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  "body": "\n## Statement\n\nLet $0 \u003c T \\le \\infty$, let $\\lambda \u003e -1$, and let $f \\colon (0, T) \\to \\C$ be measurable.\nAssume that:\n\n1. $f$ has the [[asymptotic expansion|B53THC8F79]] $f(t) \\sim \\sum_{n \\ge 0} a_n t^{n + \\lambda}$ as $t \\to 0^+$, with complex $a_n$.\n2. There is a real number $x_0$ such that $\\int_0^T e^{-x_0 t} \\abs{f(t)} \\, dt \u003c \\infty$.\n\nThen the integral $F(x) = \\int_0^T e^{-xt} f(t) \\, dt$ converges absolutely for every $x \\ge x_0$, and\n$$F(x) \\sim \\sum_{n \\ge 0} a_n \\frac{\\Gamma(n + \\lambda + 1)}{x^{n + \\lambda + 1}} \\quad \\text{as } x \\to +\\infty,$$\nwhere $\\Gamma$ is the [[gamma function|0V3FZMVB4K]].\n\n## Remarks\n\nIdea of the proof: for $x$ large, the factor $e^{-xt}$ makes the integral over $[\\delta, T)$ exponentially small in $x$, for any fixed $\\delta \u003e 0$.\nOn $(0, \\delta)$, replace $f$ by its first $N$ terms plus an error bounded by $C t^{N + \\lambda}$.\nEach term gives $\\int_0^\\infty e^{-xt} t^{n + \\lambda} \\, dt = \\Gamma(n + \\lambda + 1) x^{-n-\\lambda-1}$, up to an exponentially small change, and the error contributes $O(x^{-N-\\lambda-1})$.\nExample: $\\int_0^\\infty \\frac{e^{-xt}}{1 + t} \\, dt \\sim \\sum_{n \\ge 0} (-1)^n \\frac{n!}{x^{n+1}}$, a divergent series obtained from $\\frac{1}{1+t} = \\sum (-1)^n t^n$ near $0$.\nThe condition $\\lambda \u003e -1$ is needed: if $\\lambda \\le -1$ and $a_0 \\ne 0$, then $\\abs{f(t)}$ is of order $t^{\\lambda}$ near $0$ and the integral diverges at $t = 0$.\n\n## Explanations\n\n### L1 human\n\nFor a large parameter $x$, the weight $e^{-xt}$ is tiny except for $t$ very close to $0$.\nSo the integral only sees the behaviour of $f$ near $0$.\nWatson's lemma makes this precise: expand $f$ in powers of $t$ near $0$, integrate term by term, and the result is the correct asymptotic expansion of the integral.\n\n### L2 human\n\nThe lemma gives the large-$x$ behaviour of the [[Laplace transform|05XT5AVJBK]] $F(x)$ of $f$ from the small-$t$ behaviour of $f$.\nTogether with a change of variables it yields [[Laplace's method|9G7K8DBT9Y]] and the method of steepest descent for integrals $\\int e^{x \\phi(t)} g(t) \\, dt$.\nThe resulting series often diverge, and [[Borel summation|8DJPSYSHGJ]] can be seen as the inverse process that recovers $F$ from its series.\n",
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