Asymptotic Analysis: Linear Ordinary Differential Equations

ISBN-13: 9783642634352 / Angielski / Miękka / 2012 / 363 str.

Mikhail V. Fedoryuk; A. Rodick

Opis

This encyclopaedic book describes the developments of the last years in the area of asymptotic methods for linear ODEs and systems in the real and complex domain. Basically all main results and methods are given. Almost every known asymptotic formula is referred to. Written in a style readable for the nonspecialist in the area, the book is a guide to the extensive literature developed recently in the field. It is a much needed exposition and a comprehensive source of information for studentsand researchers in mathematics as well as in mechanics and physics.

Spis treści

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.

O autorze

Fedoryuk's book is a unique reference for the asymptotic theory of ordinary differential equations given all the important formulae of this field. It is also an indispensable guide to the literature up to the most recent research.