Revisiting the saddle-point method of Perron

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.