{
  "id": "2501.08493",
  "version": "v1",
  "published": "2025-01-14T23:50:04.000Z",
  "updated": "2025-01-14T23:50:04.000Z",
  "title": "Integration of monomials over the unit spere and unit ball in $R^n$",
  "authors": [
    "Calixto P. Calderon",
    "Alberto Torchinsky"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "We compute the integral of monomials of the form $x^{2\\beta}$ over the unit sphere and the unit ball in $R^n$ where $\\beta = (\\beta_1,...,\\beta_n)$ is a multi-index with real components $\\beta_k > -1/2$, $1 \\le k \\le n$, and discuss their asymptotic behavior as some, or all, $\\beta_k \\to\\infty$. This allows for the evaluation of integrals involving circular and hyperbolic trigonometric functions over the unit sphere and the unit ball in $ R^n$. We also consider the Fourier transform of monomials $x^\\alpha$ restricted to the unit sphere in $R^n$, where the multi-indices $\\alpha$ have integer components, and discuss their behaviour at the origin.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-14T23:50:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26B25",
      "42B99"
    ],
    "keywords": [
      "unit ball",
      "unit spere",
      "unit sphere",
      "integration",
      "hyperbolic trigonometric functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}