Hill's Equation

by Magnus, Wilhelm; Winkler, Stanley

• ISBN13: 9780486495651
• ISBN10: 0486495655
• Format: Hardcover
• Copyright: 2004-01-26
• Publisher: Dover Publications

Summary

This two-part treatment of Hill's equation encompasses the subject's most pertinent information. Part I explains the basic theory: Floquet's theorem, characteristic values and intervals of stability, analytic properties of the discriminant, infinite determinants, asymptotic behavior of the characteristic values, theorems of Liapounoff and Borg, and related topics. Part II examines numerous details: elementary formulas, oscillatory solutions, intervals of stability and instability, discriminant, coexistence, and examples. Particular attention is given to stability problems and to the question of coexistence of periodic solutions. Although intended for professionals, this book can be recommended for graduate students and advanced undergraduates. 1966 edition.

Table of Contents

PART I. GENERAL THEORY
I. Basic Concepts
1.1 Preliminary Remarks	3	(1)
1.2 Floquet's Theorem	3	(5)
1.3 The Symmetric Case Q(x) = Q(-x)	8	(3)
II. Characteristic Values and Discriminant
2.1 Characteristic Values and Intervals of Stability	11	(8)
2.2 Analytic Properties of the Discriminant	19	(9)
2.3 Infinite Determinants	28	(8)
2.4 Asymptotic Behavior of the Characteristic Values	36	(7)
2.5 Basic Results from the general Theory of Linear Differential Equations	43	(3)
2.6 Theorems of Liapounoff and Borg. Fourier Transforms	46	(5)
PART II. DETAILS
III. Elementary Formulas
3.1 Transformation into a Standard Form	51	(1)
3.2 The Liouville Transformation	51	(2)
3.3 Polar Coordinates	53	(1)
3.4 Differential Equation for the Product of Two Solutions	54	(2)
IV. Oscillatory Solutions	56	(8)
V. Intervals of Stability and Instability
5.1 Introduction	64	(2)
5.2 Regions of Absolute Stability	66	(5)
5.3 Equations with Two or More Parameters	71	(1)
5.4 Remarks on a Perturbation Method	72	(2)
5.5 Applications of the Theory of Systems of Differential Equations	74	(1)
5.6 The Instability Intervals	75	(4)
VI. Discriminant	79	(11)
VII. Coexistence
7.1 Introduction	90	(2)
7.2 Ince's Equation	92	(11)
7.3 Lame's Equation and Generalizations	103	(3)
7.4 The Whittaker-Hill Equation	106	(1)
7.5 Finite Hill's Equations	107	(4)
7.6 Extreme Cases of Coexistence	111	(3)
VIII. Examples
8.1 Impulse Functions	114	(1)
8.2 Piecewise Constant Functions	115	(2)
8.3 Piecewise Linear Functions	117	(1)
8.4 The Frequency Modulation Equation	118	(2)
List of Symbols and Notations	120	(1)
List of Theorems, Lemmas, and Corollaries	121	(1)
References	122	(5)
Additional References	127	(2)
Index	129