applying methods for deriving asymptotic bounds on the relevant integrals. We systematically use the concept of an asymptotic expansion, details of which can if necessary be found in [Wasow 2, Olver 6]. By the "formal asymptotic solution" (F.A.S.) is understood a function which satisfies the equation to some degree of accuracy. Although this concept is not precisely defined, its meaning is always clear from the context. We also note that the term "Stokes line" used in the book is equivalent to the term "anti-Stokes line" employed in the physics literature.

Table of Contents:
1. The Analytic Theory of Differential Equations.- §1. Analyticity of the Solutions of a System of Ordinary Differential Equations.- § 2. Regular Singular Points.- § 3. Irregular Singular Points.- 2. Second-Order Equations on the Real Line.- § 1. Transformations of Second-Order Equations.- § 2. WKB-Bounds.- § 3. Asymptotic Behaviour of Solutions of a Second-Order Equation for Large Values of the Parameter.- § 4. Systems of Two Equations Containing a Large Parameter.- § 5. Systems of Equations Close to Diagonal Form.- § 6. Asymptotic Behaviour of the Solutions for Large Values of the Argument.- § 7. Dual Asymptotic Behaviour.- § 8. Counterexamples.- § 9. Roots of Constant Multiplicity.- § 10. Problems on Eigenvalues.- § 11. A Problem on Scattering.- 3. Second-Order Equations in the Complex Plane.- § 1. Stokes Lines and the Domains Bounded by them.- § 2. WKB-Bounds in the Complex Plane.- § 3. Equations with Polynomial Coefficients. Asymptotic Behaviour of a Solution in the Large.- § 4. Equations with Entire or Meromorphic Coefficients.- § 5. Asymptotic Behaviour of the Eigenvalues of the Operator -d2 / dx2 + ?2q(x). Self-Adjoint Problems.- § 6. Asymptotic Behaviour of the Discrete Spectrum of the Operator -y? + ?2q(x)y. Non-Self-Adjoint Problems.- § 7. The Eigenvalue Problem with Regular Singular Points.- § 8. Quasiclassical Approximation in Scattering Problems.- § 9. Sturm-Liouville Equations with Periodic Potential.- 4. Second-Order Equations with Turning Points.- § 1. Simple Turning Points. The Real Case.- § 2. A Simple Turning Point. The Complex Case.- § 3. Some Standard Equations.- §4. Multiple and Fractional Turning Points.- § 5. The Fusion of a Turning Point and Regular Singular Point.- § 6. Multiple Turning Points. The ComplexCase.- § 7. Two Close Turning Points.- § 8. Fusion of Several Turning Points.- 5. nth-Order Equations and Systems.- § 1. Equations and Systems on a Finite Interval.- § 2. Systems of Equations on a Finite Interval.- § 3. Equations on an Infinite Interval.- § 4. Systems of Equations on an Infinite Interval.- § 5. Equations and Systems in the Complex Plane.- § 6. Turning Points.- § 7. A Problem on Scattering, Adiabatic Invariants and a Problem on Eigenvalues.- § 8. Examples.- References.