Determinant Characteristics and Argument Principle Certification for Visible Poles in Meromorphic Continuation

Jul 16, 2026

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Outward meromorphic continuation is studied from circular boundary data on the unit disk, where the unknown function is holomorphic inside and extends meromorphically to a larger disk with finitely many exterior simple poles over an unknown holomorphic background. Positive Fourier coefficients encode Taylor data, and exterior poles induce a finite exponential-sum structure. Shifted determinant characteristics yield exact factorization in a pure finite-pole model, enabling pole-reciprocal zero recovery from noiseless discrete Fourier data. With background, discretization, and noise, determinant roots provide candidate poles, while argument-principle-based local contour certification, persistence across orders and shifts, and Rouché-type perturbation bounds establish visibility conditions. Numerical experiments validate stability and show how residues, pole separation, annulus distance, shifts, and noise govern partial recovery.

LanguageEnglish

CategoryResearch & Report

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Pages27

Reading Time1 h 8 min

Abstract

Introduction

arXiv :2607 .04568v1 [math .NA] 6 Jul 2026 Determinant Characteristics and Argument-Principle Certification for Visible Poles in Meromorphic Continuation Xiaomei Yang 1 and Zhiliang Deng2* 1 School of Mathematics, Southwest Jiaotong University, No. 999, Xi’an Road, Pidu District, Chengdu, 611756, Sichuan, China. 2* School of Mathematical Science, University of Electronic Science and Technology of China, No.2006, Xiyuan Ave, West Hi-Tech Zone, Chengdu, 611731, Sichuan, China. *Corresponding author(s). E-mail(s): [dengzhl@uestc.edu.cn](dengzhl@uestc.edu.cn) ; Contributing authors: [yangxiaomath@swjtu.edu.cn](yangxiaomath@swjtu.edu.cn) ; Abstract We study outward meromorphic continuation from circular boundary data in the unit disk. The unknown function is holomorphic in the unit disk and admitsa meromorphic continuation to a larger disk, where finitely many exterior simple poles are superposed on an unknown holomorphic background. The positive Fourier coefficients of the boundary trace are Taylor coefficients at the origin, and exterior poles generate a finite exponential-sum component in these coefficients. We introduce shifted determinant characteristics and prove that, in the pure finite-pole model, the determinant for the correct order factors exactly into a nonzero constant times the polynomial whose zeros are the reciprocals of the exterior poles. The same zero set is obtained for noiseless equispaced discrete Fourier coefficients; sampling changes only the amplitudes through an aliasing factor. For data containing a holomorphic background, discretization effects, and noise, roots of a single empirical determinant are only candidate reciprocal poles. We therefore propose a root-propose and contour-certify procedure: determinant roots generate candidate regions, while local argument-principle counts, contour moments, empirical margins, and persistence over determinant orders and shifts certify visible poles. A Rouch´e-type perturbation analysis gives sufficient conditions for stable local zero counts and explains how residues, pole separation, distance to the target annulus boundary, shifts, and noise affect visibility. Numerical experiments verify the pure-pole identity, demonstrate certification under background and noise, and show that high noise, weak residues, boundary-near 1 poles, and close poles naturally lead to partial recovery of contour-certified visible poles. Keywords: numerical meromorphic continuation, determinant characteristic, argument principle, visible poles, Fourier coefficients, contour certification. 1 Introduction Numerical analytic continuation is a classical ill-conditioned problem. Although analytic continuation is unique when it exists, its numerical realization from finite and noisy data is unstable unless suitable a priori information is imposed. The severity of this instability depends strongly on both the geometry of the continuation domain and the assumed function class. For radial continuation in a disk, a bounded holomorphic prior leads to a comparatively mild conditional stability mechanism through the Hadamard three-circles theorem: the loss of accuracy is governed by the logarithmic radius. This behavior is substantially different from continuation along strips or channels, where the loss of accuracy may be much more severe; see, for example,[1] . A substantial part of the classical literature treats numerical analytic continuation as a stabilized reconstruction problem for holomorphic functions, where stability is obtained from a priori bounds, geometric estimates, rational approximation, or regularization assumptions. Cannon and Miller [2] studied stabilized continuation of bounded analytic functions, while Henrici [3] developed a constructive Weierstrasstype procedure for continuing a holomorphic function from Taylor data along a path. Approximation-theoretic foundations go back to Walsh’s work on boundary values, Chebyshev approximation, and approximat

FAQs

What problem does the paper address in meromorphic continuation?

It addresses outward meromorphic continuation from circular boundary data in the unit disk, focusing on identifying and certifying exterior simple poles rather than performing unstable pointwise extrapolation.

How does the determinant characteristic help recover exterior poles?

In the pure finite-pole model, the shifted determinant factors exactly so that its zeros are the reciprocals of the exterior poles; with noiseless equispaced discrete Fourier data, sampling preserves the same zero set up to amplitude aliasing.

How are poles certified when data include background, discretization, and noise?

The method uses a root-propose and contour-certify procedure: determinant roots generate candidate pole regions, and local argument-principle counts with contour moments, empirical margins, and persistence across determinant orders and shifts certify visible poles under perturbation.

Jul 16, 2026

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