{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "11VC51YHQJ"
    },
    {
      "field": "uses",
      "target": "11VC51YHQJ"
    },
    {
      "field": "uses",
      "target": "DQEHD56649"
    },
    {
      "field": "uses",
      "target": "KYSR7NHNFQ"
    },
    {
      "field": "uses",
      "target": "PDG1HMR8GK"
    },
    {
      "field": "defines",
      "target": "99B3B7AAVT"
    }
  ],
  "body": "\n## Summary\n\nA Weierstrass product is an infinite product of elementary factors $E_{p_n}(z/a_n)$ that converges to an [[entire function|0419794R0T]] with zeros exactly at the points $a_n$.\nEach elementary factor is $1 - z/a_n$ times an exponential correction that makes the product converge.\nWeierstrass products show that any sequence of nonzero complex numbers tending to infinity is the zero set of an entire function.\nThey are the building blocks of the [[Weierstrass factorization theorem|11VC51YHQJ]] and the [[Hadamard factorization theorem|KYSR7NHNFQ]].\n\n## Explanations\n\n### L1 human\n\nTo build a polynomial with given roots, multiply the factors $1 - z/a$.\nWith infinitely many roots, this product may diverge, as for $\\prod_{n \\ge 1} (1 - z/n)$.\nWeierstrass multiplies each factor by $\\exp(z/a + z^2/(2a^2) + \\cdots + z^p/(p a^p))$, which equals $1$ at $z = 0$ and has no zeros.\nThis correction removes the first terms of $\\log(1 - z/a)$ and makes the product converge.\nFor the roots $n = 1, 2, 3, \\dots$, the product $\\prod_n (1 - z/n) e^{z/n}$ converges, and it is closely related to the gamma function.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "30D20",
      "30D10"
    ],
    "name": "Weierstrass product",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Weierstrass_factorization_theorem"
      }
    ],
    "tag": "ACSF58HYDH"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/ACSF58HYDH/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/ACSF58HYDH.md",
  "msc": [
    "30D20",
    "30D10"
  ],
  "name": "Weierstrass product",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ACS/ACSF58HYDH.md",
  "status": null,
  "tag": "ACSF58HYDH",
  "uri": "m2://ACSF58HYDH",
  "url": "https://math2.tamnd.com/wiki/weierstrass-product/",
  "writing": {
    "tag": "ACSF58HYDH",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}