Overlapping Stokes Smoothings: Survival of the Error Function and Canonical Catastrophe Integrals

We derive doubly uniform approximations for the remainder in the optimally truncated saddle-point expansion for an integral containing a large parameter. Double uniformity means that the formulae remain valid while distant saddles responsible for the divergence of the expansion coalesce and separate (as described by catastrophe theory) and while the subdominant exponentials they contribute switch on and off (as described by the error-function smoothing of the Stokes phenomenon). Two sorts of asymptotic singularity are thereby united in a common framework. The formula for the remainder incorporates both the Stokes error function and the canonical catastrophe integrals. A numerical illustration is given, in which the distant cluster contains two saddles; the asymptotic theory gives an accurate description of the details of the fractional remainder, even when this is of order exp (-36).