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  "body": "\n## Statement\n\nLet $H = \\set{z \\in \\C : \\operatorname{Re} z \u003e 0}$ and let $F \\colon H \\to \\C$ be a [[holomorphic function|0NAHXR1FRH]] such that:\n\n1. $F(z + 1) = z F(z)$ for every $z \\in H$;\n2. $F$ is bounded on the strip $\\set{z \\in \\C : 1 \\le \\operatorname{Re} z \\le 2}$.\n\nThen $F(z) = F(1)\\,\\Gamma(z)$ for every $z \\in H$, where $\\Gamma$ is the [[gamma function|0V3FZMVB4K]].\n\n## Remarks\n\nThe gamma function satisfies both conditions: the functional equation, and $\\abs{\\Gamma(z)} \\le \\Gamma(\\operatorname{Re} z) \\le 1$ on the strip, by Euler's integral.\nFor the proof, put $v = F - F(1)\\Gamma$; it satisfies condition 1 and $v(1) = 0$.\nThe equation $v(z) = v(z + 1)/z$ extends $v$ to a meromorphic function on $\\C$, and the possible poles at $0, -1, -2, \\ldots$ are removable because $v(1) = 0$, so $v$ is entire and bounded on $0 \\le \\operatorname{Re} z \\le 1$.\nThen $u(z) = v(z) v(1 - z)$ is entire, satisfies $u(z + 1) = -u(z)$, and is bounded on that strip, so it is bounded on $\\C$ and constant by [[Liouville's theorem|0EHKVRF4J6]]; since $u(1) = 0$, $u = 0$ and hence $v = 0$.\nCondition 2 cannot be dropped: $F(z) = \\Gamma(z)\\big(1 + \\sin(2\\pi z)\\big)$ satisfies condition 1 but is not a multiple of $\\Gamma$, and it is unbounded on the strip.\n\n## Explanations\n\n### L1 human\n\nWielandt's theorem says that the gamma function is the only holomorphic solution of $F(z + 1) = z F(z)$ on the right half-plane that stays bounded on a vertical strip, up to a constant factor.\nIt is a complex analogue of the Bohr–Mollerup theorem, with boundedness in place of log-convexity.\n\n### L2 human\n\nThe theorem is a quick tool for proving identities: to show that a function $F$ equals $\\Gamma$, check the functional equation, boundedness on the strip $1 \\le \\operatorname{Re} z \\le 2$, and $F(1) = 1$.\nFor example, the [[Euler–Gauss limit formula|SKRP9NDXR4]] and the [[Legendre duplication formula|CND4E1PY9V]] can be proved this way.\nCompare with the [[Bohr–Mollerup theorem|QFF87AXWN2]], which works on the real line.\n",
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