# Revisiting the saddle-point method of Perron

## TL;DR

This paper revisits Perron's saddle-point method, providing clearer proofs, extending results with precise error bounds, and applying it to Sylvester waves, enhancing understanding of asymptotic integral expansions.

## Contribution

It offers two proofs of Perron's key asymptotic expansion result, improves error estimates, and applies the method to new problems like Sylvester waves.

## Key findings

- Two simplified proofs of Perron's main theorem
- More precise error bounds and coefficient estimates
- Application to asymptotics of Sylvester waves

## 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.