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This work aims to provide asymptotic expressions for a Cauchy transform integral related to d-bar problems.

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

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This work aims to provide asymptotic expressions for a Cauchy transform integral related to d-bar problems.
Utilized a polarization approach near the Cauchy singularity.
Applied steepest descent techniques in complex space.
Employed Stokes' theorem to decompose the integral into three evaluable parts.
Successfully derived asymptotic expressions involving special functions.
Identified cases where the stationary points approach the singularity with small parameter h.
Demonstrated the effectiveness of the polarization approach in the analysis.

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Randomized trial investigates asymptotic properties of Cauchy transforms in complex settings, suggesting new evaluation methods.

Klein et al. (2026) studied this question.

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