{
  "id": "2411.01591",
  "version": "v1",
  "published": "2024-11-03T14:58:18.000Z",
  "updated": "2024-11-03T14:58:18.000Z",
  "title": "What do sin$(x)$ and arcsinh$(x)$ have in Common?",
  "authors": [
    "Steven Finch"
  ],
  "comment": "24 pages",
  "categories": [
    "math.CA",
    "cs.DM",
    "math.CO"
  ],
  "abstract": "N. G. de Bruijn (1958) studied the asymptotic expansion of iterates of sin$(x)$ with $0 < x \\leq \\pi/2$. Bencherif & Robin (1994) generalized this result to increasing analytic functions $f(x)$ with an attractive fixed point at 0 and $x > 0$ suitably small. Mavecha & Laohakosol (2013) formulated an algorithm for explicitly deriving required parameters. We review their method, testing it initally on the logistic function $\\ell(x)$, a certain radical function $r(x)$, and later on several transcendental functions. Along the way, we show how $\\ell(x)$ and $r(x)$ are kindred functions; the same is also true for sin$(x)$ and arcsinh$(x)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-03T14:58:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "39A20",
      "11B37",
      "26A18",
      "37E05",
      "41A60",
      "65D20"
    ],
    "keywords": [
      "asymptotic expansion",
      "increasing analytic functions",
      "logistic function",
      "transcendental functions",
      "attractive fixed point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}