{
  "abstract": "For $h>0$, degree $m\\ge0$, $0\\leβ<π$, and $q>0$, let $Σ_β=\\{re^{iθ}:r\\ge0,\\ |θ|\\leβ/2\\}$ and let $E_{m,β}(q)$ denote the best uniform error for approximating $e^{-hz}$ on $Σ_β$ by rational functions whose only possible finite pole is $s=-q/h$, with multiplicity at most $m$. For the linear scaling $q=αm$ with a fixed $α>0$, a Möbius map reduces the problem to polynomial approximation on a compact lens. Write $γ_β=2-β/π$ and let $α_c(γ_β)$ denote the saddle-coalescence threshold. For $0<α<α_c(γ_β)$, let $H_β(α)\\in(0,1)$ be the exponential of the real saddle action. An exact boundary Faber integral, a global fractional-power contour deformation, and a two-saddle expansion yield the sharp two-sided order $E_{m,β}(αm)\\asymp m^{-1/2}H_β(α)^m$, uniformly on compact subsets of $0<α<α_c(γ_β)$. On this noncoalescing interval, the exponential factor $H_β(α)$ has the unique minimizer $α_β^*=1/[γ_β\\sin(π/(2γ_β))]$ and minimum $\\tan(π/(4γ_β))$, recovering the classical half-line factor $\\tan(π/8)=\\sqrt{2}-1$ at $β=0$. Numerical-range estimates give matrix dimension-independent upper bounds for projected shift-and-invert Arnoldi, while finite-degree pole searches and nonnormal experiments illustrate the distinction between the sector-uniform law and individual matrix computations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09251",
  "author_total": 2,
  "authors": [
    {
      "name": "Fei Xue"
    },
    {
      "name": "Tianqi Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5dk13wy22xmes.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09251",
  "fulltext_url": null,
  "id": "5dk13wy22xmes",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09251",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7220901383",
  "pdf_url": "https://arxiv.org/pdf/2610.09251",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5dk13wy22xmes.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.09251",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09251"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Convergence of repeated-pole rational approximation of the exponential on infinite sectors",
  "title_text": "Convergence of repeated-pole rational approximation of the exponential on infinite sectors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/5dk13wy22xmes",
  "venue": "arXiv",
  "versions": []
}