Uniform asymptotic expansions of a double integral: coalescence of two stationary points

Consider the double integral I(lambda,alpha) = integral integral(D)g(x,y,alpha)e(i lambda f(x,y,alpha))d xdy, where lambda is a large positive variable and a is an auxiliary parameter. We consider the case in which the phase function f(x, y, alpha) has two sim ple stationary points (x+(alpha),y+(alpha)) and (x_(alpha),y_(alpha)) in D, which coalesce at a point (x(0),y(0)) as alpha approaches a critical value cue. The point (x(0), y(0)) can either be an interior point of D or a boun dary point of D. Asymptotic expansions are derived in both cases, which hol d uniformly in a neighbourhood of cro. Our derivation is mathematically rig orous.