Asymptotic expansions of one oscillatory double integral with large parameter (Q2792141)

The author considers integrals of the type NEWLINE\[NEWLINE I(x,y,k)\equiv \int \int_{D}g(u,v,x,y)e^{ikf(u,v,x,y)}\, du\,dv\, , NEWLINE\]NEWLINE where NEWLINE\[NEWLINE f(u,v,x,y)=2(x-u)\cos v - 2y \sin v\, , \qquad g(u,v,x,y)=1\, , NEWLINE\]NEWLINE and NEWLINE\[NEWLINE (u,v) \in D \equiv [a,b]\times [0,\pi/2]\, , \qquad (x,y) \in \Omega \equiv \{(x,y): x \geq 0\, , y \geq 0 \}\, . NEWLINE\]NEWLINE Integrals of this type appear in several boundary value problems, e.g., in diffraction theory, and many numerical methods have been exploited for their computation. The aim of this paper is to provide asymptotic expansions as the parameter \(k\) tends to \(+\infty\). For example, if \((x,y) \in \Omega_1\equiv \{(x,y): 0\leq x<a\, ,y \geq 0\}\), the author proves an expansion of the type NEWLINE\[NEWLINE I(x,y,k)=\Big[\frac{i}{2}e^{-2iyk}\log \frac{a-x}{b-x}\Big]\frac{1}{k}+C_I \frac{1}{k^{3/2}}+O\Big(\frac{1}{k^2}\Big)\, , NEWLINE\]NEWLINE as \(k\to +\infty\), where NEWLINE\[NEWLINE C_I\equiv \frac{i\sqrt{\pi}}{2}\Big[\frac{\sqrt{r_b}}{b-x}e^{-2ikr_b+i\pi/4}-\frac{\sqrt{r_a}}{a-x}e^{-2ikr_a+i\pi/4}\Big]\, , NEWLINE\]NEWLINE with NEWLINE\[NEWLINE r_b\equiv \sqrt{(b-x)^2+y^2} \, ,\qquad r_a\equiv \sqrt{(a-x)^2+y^2} \, . NEWLINE\]