Abstract

We consider a large class of physical wave fields u written as double inverse Fourier transforms of some functions F of two complex variables. Such integrals occur very often in practice, especially in diffraction theory. Our aim is to provide a closed-form far-field asymptotic expansion of u. In order to do so, we need to generalise the well-established complex analysis notion of contour indentation to integrals of functions of two complex variables. It is done by introducing the so-called bridge and arrow notation. Thanks to another integration surface deformation, we show that, to achieve our aim,  we only need to study a finite number of real points in the Fourier space: the contributing points. This is called the locality principle. We provide an extensive set of results allowing one to decide whether a point is contributing or not and derive asymptotic formulae for each contributing points. Time permitting, we will show how this theory developed for double Fourier transforms can be efficiently adapted to more realistic Fourier-like integrals.The main part of this talk is based on the following paper:R.C. Assier, A.V. Shanin, A.I. Korolkov (2022), A contribution to the mathematical theory of diffraction. A note on double Fourier integrals., Q. J. Mech. Appl. Math., hbac017