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Uniform asymptotic expansions of a double integral: Coalescence of two stationary points. (English) Zbl 0966.41020

The double integral

I(λ,α)= D g(x,y,α)e iλf(x,y,α) dxdy

where λ is a large positive variable and α is an auxiliary parameter is considered. The case where the function f has two simple stationary points (x + (α),y + (α)) and (x - (α),y - (α)) in D, which coalesce at a point (x 0 ,y 0 ) (which can either be an interior or a boundary point of D) as α approaches a critical value α 0 . Asymptotic expansions are derived in both cases. The considered integral can be related with certain integral transforms. The concept of coalescence is interesting.

MSC:
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)