Revisiting the saddle-point method of Perron

Cormac O’Sullivan

Vol. 298 (2019), No. 1, 157–199

DOI: 10.2140/pjm.2019.298.157

Abstract

Perron’s saddle-point method gives a way to find the complete asymptotic expansion of certain integrals that depend on a parameter going to infinity. We give two proofs of the key result. The first is a reworking of Perron’s original proof, showing the clarity and simplicity that has been lost in some subsequent treatments. The second proof extends the approach of Olver which is based on Laplace’s method. New results include more precise error terms and bounds for the expansion coefficients. We also treat Perron’s original examples in greater detail and give a new application to the asymptotics of Sylvester waves.

Keywords

asymptotics, saddle-point method, Sylvester waves

Mathematical Subject Classification 2010

Primary: 41A60

Secondary: 11P82

Milestones

Received: 14 June 2017

Revised: 23 February 2018

Accepted: 23 March 2018

Published: 2 February 2019