On the use of Hadamard expansions in hyperasymptotic evaluation of Laplace-type integrals—III: Clusters of saddle points

It is shown how the recently developed Hadamard expansion procedure can be applied to the hyperasymptotic evaluation of Laplace-type integrals containing a large variable when the phase function has a cluster of close-lying saddle points. The modification to this procedure that is required when the saddles in the cluster coalesce to form a single higher-order saddle is discussed. An example is also considered in which there is both a coalescence of saddles and a Stokes phenomenon as the phase of the large variable is allowed to vary. Numerical examples are given to illustrate the accuracy that can be obtained with this new procedure.