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SUMMARY:Using multidimensional complex analysis to estimate double Fourier
 -like integrals arising in diffraction theory - Raphael Assier (University
  of Manchester)
DTSTART:20230207T110000Z
DTEND:20230207T114500Z
UID:TALK194755@talks.cam.ac.uk
DESCRIPTION:We consider a large class of physical wave fields u written as
  double inverse Fourier transforms of some functions F of two complex vari
 ables. Such integrals occur very often in practice\, especially in diffrac
 tion theory. Our aim is to provide a closed-form far-field asymptotic expa
 nsion 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 a
 nd arrow notation. Thanks to another integration surface deformation\, we 
 show that\, to achieve our aim\,&nbsp\; we only need to study a finite num
 ber of real points in the Fourier space: the contributing points. This is 
 called the locality principle. We provide an extensive set of results allo
 wing one to decide whether a point is contributing or not and derive asymp
 totic formulae for each contributing points. Time permitting\, we will sho
 w how this theory developed for double Fourier transforms can be efficient
 ly 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. Koro
 lkov&nbsp\;(2022)\, A contribution to the mathematical theory of diffracti
 on. A note on double Fourier integrals.\, Q. J. Mech. Appl. Math.\, hbac01
 7
LOCATION:Seminar Room 1\, Newton Institute
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