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  "body": "\n## Statement\n\nLet $\\Gamma$ be the [[gamma function|0V3FZMVB4K]], extended to a [[meromorphic function|0NSD1TPHR0]] on $\\C$.\nFor every $z \\in \\C \\setminus \\Z$,\n$$\\Gamma(z)\\,\\Gamma(1 - z) = \\frac{\\pi}{\\sin(\\pi z)},$$\nwhere $\\sin$ is the [[complex sine|0A7YDKRNR6]].\n\n## Remarks\n\nFor $0 \u003c \\operatorname{Re} z \u003c 1$, the product $\\Gamma(z)\\Gamma(1-z)$ equals the beta integral $\\int_0^\\infty \\frac{t^{z-1}}{1+t}\\,dt$.\nThis integral is computed by the residue theorem, or for real $z$ by a substitution, and equals $\\pi / \\sin(\\pi z)$.\nBoth sides are meromorphic on $\\C$, so the identity extends from the strip to all of $\\C \\setminus \\Z$ by the identity theorem.\nAnother proof multiplies the Weierstrass products of $1/\\Gamma(z)$ and $1/\\Gamma(1-z)$ and compares with [[Euler's sine product|DQEHD56649]].\nAt $z = 1/2$ the formula gives $\\Gamma(1/2)^2 = \\pi$, so $\\Gamma(1/2) = \\sqrt{\\pi}$, because $\\Gamma(1/2) \u003e 0$.\nThe integers are excluded: at each $n \\in \\Z$ one factor on the left has a [[pole|0WPQYW8MTE]] and $\\sin(\\pi n) = 0$.\nIn Mathlib it is `Complex.Gamma_mul_Gamma_one_sub`.\n\n## Explanations\n\n### L1 human\n\nThe gamma function at $z$ times the gamma function at $1 - z$ is a simple expression in the sine function.\nThe two points $z$ and $1 - z$ are symmetric about the point $1/2$, so the formula relates the values of $\\Gamma$ on the two sides of $1/2$, hence the name.\nThe formula explains why $\\Gamma(1/2) = \\sqrt{\\pi}$.\n\n### L2 human\n\nThe right side $\\pi/\\sin(\\pi z)$ is never $0$, so the formula shows that $\\Gamma$ has no zeros off the integers.\nSince also $\\Gamma(n) = (n-1)! \\ne 0$ for $n \\ge 1$, the function $1/\\Gamma$ is entire with zeros exactly at $0, -1, -2, \\ldots$.\nTogether with the functional equation $\\Gamma(z+1) = z\\Gamma(z)$ and the duplication formula, it is one of the basic identities of the gamma function.\nIt is used, for example, in the functional equation of the Riemann zeta function, where values of $\\Gamma$ at $s/2$ and $(1-s)/2$ appear.\n",
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