Asymptotic expansions of Laplace-type integrals. III

We consider the asymptotic expansion for large λ of Laplace-type integrals of the form∫0∞∫0∞g(x,y)e−λf(x,y) dx dyfor a wide class of amplitude functions g(x,y) and ‘polynomial’ (noninteger powers are permitted) phases f(x,y) possessing an isolated, though possibly degenerate, critical point at the origin. The resulting algebraic expansions valid in a certain sector of the complex λ plane are based on recent results obtained in Kaminski and Paris (Philos. Trans. Roy. Soc. London A 356 (1998) 583–623; 625–667) when g(x,y)≡1. The limitation of the validity of the algebraic expansion to this sector as certain coefficients in the phase function are allowed to take on complex values is due to the appearance of exponential contributions. This is examined in detail in the special case when the phase function corresponds to a single internal point in the associated Newton diagram. Numerical examples illustrating the accuracy of the expansions are discussed.