On the asymptotic properties of an interesting integral

Raphael C. Assier, I. David Abrahams

Published 2020-02-29Version 1

We introduce and study a new special integral, denoted $I_{+-}^{\varepsilon}$, depending on two complex parameters $\alpha_1$ and $\alpha_2$. 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 $|\alpha_1|$ and $|\alpha_2 |$ 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, 2019). 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.