On the existence of nonoscillatory phase functions for second order ordinary differential equations in the high-frequency regime. (English) Zbl 1349.34037

Summary: 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 several 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.


34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
33E30 Other functions coming from differential, difference and integral equations
33F05 Numerical approximation and evaluation of special functions
65L99 Numerical methods for ordinary differential equations


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