On the existence of nonoscillatory phase functions for second order differential equations in the high-frequency regime

TL;DR

This paper demonstrates that solutions to certain highly oscillatory second order differential equations can be approximated by nonoscillatory phase functions, enabling efficient evaluation of special functions like Bessel functions with constant complexity.

Contribution

It develops an analytical framework showing the existence of nonoscillatory phase functions for high-frequency second order ODEs, facilitating accurate and efficient computation of special functions.

Findings

Solutions can be approximated using nonoscillatory phase functions

Special functions like Bessel functions can be evaluated with O(1) operations in the order 

Numerical experiments illustrate the practical implications of the theory

Abstract

We observe that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using nonoscillatory phase functions. In addition, we describe numerical experiments which illustrate important implications of this fact. For example, that many special functions of great interest --- such as the Bessel functions $J_\nu$ and $Y_\nu$ --- can be evaluated accurately using a number of operations which is $O(1)$ in the order $\nu$. The present paper is devoted to the development of an analytical apparatus. Numerical aspects of this work will be reported at a later date.