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SUMMARY:A note on double Fourier Integrals with applications to diffractio
 n theory - Raphael Assier (University of Manchester)
DTSTART:20211129T150000Z
DTEND:20211129T160000Z
UID:TALK166531@talks.cam.ac.uk
CONTACT:Alistair Hales
DESCRIPTION:One dimensional complex analysis and Fourier transforms are ve
 ry successful tools in diffraction theory. They led to many innovative mat
 hematical methods such as the Wiener-Hopf technique. Important canonical p
 roblems such as the diffraction by a half-plane can be dealt with efficien
 tly and elegantly with those. The resulting wave fields are often written 
 as an inverse Fourier transform integral over the real axis of the complex
  Fourier space. It is often convenient to assume that the wavenumber has a
  small positive imaginary part so that this integral is actually well defi
 ned\, and one then has to let the imaginary part of the wavenumber tend to
  zero somehow. This process is not straightforward and necessitates an ind
 entation of the integration contour. However\, even if they are very succe
 ssful\, such techniques are mainly applicable to “simple” two-dimensio
 nal problems.\nFor intrinsically three-dimensional problems (e.g. quarter-
 plane) or complicated two-dimensional problems (e.g. penetrable wedge)\, i
 t is not sufficient. One approach is to use double Fourier transforms toge
 ther with complex analysis in two complex variables.\nIn this talk\, we wi
 ll consider physical fields defined as double inverse Fourier integrals of
  a special class of spectral functions: those with the so-called real-prop
 erty. We will generalise the concept of contour indentation to higher dime
 nsions\, and provide a concise notation\, the bridge and arrow notation\, 
 that specifies the relative position of the integration surface and the si
 ngularities of the spectral functions. Using this we will aim to shed some
  light on the far-field asymptotics of such physical-field and show that i
 t can be reduced to the local study of a finite set of very specific point
 s in the double Fourier space.\nWe will illustrate this method by applying
  it to the long-standing problem of diffraction by a quarter-plane. This i
 s some ongoing joint work with A.V. Shanin and A.I. Korolkov.\n
LOCATION:Zoom
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