Overview

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 Complex Case.- 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.