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  "body": "\n## Statement\n\nLet $0 \u003c r \\le \\infty$ and $\\alpha \u003c \\beta$ be real numbers, and let\n$$S = \\set{z \\in \\C : 0 \u003c \\abs{z} \u003c r, \\ \\alpha \u003c \\arg z \u003c \\beta}$$\nbe an open sector with vertex $0$, of any finite opening $\\beta - \\alpha$.\nLet $\\sum_{n \\ge 0} a_n z^n$ be a [[formal power series|072W67KFYK]] with complex coefficients.\nThen there is a [[holomorphic function|0DG0Z72CQD]] $f$ on $S$ with the following property.\nOn every closed subsector $\\set{z : 0 \u003c \\abs{z} \\le \\rho, \\ \\alpha' \\le \\arg z \\le \\beta'}$ with $\\rho \u003c r$ and $\\alpha \u003c \\alpha' \u003c \\beta' \u003c \\beta$,\n$$f(z) \\sim \\sum_{n \\ge 0} a_n z^n \\quad \\text{as } z \\to 0,$$\nin the sense of [[asymptotic expansions|B53THC8F79]] in powers of $z$.\n\n## Remarks\n\nIdea of the proof: after a rotation, one picks $\\gamma \u003e 0$ so small that $\\operatorname{Re}(z^{-\\gamma}) \u003e 0$ on $S$ for a branch of $z^{-\\gamma}$.\nThen one sets $f(z) = \\sum_{n} a_n z^n \\left(1 - e^{-b_n z^{-\\gamma}}\\right)$ with positive numbers $b_n$ that tend to $0$ fast.\nEach factor $1 - e^{-b_n z^{-\\gamma}}$ is bounded on $S$ and at most $b_n \\abs{z}^{-\\gamma}$ in absolute value, which makes the series converge and keeps the tail small near $0$.\nThe function $f$ is never unique: adding $e^{-z^{-\\gamma}}$ for a suitable $\\gamma$ does not change the expansion.\nThe real analogue, with smooth functions on an interval, is [[Borel's lemma|KZHNA19FSS]].\nIf the opening of $S$ is larger than $2\\pi$, $S$ is understood on the Riemann surface of the logarithm, and the theorem still holds.\n\n## Explanations\n\n### L1 human\n\nAny formal power series, even one that diverges everywhere, is the asymptotic expansion of some holomorphic function on a sector.\nSo asymptotic expansions on sectors carry no growth condition on their coefficients.\nThis is the complex counterpart of the fact that every sequence of numbers is the sequence of derivatives at $0$ of a smooth function.\n\n### L2 human\n\nThe theorem shows that the map from functions with an asymptotic expansion on a sector to formal power series is onto.\nIts kernel consists of functions that are flat at $0$ in the sector, such as $e^{-1/z}$ on $\\abs{\\arg z} \u003c \\pi/2$.\nTo recover uniqueness one needs stronger conditions, such as Gevrey estimates on a sector of opening larger than $\\pi/k$.\nThis is the starting point of $k$-summability and of [[Borel summation|8DJPSYSHGJ]] for divergent solutions of differential equations.\n",
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