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  "body": "\n## Summary\n\nA function $f$ on $(0, \\infty)$ is Mellin convergent at a complex number $s$ when $x^{s-1} f(x)$ is integrable on $(0, \\infty)$.\nThis is exactly the condition under which the [[Mellin transform|N5PVGRJE1T]] $\\mathcal{M}f(s)$ is defined as an absolutely convergent integral.\nIt depends only on $\\operatorname{Re} s$, and the set of real parts where it holds is an interval.\nIts interior gives the vertical strip on which the Mellin transform is holomorphic.\n\n## Explanations\n\n### L1 human\n\nThe power $x^{s-1}$ is large near $0$ when $\\operatorname{Re} s \u003c 1$ and large near $\\infty$ when $\\operatorname{Re} s \u003e 1$.\nSo Mellin convergence is a balance: $f$ must be small enough near $0$ to tame one end and small enough near $\\infty$ to tame the other.\nFor example, $f(x) = 1/(1 + x)$ behaves like $1$ near $0$ and like $1/x$ near $\\infty$.\nIt is Mellin convergent exactly when $0 \u003c \\operatorname{Re} s \u003c 1$.\n",
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