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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 $z \\in \\C$ with $2z \\notin \\set{0, -1, -2, \\ldots}$,\n$$\\Gamma(z) \\, \\Gamma\\!\\left(z + \\tfrac12\\right) = 2^{1 - 2z} \\sqrt{\\pi} \\; \\Gamma(2z),$$\nwhere $2^{1-2z} = e^{(1-2z)\\ln 2}$.\n\n## Remarks\n\nOne proof applies the [[Bohr–Mollerup theorem|QFF87AXWN2]] to $f(x) = 2^{x-1}\\Gamma(\\tfrac{x}{2})\\Gamma(\\tfrac{x+1}{2})/\\sqrt{\\pi}$ for $x \u003e 0$: it satisfies $f(1) = 1$, $f(x+1) = x f(x)$, and $\\ln f$ is convex, so $f = \\Gamma$; then put $x = 2z$.\nAnother proof uses the [[beta function|QHNWVSC6AQ]]: the substitution $t \\mapsto 4t(1-t)$ gives $B(z, z) = 2^{1-2z} B(z, \\tfrac12)$.\nThe identity theorem extends the formula from $(0, \\infty)$ to all allowed complex $z$.\nCheck at $z = 1$: the left side is $\\Gamma(1)\\Gamma(\\tfrac32) = \\tfrac{\\sqrt{\\pi}}{2}$, and the right side is $2^{-1}\\sqrt{\\pi}\\,\\Gamma(2) = \\tfrac{\\sqrt{\\pi}}{2}$.\nIt is the case $m = 2$ of the [[Gauss multiplication formula|Y4759V8CY6]].\nIn Mathlib the statement is `Complex.Gamma_mul_Gamma_add_half`.\n\n## Explanations\n\n### L1 human\n\nThe duplication formula expresses $\\Gamma$ at a doubled argument through two values of $\\Gamma$ at half that size, half a unit apart.\nFor whole numbers it is a statement about factorials: $(2n)! = \\frac{4^n}{\\sqrt{\\pi}}\\, n! \\, \\Gamma(n + \\tfrac12)$.\n\n### L2 human\n\nThe formula is how one computes $\\Gamma$ at half-integers, giving $\\Gamma(n + \\tfrac12) = \\frac{(2n)!}{4^n n!}\\sqrt{\\pi}$.\nIt appears in the functional equation of the Riemann zeta function, where it converts between the symmetric form with $\\Gamma(s/2)$ and the form with $\\Gamma(s)$ and a cosine.\nIt also gives the volume of the unit ball in odd dimensions in closed form.\n",
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