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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.16542 |
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| _version_ | 1866910232756617216 |
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| author | Klein, Christian Sjöstrand, Johannes Zerzeri, Maher |
| author_facet | Klein, Christian Sjöstrand, Johannes Zerzeri, Maher |
| contents | 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 $\mathbb{C}^{2}$ ($\widetildeω:=\overlineω$ 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 asymptotic expressions in terms of special functions are given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16542 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stationary phase with Cauchy singularity. A critical point of signature $(+,-)$ Klein, Christian Sjöstrand, Johannes Zerzeri, Maher Analysis of PDEs Mathematical Physics 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 $\mathbb{C}^{2}$ ($\widetildeω:=\overlineω$ 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 asymptotic expressions in terms of special functions are given. |
| title | Stationary phase with Cauchy singularity. A critical point of signature $(+,-)$ |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2601.16542 |