On the coefficients that arise from Laplace's method

Abstract

Laplace's method is one of the best-known techniques in the asymptotic approximation of integrals. The salient step in the technique's historical development was Erdélyi's use of Watson's Lemma to obtain an infinite asymptotic expansion valid for any Laplace-type integral, published in 1956. Erdélyi's expansion contains coefficients c s that must be calculated in each application of Laplace's method, a tedious process that has traditionally involved the reversion of a series. This paper shows that the coefficients c s in fact have a very simple general form. In effect, we extend Erdélyi's theorem. Our results greatly simplify calculation of the c s in any particular application and clarify the theoretical basis of Erdélyi's expansion: it turns out that Faà di Bruno's formula has always played a central role in it. We prove or derive the following: • The correct dimensionless groups. Erdélyi's expansion is properly expressed in terms of scaled coefficients c s * . • Two explicit expressions for c s * in terms of combinatorial objects called partial ordinary Bell polynomials. This form is probably computationally optimal and makes checking for correctness a relatively straightforward process. • A recursive expression for c s * . • Each coefficient c s * can be expressed as a polynomial in ( α + s ) / μ , where α and μ are quantities in Erdélyi's formulation. The main insight that emerges is that the traditional approach to Laplace's method, involving reversion of a series, is less efficient and need only be invoked if one is interested in the role of the reversion coefficients in Erdélyi's expansion—a point which Erdélyi himself alluded to. We consider as an example an integral that occurs in a variational approach to finding the binding energy of helium dimers. We also present a three-line computer code to generate the coefficients c s * exactly in the general case. In a sequel paper (to be published in SIAM Review), a new representation for the gamma function is obtained, and the link with Faà di Bruno's formula is explained.