ИСТИНА

A 2D Fourier integral is studied, whose transformant is a function having algebraic growth at infinity and holomorphic everywhere except some polar and branching sets of complex codimension 1. The integration surface is the real plane, possibly slightly shifted to avoid the singularities of the transformant. The aim of the talk is to build the deformation of the integration surface into some other surface to make the exponential factor of the Fourier integral decaying as fast as possible on it. According to the multidimensional Cauchy's theorem, a deformation (homotopy) not hitting the singular sets does not change the value of the integral. At the same time, a proper deformation makes the integral more suitable for numerical evaluation or for asymptotical investigation. A general procedure of deformation of the integration surface is proposed in the talk. The resulting surface is a sum of components stemming from the special points of the singularities of the transformant, such as the saddle on singularities or the crossings (see [1]). The components have topological structure dictated by the special point type. Thus, the field becomes splitted explicitly into terms corresponding to the special points of the singularities. The work is being done in collaboration with Raphael C. Assier and Andrey I. Korolkov from the University of Manchester. References 1. R.C.Assier, A.V.Shanin, A.I.Korolkov, A contribution to the mathematical theory of diffraction. Part I: A note on double Fourier integrals // QJMAM, 76(2):211-241, 2022.