{
  "id": "1006.3105",
  "version": "v4",
  "published": "2010-06-15T22:59:15.000Z",
  "updated": "2011-02-28T18:19:10.000Z",
  "title": "Holomorphic functions on subsets of C",
  "authors": [
    "Buma L. Fridman",
    "Daowei Ma"
  ],
  "comment": "12 pages",
  "categories": [
    "math.CV"
  ],
  "abstract": "Let $\\Gamma $ be a $C^\\infty $ curve in $\\Bbb{C}$ containing 0; it becomes $\\Gamma_\\theta $ after rotation by angle $\\theta $ about 0. Suppose a $C^\\infty $ function $f$ can be extended holomorphically to a neighborhood of each element of the family $\\{\\Gamma_\\theta \\}$. We prove that under some conditions on $\\Gamma $ the function $f$ is necessarily holomorphic in a neighborhood of the origin. In case $\\Gamma $ is a straight segment the well known Bochnak-Siciak Theorem gives such a proof for \\textit{real analyticity}. We also provide several other results related to testing holomorphy property on a family of certain subsets of a domain in $\\Bbb{C}$.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2011-02-28T18:19:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30E99"
    ],
    "keywords": [
      "holomorphic functions",
      "neighborhood",
      "straight segment",
      "bochnak-siciak theorem",
      "testing holomorphy property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1006.3105F"
    }
  }
}