Introduction

In Chapter 3 we introduced two basic modes of expansion using Hadamard series, namely the forward expansion Scheme A and the forward-reverse expansion Scheme B. The first scheme uses the Hadamard series Sn(z), defined in (3.2.9), and is suitable for isolated saddle points when adjacent saddles or other singularities are sufficiently remote to result in a sequence of well-separated exponential levels. If maximal exponential separation is employed, the resulting convergence of the Hadamard series has to be accelerated through use of the modified form of the series. This involves the computation of coefficients expressed in terms of one-dimensional integrals in (3.2.18) of a common form at each level of the expansion. If one is prepared to accept a reduced exponential separation, however, it is possible, through judicious choice of the expansion points Ωn, to produce Hadamard series that converge rapidly at a geometric rate without the need for the computationally more expensive modified form.

In Scheme B, the zeroth interval is dealt with by forward expansion as in Scheme A, but with forward-reverse expansion about the points Ωn for the intervals with n ≥ 1. This has the advantage of covering a given interval on the integration path with fewer evaluations of the inversion expansions. By careful choice of the Ωn it is similarly possible to arrange for the Hadamard series at all levels to converge at a geometric rate.