{
  "abstract": "We introduce an $O(M)$ algorithm for evaluating the azimuthal Fourier modes $G_{k,m}$, $m = 0, 1, ..., M$, of the three-dimensional Helmholtz Green's function with real wavenumber $k$, together with all their first- and second-order derivatives with respect to the cylindrical source and target coordinates. The cost is independent of both the wavenumber and the source-target separation, and high relative accuracy is retained even for modes whose magnitude is exponentially small. The method combines contour deformation at a few boundary modes with a boundary-value formulation of the five-term recurrence in the mode index. Derivative quantities are obtained from stable recurrences, adding only a small constant factor to the cost of $G_{k,m}$ alone. Numerical experiments demonstrate high relative accuracy, linear scaling in $M$, and applications to modal boundary integral equation solvers for axisymmetric acoustic scattering, where the $k$-independent kernel evaluator makes dense per-mode linear algebra the dominant cost.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.09952",
  "author_total": 1,
  "authors": [
    {
      "name": "Hanwen Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/39r3hvnzhac9s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.09952",
  "fulltext_url": null,
  "id": "39r3hvnzhac9s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.09952",
  "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",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "physics.comp-ph"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2605.09952",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/39r3hvnzhac9s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:01Z",
      "native_id": "2605.09952",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.09952"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "physics",
      "origin": "arxiv:physics.comp-ph",
      "subfield": "computational"
    }
  ],
  "title": "Fast Evaluation of the Azimuthal Fourier Modes of the 3D Helmholtz Green's Function and Their Derivatives",
  "title_text": "Fast Evaluation of the Azimuthal Fourier Modes of the 3D Helmholtz Green's Function and Their Derivatives",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/39r3hvnzhac9s",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.1016/j.jcp.2026.115464",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}