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  "body": "\n## Statement\n\nFor every $z \\in \\C$,\n$$\\sin(\\pi z) = \\pi z \\prod_{n=1}^{\\infty} \\left(1 - \\frac{z^2}{n^2}\\right),$$\nwhere $\\sin$ is the [[complex sine|0A7YDKRNR6]].\nThe product converges absolutely for every $z$, and [[uniformly|08FPABDQ4C]] on each compact subset of $\\C$.\n\n## Remarks\n\nThe convergence follows from $\\sum_n \\abs{z}^2 / n^2 \u003c \\infty$.\nBoth sides are [[entire functions|0419794R0T]] with simple zeros exactly at the integers.\nBy the [[Weierstrass factorization theorem|11VC51YHQJ]], their quotient is $e^{g(z)}$ for an entire function $g$.\nOne shows that $g$ is constant, for example by comparing logarithmic derivatives with the partial fraction expansion $\\pi \\cot(\\pi z) = \\frac{1}{z} + \\sum_{n \\ge 1} \\frac{2z}{z^2 - n^2}$; the limit at $z = 0$ then gives $e^{g} = 1$.\nComparing the coefficients of $z^3$ on both sides gives $\\sum_{n \\ge 1} 1/n^2 = \\pi^2/6$, the [[Basel problem|0DMY75D6FV]].\nAt $z = 1/2$ it gives the [[Wallis product|0XCYA4YERS]] $\\frac{2}{\\pi} = \\prod_{n \\ge 1} \\left(1 - \\frac{1}{4n^2}\\right)$.\nIn Mathlib it is `Complex.tendsto_euler_sin_prod`.\n\n## Explanations\n\n### L1 human\n\nA polynomial is determined, up to a constant, by its roots.\nThe sine function behaves like a polynomial of infinite degree with roots at all the integers.\nEuler's formula writes $\\sin(\\pi z)$ as the product of the factors $(1 - z/n)(1 + z/n)$ over all its roots, times $\\pi z$.\n\n### L2 human\n\nThe formula is the standard example of a [[Weierstrass product|ACSF58HYDH]].\nHere no exponential correction factors are needed, because the factors $1 - z/n$ and $1 + z/n$ are paired.\nTaking the logarithmic derivative gives the partial fraction expansion of $\\pi\\cot(\\pi z)$.\nCombined with the product for $1/\\Gamma$, it yields the [[reflection formula|WF7KQAQTS2]] for the gamma function.\n",
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