Stationary phase with Cauchy singularity. A critical point of signature $(+,-)$

TL;DR

The paper develops a rigorous small-$h$ asymptotic framework for a solid Cauchy transform with a Cauchy-type singularity near a stationary point of oscillatory phase. When the pole lies close to a stationary point, a polarization method in $\mathbb{C}^2$ combined with Stokes' theorem yields a three-term decomposition of the integral, with the leading terms described by Dawson-type special functions and explicit $h^{1/2}$-scaling factors. The first term $I(\zeta,1)$ is analyzed via a 1D stationary-phase reduction on a Cartesian contour, the second term $II(\zeta,1)$ via an auxiliary $t$-integral representation with detailed intermediate and small-$\varepsilon$ regime analysis, and the third term $III(\zeta,1)$ is shown to be $O(h^{\infty})$. Collectively, the results provide concrete, rotation-invariant leading-order expressions for the integral across regimes $0<|\zeta|<h^{1/2+\delta}$ and offer insight into d-bar problems in 2D integrable systems and related inverse problems. The approach highlights the utility of complex contour deformations, polarization, and special-function representations in high-frequency asymptotics.

Abstract

Asymptotic expressions for an integral appearing in the solution of a d-bar problem are presented. The integral is a solid Cauchy transform of a function with a rapidly oscillating phase with a small parameter $h$, $0<h\ll 1$. Whereas standard steepest descent approaches can be applied to the case where the stationary points of the phase $ω_{k}$, $k=1,\ldots, N$ are far from the singularity $ζ$ of the integrand, a polarization approach is proposed for the case that $|ζ-ω_{k}|<\mathcal{O}(\sqrt{h})$ for some $k$. In this case the problem is studied in $\mathbb{C}^{2}$ ($\tildeω:=\barω$ is treated as an independent variable) on steepest descent contours. An application of Stokes' theorem allows for a decomposition of the integral into three terms for which asymptotics expressions in terms of special functions are given.

Theorems & Definitions (23)

• Proposition 1.1
• Theorem 1.2
• Remark 1.3
• Proposition 2.1
• Lemma 3.1
• Remark 3.2
• Remark 3.3
• Lemma 3.4
• Proposition 3.5
• Lemma 4.1