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  "body": "\n## Definition\n\nLet $\\gamma$ be the [[Euler–Mascheroni constant|00C5P9JZBK]].\nThe *Barnes G-function* is the [[entire function|0G4F0XB9X2]] $G$ given for $z \\in \\C$ by\n$$G(1 + z) = (2\\pi)^{z/2} \\exp\\left(-\\frac{z + (1+\\gamma) z^2}{2}\\right) \\prod_{k=1}^\\infty \\left(1 + \\frac{z}{k}\\right)^k \\exp\\left(\\frac{z^2}{2k} - z\\right).$$\n\n## Remarks\n\nThe product converges uniformly on compact sets.\nThe function $G$ satisfies $G(1) = 1$ and\n$$G(z + 1) = \\Gamma(z)\\, G(z)$$\nfor all $z$, where $\\Gamma$ is the [[gamma function|0H5SYBSED9]].\nAt a pole of $\\Gamma$ the right side is read as a limit, since $G$ has a zero there.\n\nExample: the recurrence gives $G(2) = \\Gamma(1) G(1) = 1$, $G(3) = \\Gamma(2) G(2) = 1$, $G(4) = \\Gamma(3) G(3) = 2$ and $G(5) = \\Gamma(4) G(4) = 12$.\nIn general $G(n) = \\prod_{k=0}^{n-2} k!$ for integers $n \\ge 2$.\n\nNon-example: the function $F(z) = G(z) \\cdot (1 + \\sin^2 \\pi z)$ also satisfies $F(1) = 1$ and $F(z+1) = \\Gamma(z) F(z)$, but it is not $G$.\nSo the recurrence alone does not determine $G$; one needs an extra condition, for example that $\\frac{d^3}{dx^3} \\log G(x) \\ge 0$ for $x \u003e 0$.\n\nFrom the product, $G$ has zeros exactly at $z = -n$ for integers $n \\ge 0$, of order $n + 1$.\n",
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