An extension of the method of steepest descents

ABSTRACT 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 a (say α = 0) two saddle points coincide (say at z = 0) the ordinary method of steepest descents gives expansions which are not uniformly valid for small α . In the present paper we consider this case of two nearly coincident saddle points and construct uniform expansions as follows. A new complex variable u is introduced by the implicit relation where the parameters ζ ( α ), A ( α ) are determined explicitly from the condition that the ( u , z ) transformation is uniformly regular near z = 0, α = 0 (see § 2 below). We show that with these values of the parameters there is one branch of the transformation which is uniformly regular. By taking u on this branch as a new variable of integration we obtain for the integral uniformly asymptotic expansions of the form where Ai and Ai′ are the Airy function and its derivative respectively, and A ( α ), ζ ( α ) are the parameters in the transformation. The application to Bessel functions of large order is briefly described.