{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "KYSR7NHNFQ"
    }
  ],
  "body": "\n## Statement\n\nFor an integer $p \\ge 0$, let $E_p$ be the elementary factor\n$$E_0(z) = 1 - z, \\qquad E_p(z) = (1 - z) \\exp\\left(z + \\frac{z^2}{2} + \\cdots + \\frac{z^p}{p}\\right) \\quad (p \\ge 1).$$\n\n1. Let $(a_n)$ be a finite or infinite sequence of nonzero complex numbers with $\\abs{a_n} \\to \\infty$ if it is infinite. Then there is an [[entire function|0419794R0T]] whose [[zeros|0CEGEASY49]] are exactly the points $a_n$. The multiplicity of each zero is the number of times it occurs in the sequence.\n2. Let $f$ be an entire function that is not identically zero. Let $m \\ge 0$ be the order of the zero of $f$ at $0$. Let $(a_n)$ be the nonzero zeros of $f$, repeated according to multiplicity. Then there are an entire function $g$ and integers $p_n \\ge 0$ such that\n$$f(z) = z^m e^{g(z)} \\prod_{n} E_{p_n}\\!\\left(\\frac{z}{a_n}\\right),$$\nwhere the product converges uniformly on compact subsets of $\\C$.\n\n## Remarks\n\nThe key estimate is $\\abs{1 - E_p(z)} \\le \\abs{z}^{p+1}$ for $\\abs{z} \\le 1$.\nChoose the $p_n$ so that $\\sum_n (r / \\abs{a_n})^{p_n + 1} \u003c \\infty$ for every $r \u003e 0$, for example $p_n = n - 1$.\nThen the [[Weierstrass product|ACSF58HYDH]] converges and has the prescribed zeros.\nFor part 2, the quotient of $f$ by $z^m$ times this product is entire and has no zeros.\nSo it is $e^{g}$ for an entire $g$, because $\\C$ is simply connected.\nA classical example is [[Euler's sine product|DQEHD56649]] $\\sin(\\pi z) = \\pi z \\prod_{n \\ge 1} (1 - z^2/n^2)$.\nThe convergence factors are needed: for $a_n = n$ the product $\\prod_n (1 - z/n)$ diverges, while $\\prod_n (1 - z/n) e^{z/n}$ converges.\n\n## Explanations\n\n### L1 human\n\nA polynomial is determined, up to a constant factor, by its roots: it is a constant times a product of factors $z - a$.\nThe theorem extends this to entire functions, which can have infinitely many zeros.\nThe plain factors $1 - z/a_n$ may give a product that does not converge, so each factor gets an exponential correction that is close to $1$ near $0$.\nWhat is left after removing all zeros is a function without zeros, which is the exponential of an entire function.\n\n### L2 human\n\nPart 1 shows that any sequence tending to infinity is the zero set of some entire function.\nAs a consequence, every [[meromorphic function|0NSD1TPHR0]] on $\\C$ is a quotient of two entire functions: multiply it by an entire function with zeros at its poles.\nThe theorem leaves $g$ and the $p_n$ very free.\nFor functions of finite order the [[Hadamard factorization theorem|KYSR7NHNFQ]] makes both choices precise.\nThe companion result for poles and principal parts is [[Mittag-Leffler's theorem|X4R2GY1F6J]].\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "0419794R0T",
      "ACSF58HYDH"
    ],
    "kind": "statement",
    "msc": [
      "30D20",
      "30D10"
    ],
    "name": "Weierstrass factorization theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q1330788",
      "wikipedia": "Weierstrass factorization theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Weierstrass_factorization_theorem"
      }
    ],
    "tag": "11VC51YHQJ",
    "uses": [
      "0419794R0T",
      "0CEGEASY49",
      "ACSF58HYDH",
      "0RMVS7RXZX"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/11VC51YHQJ/",
  "links": [
    {
      "field": "about",
      "target": "0419794R0T"
    },
    {
      "field": "about",
      "target": "ACSF58HYDH"
    },
    {
      "field": "uses",
      "target": "0419794R0T"
    },
    {
      "field": "uses",
      "target": "0CEGEASY49"
    },
    {
      "field": "uses",
      "target": "ACSF58HYDH"
    },
    {
      "field": "uses",
      "target": "0RMVS7RXZX"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/11VC51YHQJ.md",
  "msc": [
    "30D20",
    "30D10"
  ],
  "name": "Weierstrass factorization theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/11V/11VC51YHQJ.md",
  "status": null,
  "tag": "11VC51YHQJ",
  "uri": "m2://11VC51YHQJ",
  "url": "https://math2.tamnd.com/wiki/weierstrass-factorization-theorem/",
  "writing": {
    "tag": "11VC51YHQJ",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}