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  "body": "\n## Definition\n\nLet $\\mathbb{H}$ be the [[upper half-plane|X6XGGH5FPW]].\nThe *Jacobi theta function* is the function $\\vartheta \\colon \\C \\times \\mathbb{H} \\to \\C$ given by\n$$\\vartheta(z, \\tau) = \\sum_{n = -\\infty}^{\\infty} \\exp(\\pi i n^2 \\tau + 2 \\pi i n z).$$\nIts value at $z = 0$ is the theta constant $\\theta(\\tau) = \\vartheta(0, \\tau) = \\sum_{n \\in \\Z} e^{\\pi i n^2 \\tau}$.\n\n## Remarks\n\nFor $\\operatorname{Im} \\tau \u003e 0$ the terms decay like $e^{-\\pi n^2 \\operatorname{Im} \\tau}$ times a factor locally bounded in $z$, so the series converges absolutely and locally uniformly.\nSo $\\vartheta$ is [[holomorphic|0NAHXR1FRH]] in both variables.\nIt satisfies $\\vartheta(z + 1, \\tau) = \\vartheta(z, \\tau)$ and $\\vartheta(z + \\tau, \\tau) = \\exp(-\\pi i \\tau - 2 \\pi i z)\\, \\vartheta(z, \\tau)$.\nCheck of the second rule: replace $n$ by $m - 1$ and use $n^2 + 2n = (n + 1)^2 - 1$.\nThe theta constant satisfies $\\theta(-1/\\tau) = \\sqrt{-i\\tau}\\, \\theta(\\tau)$, by the Poisson summation formula.\nExample: $\\theta(\\tau)^4 = \\sum_{n \\ge 0} r_4(n) e^{\\pi i n \\tau}$, where $r_4(n)$ counts the ways to write $n$ as a sum of four squares.\nNon-example: for real $\\tau$ the series diverges, because every term has absolute value $1$ when $z$ is also real; this is why $\\tau$ must lie in $\\mathbb{H}$.\nOther authors use the nome $q = e^{\\pi i \\tau}$ and the four functions $\\vartheta_1, \\ldots, \\vartheta_4$; the function above is then $\\vartheta_3$.\n",
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