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  "body": "\n## Definition\n\nLet $\\omega_1, \\omega_2 \\in \\C$ be linearly independent over $\\R$, and let $\\Lambda = \\set{m\\omega_1 + n\\omega_2 : m, n \\in \\Z}$ be the lattice they span.\nAn *elliptic function* for $\\Lambda$ is a [[meromorphic function|0NSD1TPHR0]] $f$ on $\\C$ such that\n$$f(z + \\omega_1) = f(z) \\quad \\text{and} \\quad f(z + \\omega_2) = f(z)$$\nfor all $z \\in \\C$ that are not poles of $f$.\nA function is *elliptic* if it is elliptic with respect to some such lattice.\n\n## Remarks\n\nThen $f(z + \\omega) = f(z)$ for every $\\omega \\in \\Lambda$, so $f$ is [[periodic|089KWVYD69]] with each lattice vector as a period.\nA *fundamental parallelogram* is a set $\\set{a + s\\omega_1 + t\\omega_2 : 0 \\le s, t \u003c 1}$ with $a \\in \\C$; an elliptic function is determined by its values there.\nExample: the [[Weierstrass elliptic function|361NAK68QK]] $\\wp$ of $\\Lambda$ is elliptic with respect to $\\Lambda$, and so is its derivative $\\wp'$.\nConstant functions are elliptic for every lattice.\nNon-example: $e^z$ is meromorphic and periodic, but its periods are exactly the numbers $2\\pi i k$ with $k \\in \\Z$.\nThey all lie on one line through $0$, so there is no second period independent over $\\R$.\nEquivalently, the elliptic functions for $\\Lambda$ are the meromorphic functions on the compact Riemann surface $\\C / \\Lambda$.\n",
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