论文标题
在规范衍射积分的渐近特性上
On the asymptotic properties of a canonical diffraction integral
论文作者
论文摘要
我们介绍和研究一个新的规范积分，表示为$ i _ {+ - }^{\ varepsilon} $，具体取决于两个复杂参数$α_1$和$α_2$。它是由四分之一平面的波衍射的规范问题引起的，并且是启发式构造的，以捕获尖端和边缘附近的复杂场。我们在$ \ mathbb {c}^2 $中建立了该积分的分析性区域，并将其丰富的渐近行为推导为$ |α_1| $和$ |α_2| $倾向于无限。我们还研究了通过将特定的双重库奇积分算子应用于该积分而获得的函数的衰减特性。这些结果使我们能够证明该积分分享了关键未知函数所期望的所有渐近属性$ g _ {+ - } $在通过两种复合变量wiener--hopf技术研究四分之一平面衍射问题时会产生（请参阅Assier \＆Abrxiv，Arxiv，Arxiv，Arxiv：1905.05363，2020，2020）。结果，可以使用积分$ i _ {+ - }^{\ varepsilon} $来模仿未知函数$ g _ {+ - } $，并构建有效的“受过教育的”近似值对四分之一平面问题。
We introduce and study a new canonical integral, denoted $I_{+-}^{\varepsilon}$, depending on two complex parameters $α_1$ and $α_2$. It arises from the canonical 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 $\mathbb{C}^2$, 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, arXiv:1905.03863, 2020). As a result, the integral $I_{+ -}^{\varepsilon}$ can be used to mimic the unknown function $G_{+ -}$ and to build an efficient `educated' approximation to the quarter-plane problem.
