On the coefficients that arise from Laplaceʹs method

Laplaceʹs method is one of the best-known techniques in the asymptotic approximation of integrals. The salient step in the techniqueʹs historical development was Erdélyiʹs use of Watsonʹs Lemma to obtain an infinite asymptotic expansion valid for any Laplace-type integral, published in 1956. Erdélyiʹs expansion contains coefficients c s that must be calculated in each application of Laplaceʹs 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 Erdélyiʹs theorem. Our results greatly simplify calculation of the c s in any particular application and clarify the theoretical basis of Erdélyiʹs expansion: it turns out that Faà di Brunoʹs formula has always played a central role in it. ve or derive the following: • rrect dimensionless groups. Erdélyiʹs expansion is properly expressed in terms of scaled coefficients c s * . plicit 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. rsive expression for c s * . oefficient c s * can be expressed as a polynomial in ( α + s ) / μ , where α and μ are quantities in Erdélyiʹs formulation. in insight that emerges is that the traditional approach to Laplaceʹs 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 Erdélyiʹs expansion—a point which Erdélyi himself alluded to. sider 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 Brunoʹs formula is explained.