Asymptotic Analysis of Diffraction Integrals in Gevrey Spaces

We present an overview on the state of the art of the research on the asymptotic behavior of diffraction integrals, i.e., the oscillatory integrals employed in the Fresnel theory of optics. We focus on the behavior of such integrals in the presence of standard caustics, in particular the elliptic and the hyperbolic umbilics, adopting the functional setting of Gevrey spaces. We also derive new estimates for the shadow region of the hyperbolic umbilic in terms of the distance from the caustic under symmetry condition in the space of parameters.