Integrals with a large parameter. The continuation of uniformly asymptotic expansions
Authors: Ursell F. 
Published: November 2008

In the integral the functions g(z), f(z, α) are analytic functions of their arguments, and N is a large positive parameter. When N tends to ∞, asymptotic expansions can usually be found by the method of steepest descents, which shows that the principal contributions arise from the saddle-points, i.e. the values of z at which ∂f/∂z = 0. The position of the saddle-points varies with α, and if for some α (say α = 0) two saddle-points z1(α), z2(α) coincide (say at z = 0) the ordinary method of steepest descents gives expansions which are not uniformly valid for small α. In an earlier paper (Chester, Friedman and Ursell (1)), a uniform asymptotic expansion of the form

was obtained, where Ai and Ai' are the Airy function and its derivative respectively, and where the regular functions A(α) and ζ(α) are given by

The coefficient functions As(α), Bs(α) are also regular functions of a for which, however, no explicit expressions are known. This Airy-function expansion was shown to be valid in a circle |α| ≥ Rα, independent of N. Airy-function expansions of the same form but involving slightly different argument functions instead of our A(α) and ζ(α) have long been known but are valid only in a region which shrinks to the point α = 0 as N → ∞. This improvement in the region of validity greatly simplifies the matching of steepest-descents and Airy-function expansions across the common region of validity.

In the present paper a further improvement is obtained. The validity of our Airyfunction expansion is extended to a still larger region which may be unbounded and which in many practical cases covers the whole region of interest, so that no matching with other expansions is needed. For this purpose the relation between steepestdescents and Airy-function expansions is investigated.

It is easy to see that by a process of matching the steepest-descents coefficients can be expressed in terms of the Airy-function coefficients. It is now shown that conversely the Airy-function coefficients can be expressed in terms of the steepestdescents coefficients, and that they involve the two saddle-points symmetrically.It is thus possible to infer desired properties of the Airy-function coefficients (e.g.analytic continuation and boundedness) from the corresponding properties of the steepest-descents coefficients, and hence to infer the equivalence of the two expansions (except near α = 0).

The following result is typical. Suppose that the steepest-descents expansion can be shown to be valid in a region (excluding a neighbourhood of α = 0) of the α-plane, and suppose further that the steepest-descents coefficients and the functions ζ(α) and A (α) can be shown to satisfy certain simple conditions of regularity and boundedness.(In practice it is usually not difficult to verify these.) Then it is shown that our Airy-function expansions can be continued into the same region and that it is a valid asymptotic expansion of the integral there.

These conditions of boundedness are not satisfied in a certain integral arising in the study of Kelvin's ship-wave pattern, where the coefficients at one saddle-point become unbounded near the track of the disturbance. The argument is modified to show that, nevertheless, the Airy-function expansion holds uniformly up to the track of the disturbance.