{
  "abstract": "The functions $φ_\\ell(z)=(e^z-s_\\ell(z))/z^\\ell$, with $s_\\ell$ the Taylor polynomial of $e^z$ of degree $\\ell-1$, are what exponential integrators evaluate, and they evaluate them through rational approximations on the negative real axis $\\mathbb{R}_-$. Schmelzer and Trefethen (2007/08) conjectured that the best such approximations converge at the rate of the classical `1/9'-problem: $E_n(φ_\\ell)^{1/n}\\to H$, Halphen's constant, for every $\\ell$. We prove it. The theorem behind it gives the same rate for $f=u_0+u_1\\exp$ with $u_0$ rational and $u_1\\not\\equiv0$ meromorphic with finitely many poles and $\\log|u_1(z)|=o(|z|)$, the lower bound required only away from $\\mathbb{R}_-$, whenever $f$ has no pole on $\\mathbb{R}_-$ (Theorem~4.4). For rational $u_1$ this is Theorem 1 of Stahl and Schmelzer (2009), announced with its proof deferred to a manuscript that never appeared; the hypotheses admit, for instance, $u_1(z)=\\sinh(π\\sqrt z)/(π\\sqrt z)$, whose zeros are infinite in number. The proof is a reading of Gonchar and Rakhmanov (1989) rather than an extension of them. Their theorem concerns a sequence given by a Cauchy-type integral with a varying weight $Φ_n$, and a contour identity of Schmelzer and Trefethen presents $u_1\\exp$, once its principal parts are removed, in exactly that form, with $Φ_n(t)=u_1(-nt)e^{-nt}$. The factor $u_1(-nt)$ grows subexponentially in $n$, and their hypothesis on $Φ_n$ divides by $2n$, so it drops out of the external field: field, extremal arc, $S$-property and constant are all theirs, unchanged. For $φ_\\ell$ the factor is $t^{-\\ell}$, whose pole sits at the origin, a point of the approximation set and hence at positive distance from every admissible contour.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14489",
  "author_total": 1,
  "authors": [
    {
      "name": "Thomas Schmelzer"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9n0mp34kbnjae.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14489",
  "fulltext_url": null,
  "id": "9n0mp34kbnjae",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14489",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7213268303",
  "pdf_url": "https://arxiv.org/pdf/2609.14489",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9n0mp34kbnjae.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14489",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14489"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The `1/9'-problem for the $φ$-functions",
  "title_text": "The `1/9'-problem for the φ-functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/9n0mp34kbnjae",
  "venue": "arXiv",
  "versions": []
}