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  "body": "\n## Summary\n\nThe poles of the [[Mellin transform|N5PVGRJE1T]] of a function $f$ control the behaviour of $f(x)$ as $x \\to 0^+$ and as $x \\to \\infty$.\nA pole at $s = \\rho$ left of the [[fundamental strip|1JAA21FHHQ]] gives a term $x^{-\\rho}$ in the [[asymptotic expansion|BJQTEPEZ9G]] of $f$ near $0$.\nA multiple pole adds powers of $\\log x$ to this term.\nPoles to the right of the strip give the expansion of $f$ near $\\infty$ in the same way.\nThe method turns asymptotic questions about sums and integrals into the study of a [[meromorphic function|0ZZ7ZYDR8W]].\nIt is a basic tool in analytic number theory and in the analysis of algorithms.\n\n## Explanations\n\n### L1 human\n\nThe Mellin transform sends a function of $x \u003e 0$ to a function of a complex variable $s$.\nEach singular point of the transform is a fingerprint of one term of $f$: a simple pole at $s = \\rho$ with residue $c$ corresponds to a term $c\\, x^{-\\rho}$.\nTo find these terms one moves the line of the inverse Mellin integral across the poles and collects the [[residues|0PHN07KVQT]].\nFor example, the Mellin transform of $e^{-x}$ is the [[gamma function|0H5SYBSED9]] $\\Gamma(s)$.\nIts poles at $s = -n$ have residues $(-1)^n / n!$, and they give back the Taylor series $e^{-x} = \\sum_{n \\ge 0} (-1)^n x^n / n!$.\n",
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  "name": "Asymptotic expansion from poles of a Mellin transform",
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