Assier Raphaël C, Abrahams I David
Department of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK.
Isaac Newton Institute, University of Cambridge, 20 Clarkson Road, Cambridge CB3 0EH, UK.
Proc Math Phys Eng Sci. 2020 Oct;476(2242):20200150. doi: 10.1098/rspa.2020.0150. Epub 2020 Oct 14.
We introduce and study a new canonical integral, denoted , depending on two complex parameters and . 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 , and derive its rich asymptotic behaviour as | | and | | 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 arising when the quarter-plane diffraction problem is studied via a two-complex-variables Wiener-Hopf technique (see Assier & Abrahams, , in press). As a result, the integral can be used to mimic the unknown function and to build an efficient 'educated' approximation to the quarter-plane problem.
我们引入并研究一种新的规范积分，记为 ，它依赖于两个复参数 和 。它源自四分之一平面的波衍射问题，并且是通过启发式构造来捕捉尖端和边缘附近的复场。我们在 中建立了这个积分的一些解析区域，并推导了当 和 趋于无穷时它丰富的渐近行为。我们还研究了对这个积分应用特定的双柯西积分算子所得到的函数的衰减性质。这些结果使我们能够表明，这个积分具有通过双复变量维纳 - 霍普夫技术研究四分之一平面衍射问题时关键未知函数 所预期的所有渐近性质（见阿西耶与亚伯拉罕斯，即将发表）。因此，积分 可用于模拟未知函数 并为四分之一平面问题构建一个有效的“有根据的”近似。
