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  "body": "\n## Summary\n\nThe Mellin transform sends a function $f$ on $(0, \\infty)$ to the function $\\mathcal{M}f(s) = \\int_0^\\infty x^{s-1} f(x) \\, dx$ of a complex variable $s$.\nIt is the integral transform adapted to the multiplicative group $(0, \\infty)$, in the way the [[Fourier transform|DPVM34ADZV]] is adapted to the additive group $\\R$.\nIt usually converges on a vertical strip of the complex plane, and it turns dilations $x \\mapsto ax$ into multiplication by $a^{-s}$.\nIt links power series and Dirichlet series with the [[gamma function|0H5SYBSED9]] and the [[Riemann zeta function|1XWBY4N0VT]], and it is a main tool for asymptotic expansions.\n\n## Explanations\n\n### L1 human\n\nThe Mellin transform measures how a function on the positive reals behaves against the powers $x^{s-1}$.\nChanging the variable to $x = e^{u}$ turns it into a Laplace or Fourier transform in $u$, so it is the Fourier transform of a function viewed on a logarithmic scale.\nIts basic example is $f(x) = e^{-x}$, whose Mellin transform is $\\Gamma(s)$ for $\\operatorname{Re} s \u003e 0$.\nScaling the input, $x \\mapsto 2x$, only multiplies the transform by $2^{-s}$.\n",
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