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  "body": "\n## Statement\n\nLet $f \\colon (0, \\infty) \\to (0, \\infty)$ be a function such that:\n\n1. $f(1) = 1$;\n2. $f(x + 1) = x f(x)$ for every $x \u003e 0$;\n3. $\\ln f$ is [[convex|0VV0HSW1ZS]] on $(0, \\infty)$.\n\nThen $f(x) = \\Gamma(x)$ for every $x \u003e 0$, where $\\Gamma$ is the [[gamma function|0V3FZMVB4K]].\n\n## Remarks\n\nThe gamma function satisfies the three conditions; its [[log-convexity|8FRWATNNMX]] follows from Hölder's inequality applied to $\\Gamma(x) = \\int_0^\\infty t^{x-1} e^{-t}\\, dt$.\nFor uniqueness, conditions 1 and 2 give $f(n + 1) = n!$, so the values at integers are fixed.\nFix $0 \u003c x \\le 1$ and an integer $n \\ge 2$.\nConvexity of $\\ln f$ compares its slope on $[n, n + x]$ with its slopes on $[n - 1, n]$ and $[n, n + 1]$.\nThis squeezes $f(n + x)$ between two explicit bounds, and condition 2 turns them into bounds for $f(x)$.\nLetting $n \\to \\infty$ yields $f(x) = \\lim_{n \\to \\infty} \\frac{n!\\, n^x}{x (x + 1) \\cdots (x + n)}$, the same limit for every such $f$.\nWithout condition 3 uniqueness fails: $f(x) = \\Gamma(x)\\,(1 + \\frac12 \\sin(2\\pi x))$ satisfies conditions 1 and 2, takes positive values, and differs from $\\Gamma$ at $x = \\frac14$.\n\n## Explanations\n\n### L1 human\n\nMany functions interpolate the factorials, taking the value $(n-1)!$ at each positive integer $n$.\nThe theorem says that the gamma function is the only one that obeys the rule $f(x + 1) = x f(x)$ and whose logarithm bends upward everywhere.\nSo the gamma function is not an arbitrary choice: it is the natural extension of the factorial.\n\n### L2 human\n\nThe theorem is a convenient way to prove identities for $\\Gamma$: show that the other side satisfies the three conditions.\nFor example, it gives the formula $B(x, y) = \\Gamma(x)\\Gamma(y)/\\Gamma(x + y)$ for the [[beta function|QHNWVSC6AQ]], by applying the theorem to $x \\mapsto B(x, y)\\,\\Gamma(x + y)/\\Gamma(y)$ for fixed $y$.\nIt also gives the Gauss product formula for $\\Gamma$ and the Legendre duplication formula.\nIt links the theory of [[convex functions|0VV0HSW1ZS]] with special functions.\n",
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