On the coefficients that arise from Laplace's method
Journal of Computational and Applied Mathematics
Journal of Approximation Theory
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Laplace's method is a preeminent technique in the asymptotic approximation of integrals. Its utility was enhanced enormously in 1956 when Erdélyi found a way to apply Watson's lemma and thereby obtain an infinite asymptotic expansion valid, in principle, for any integral of Laplace type. Erdélyi's formulation requires tedious computation of coefficients $c_{s}$ for each specific application of the method, and traditionally this has involved reverting a series. Recently, it was shown that the coefficients $c_{s}$ can be computed via a simple, explicit expression that is probably computationally optimal, which avoids the reversion approach altogether. The formula is made possible by recognizing the central role of Faà di Bruno's formula, alongside Watson's lemma, in Erdélyi's formulation of Laplace's classical method. Laplace's method can now be implemented cleanly and relatively quickly, provided one has the luck and the patience to get to the point where implementation becomes automatic.The present paper outlines the recent discovery of the role of Faà di Bruno's formula in Laplace's method, gives examples of the application of the explicit expression for the coefficients $c_{s},$ and provides grounds for a possible generalization of the result.