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  "body": "\n## Definition\n\nLet $\\omega_1, \\omega_2 \\in \\C$ be linearly independent over $\\R$, and let $\\Lambda = \\Z\\omega_1 + \\Z\\omega_2$ be the [[lattice|WN9V7WSJKF]] they span.\nThe *Weierstrass sigma function* of $\\Lambda$ is the [[infinite product|JYV1Y8XX16]]\n$$\\sigma(z) = z \\prod_{\\omega \\in \\Lambda \\setminus \\set{0}} \\left(1 - \\frac{z}{\\omega}\\right) \\exp\\!\\left(\\frac{z}{\\omega} + \\frac{z^2}{2\\omega^2}\\right), \\qquad z \\in \\C.$$\nThe *Weierstrass zeta function* of $\\Lambda$ is the meromorphic function\n$$\\zeta(z) = \\frac{\\sigma'(z)}{\\sigma(z)} = \\frac{1}{z} + \\sum_{\\omega \\in \\Lambda \\setminus \\set{0}} \\left(\\frac{1}{z - \\omega} + \\frac{1}{\\omega} + \\frac{z}{\\omega^2}\\right), \\qquad z \\in \\C \\setminus \\Lambda.$$\n\n## Remarks\n\nThe product converges absolutely and locally uniformly, because each factor is $1 + O(\\abs{z/\\omega}^3)$ and $\\sum_{\\omega \\ne 0}\\abs{\\omega}^{-3} \u003c \\infty$.\nSo $\\sigma$ is an [[entire function|0G4F0XB9X2]], odd, with simple zeros exactly at the points of $\\Lambda$.\nAlso $\\zeta'(z) = -\\wp(z)$, where $\\wp$ is the [[Weierstrass elliptic function|361NAK68QK]] of $\\Lambda$.\nSince $\\wp$ is periodic, $\\eta_j = \\zeta(z + \\omega_j) - \\zeta(z)$ is a constant, the *quasi-period*, for $j = 1, 2$.\nThen $\\sigma(z + \\omega_j) = -\\exp\\big(\\eta_j(z + \\omega_j/2)\\big)\\,\\sigma(z)$.\nIf $\\operatorname{Im}(\\omega_2/\\omega_1) \u003e 0$, the Legendre relation $\\eta_1\\omega_2 - \\eta_2\\omega_1 = 2\\pi i$ holds; it follows by integrating $\\zeta$ around a fundamental parallelogram, where $\\zeta$ has one simple pole of residue $1$.\nExample: replace $\\Lambda$ by the one-dimensional group $\\Z\\pi$ in the same formulas.\nSince $\\sum_{n \\ne 0} 1/(n\\pi)^2 = 1/3$, they give $\\zeta(z) = \\cot z + z/3$ and $\\sigma(z) = e^{z^2/6}\\sin z$.\nThis is the limit of the lattice case when $\\omega_1 = \\pi$ is fixed and $\\omega_2 \\to i\\infty$.\nNon-example: $\\zeta$ is not an elliptic function.\nA nonconstant elliptic function cannot have only one simple pole in a period parallelogram, and indeed the Legendre relation shows that $\\eta_1, \\eta_2$ are not both $0$.\n",
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