@article{f653f403c8194b7a93e20adea0a35164,

title = "On the asymptotic properties of a canonical diffraction integral: A canonical diffraction integral",

abstract = "We introduce and study a new canonical integral, denoted I+-ϵ, depending on two complex parameters α 1 and α 2. It arises from the problem of wave diffraction by a quarter-plane and is heuristically constructed to capture the complex field near the tip and edges. We establish some region of analyticity of this integral in C2, and derive its rich asymptotic behaviour as |α 1 | and |α 2 | tend to infinity. We also study the decay properties of the function obtained from applying a specific double Cauchy integral operator to this integral. These results allow us to show that this integral shares all of the asymptotic properties expected from the key unknown function G +- arising when the quarter-plane diffraction problem is studied via a two-complex-variables Wiener-Hopf technique (see Assier & Abrahams, SIAM J. Appl. Math., in press). As a result, the integral I+-ϵ can be used to mimic the unknown function G +- and to build an efficient 'educated' approximation to the quarter-plane problem.",

keywords = "Cauchy integrals, Wiener-Hopf, diffraction, quarter-plane",

author = "Raphael Assier and Abrahams, {I David}",

note = "Funding Information: Data accessibility. This article does not contain any additional data. Authors{\textquoteright} contributions. R.C.A. developed the mathematics, performed the numerical implementation, produced the figures and wrote the article. I.D.A. contributed through mathematical discussions and provided his invaluable insight throughout the development of this work. He also contributed to the final edits to the manuscript and gave his approval for submission. Competing interests. We declare we have no competing interests. Funding. This work was supported by the EPSRC grant no. EP/R014604/1. Abrahams also acknowledges the support of UKRI/EPSRC grant no. EP/K032208/1. Acknowledgements. Both authors thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme {\textquoteleft}Bringing pure and applied analysis together via the Wiener– Hopf technique, its generalizations and applications{\textquoteright} where some work on this paper was undertaken. Publisher Copyright: {\textcopyright} 2020 The Author(s).",

year = "2020",

month = oct,

doi = "10.1098/rspa.2020.0150",

language = "English",

volume = "476",

journal = "Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science",

issn = "1471-2946",

publisher = "Royal Society",

number = "2242",

}