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Uniform asymptotic expansions of a double integral: Coalescence of two stationary points. (English) Zbl 0966.41020
The double integral $$I(\lambda, \alpha)= \iint_Dg(x,y, \alpha) e^{i\lambda f(x,y,\alpha)} dxdy$$ where $\lambda$ is a large positive variable and $\alpha$ is an auxiliary parameter is considered. The case where the function $f$ has two simple stationary points $(x_+(\alpha), y_+(\alpha))$ and $(x_-(\alpha), y_-(\alpha))$ in $D$, which coalesce at a point $(x_0,y_0)$ (which can either be an interior or a boundary point of $D)$ as $\alpha$ approaches a critical value $\alpha_0$. Asymptotic expansions are derived in both cases. The considered integral can be related with certain integral transforms. The concept of coalescence is interesting.
Reviewer: F.Perez Acosta (Laguna)
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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