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  "body": "\n## Definition\n\nLet $\\Gamma$ be the [[gamma function|0H5SYBSED9]] and $\\ln$ the [[natural logarithm|04DXAFN2ES]].\nFor a real number $x \u003e 0$, the *K-function* is\n$$K(x) = (2\\pi)^{-(x-1)/2} \\exp\\left(\\binom{x}{2} + \\int_0^{x-1} \\ln \\Gamma(t + 1)\\, dt\\right),$$\nwhere $\\binom{x}{2} = \\frac{x(x-1)}{2}$.\n\n## Remarks\n\nRaabe's formula $\\int_a^{a+1} \\ln \\Gamma(u)\\, du = \\frac12 \\ln 2\\pi + a \\ln a - a$ for $a \u003e 0$ gives\n$$K(x + 1) = x^x K(x) \\qquad (x \u003e 0).$$\n\nExample: $\\int_0^1 \\ln \\Gamma(t+1)\\, dt = \\int_0^1 (\\ln t + \\ln \\Gamma(t))\\, dt = -1 + \\frac12 \\ln 2\\pi$.\nHence $K(2) = (2\\pi)^{-1/2} \\exp(1 - 1 + \\frac12 \\ln 2\\pi) = 1$.\nThe recurrence then gives $K(3) = 2^2 = 4$, $K(4) = 4 \\cdot 3^3 = 108$, and in general $K(n + 1) = \\prod_{k=1}^n k^k$.\n\nNon-example: the function $x \\mapsto x^x$ does not satisfy the recurrence; for instance $3^3 = 27 \\ne 2^2 \\cdot 2^2 = 16$.\n\nThe formula above is stated for $x \u003e 0$, where $t + 1 \u003e 0$ on the path of integration.\nThe same formula, with the principal branch of $\\ln \\Gamma$, defines $K$ as a holomorphic function on $\\C \\setminus (-\\infty, 0]$.\n",
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