Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Revisiting second-order linear differential equations over Hardy fields

Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven

Abstract

We review second-order homogeneous linear differential equations with coefficient functions whose germs lie in a Hardy field (and hence are strongly non-oscillating). We prove a conjecture of Boshernitzan (1982): the oscillating solutions to such an equation are given by amplitude and phase functions with germs in a bigger Hardy field, and hence oscillate in a very regular way. We give sharp conditions for the uniqueness of such germs, study their asymptotic behavior, and use this to obtain information about the zeros and critical points of oscillating solutions.

Revisiting second-order linear differential equations over Hardy fields

Abstract

We review second-order homogeneous linear differential equations with coefficient functions whose germs lie in a Hardy field (and hence are strongly non-oscillating). We prove a conjecture of Boshernitzan (1982): the oscillating solutions to such an equation are given by amplitude and phase functions with germs in a bigger Hardy field, and hence oscillate in a very regular way. We give sharp conditions for the uniqueness of such germs, study their asymptotic behavior, and use this to obtain information about the zeros and critical points of oscillating solutions.
Paper Structure (7 sections, 104 theorems, 153 equations, 1 figure)

This paper contains 7 sections, 104 theorems, 153 equations, 1 figure.

Table of Contents

  1. Differential Algebra
  2. Asymptotic Differential Algebra
  3. Functions and Germs
  4. Hardy Fields
  5. Second-Order Linear Differential Equations over Hardy Fields
  6. Distribution of Zeros and Critical Points
  7. When is the Perfect Hull of a Hardy Field $\upomega$-free?

Key Result

Lemma 1.1

Suppose $K$ is closed under integration and $K^\dagger=K$. Then

Figures (1)

  • Figure 1: Plot of $y$, $-y$, $v$

Theorems & Definitions (184)

  • Example : Chebyshev's equation Kamke
  • Example : Coulomb wave equation
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • ...and 174 more