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  "body": "\n## Statement\n\nLet $f$ be a continuous function on the closed half-plane $\\operatorname{Re} z \\ge 0$ that is [[holomorphic|0DG0Z72CQD]] on the open half-plane $\\operatorname{Re} z \u003e 0$.\nAssume that:\n\n1. there are constants $C, \\tau \u003e 0$ such that $\\abs{f(z)} \\le C e^{\\tau \\abs{z}}$ for $\\operatorname{Re} z \\ge 0$;\n2. there are constants $C' \u003e 0$ and $c \u003c \\pi$ such that $\\abs{f(iy)} \\le C' e^{c \\abs{y}}$ for all real $y$;\n3. $f(n) = 0$ for every integer $n \\ge 0$.\n\nThen $f$ is identically zero.\n\n## Remarks\n\nCondition 1 says that $f$ has [[exponential type|9P8D8YAHQS]] in the half-plane.\nThe proof compares $f$ with $\\sin(\\pi z)$, which vanishes at the integers, and uses the Phragmén–Lindelöf principle to bound $f(z)/\\sin(\\pi z)$.\nThe bound $c \u003c \\pi$ in condition 2 is sharp: $f(z) = \\sin(\\pi z)$ satisfies conditions 1 and 3, and $\\abs{\\sin(\\pi i y)} = \\abs{\\sinh(\\pi y)}$, which grows like $e^{\\pi \\abs{y}}/2$.\nApplied to a difference, the theorem shows that two functions of this kind that agree at all nonnegative integers are equal.\n\n## Explanations\n\n### L1 human\n\nA function on the integers can be extended to the complex plane in many ways, for example by adding $\\sin(\\pi z)$.\nCarlson's theorem says that the extension becomes unique once its growth is limited.\nThe function may grow exponentially, but on the imaginary axis more slowly than $\\sin(\\pi z)$.\n\n### L2 human\n\nThe theorem justifies many formulas that are first proved for integer values and then used for complex values.\nFor example, two expressions in $z$ that agree for $z = 0, 1, 2, \\dots$ and satisfy the growth conditions agree everywhere.\nIt is used for interpolation of sequences and for questions of uniqueness of Newton series.\nLike the [[Paley–Wiener theorem|THY12GPZ8C]], it shows how the exponential type of an entire function controls its behaviour.\n",
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