{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $f \\colon (0, \\infty) \\to \\C$ be measurable, let $a \\in \\R$ with $a \\ne 0$, and let $g(x) = f(x^{a})$ for $x \u003e 0$.\nLet $s \\in \\C$.\nThen $g$ is [[Mellin convergent|SGZZ4Y0D36]] at $s$ if and only if $f$ is Mellin convergent at $s/a$.\nIn that case\n$$\\mathcal{M}g(s) = \\frac{1}{\\abs{a}} \\, \\mathcal{M}f\\left(\\frac{s}{a}\\right),$$\nwhere $\\mathcal{M}$ is the [[Mellin transform|N5PVGRJE1T]].\n\n## Remarks\n\nFor $a \u003e 0$, substitute $u = x^{a}$, so $x = u^{1/a}$ and $dx = \\frac{1}{a} u^{1/a - 1} \\, du$:\n$$\\int_0^\\infty x^{s-1} f(x^{a}) \\, dx = \\frac{1}{a} \\int_0^\\infty u^{(s-1)/a} \\, u^{1/a - 1} f(u) \\, du = \\frac{1}{a} \\int_0^\\infty u^{s/a - 1} f(u) \\, du.$$\nFor $a \u003c 0$ the substitution reverses the orientation of $(0, \\infty)$, which turns the factor $1/a$ into $1/\\abs{a}$.\nThe case $a = -1$ is the rule for the inversion $x \\mapsto 1/x$.\nExample: $f(x) = e^{-x}$ and $a = 2$ give $\\int_0^\\infty x^{s-1} e^{-x^2} \\, dx = \\frac{1}{2} \\Gamma(s/2)$ for $\\operatorname{Re} s \u003e 0$; at $s = 1$ this is $\\int_0^\\infty e^{-x^2} \\, dx = \\frac{1}{2}\\Gamma(1/2) = \\frac{\\sqrt{\\pi}}{2}$.\nThe hypothesis $a \\ne 0$ is needed: for $a = 0$, $g$ is the constant $f(1)$, which is Mellin convergent at no $s$ unless $f(1) = 0$.\n\n## Explanations\n\n### L1 human\n\nReplacing $x$ by a power $x^{a}$ rescales the Mellin variable: $s$ becomes $s/a$, and a factor $1/\\abs{a}$ appears.\nIt is the logarithmic version of the rule that stretching a function by a factor rescales its Fourier transform.\n\n### L2 human\n\nThis rule computes many transforms from a few, for instance Gaussian integrals from the gamma function.\nIt is used in the Riemann proof of the functional equation, where the theta function $\\sum_n e^{-\\pi n^2 x}$ meets $\\Gamma(s/2)$ rather than $\\Gamma(s)$.\nCombined with the rules for dilations and for multiplication by powers, it describes the Mellin transform of $x^{b} f(c x^{a})$ for all $a \\ne 0$, $b \\in \\C$ and $c \u003e 0$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "N5PVGRJE1T"
    ],
    "kind": "statement",
    "msc": [
      "44A15"
    ],
    "name": "Mellin transform under a power substitution",
    "role": "proposition",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Mellin_transform"
      }
    ],
    "tag": "CG7BFJ04YE",
    "uses": [
      "SGZZ4Y0D36",
      "N5PVGRJE1T"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/CG7BFJ04YE/",
  "links": [
    {
      "field": "about",
      "target": "N5PVGRJE1T"
    },
    {
      "field": "uses",
      "target": "SGZZ4Y0D36"
    },
    {
      "field": "uses",
      "target": "N5PVGRJE1T"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/CG7BFJ04YE.md",
  "msc": [
    "44A15"
  ],
  "name": "Mellin transform under a power substitution",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CG7/CG7BFJ04YE.md",
  "status": null,
  "tag": "CG7BFJ04YE",
  "uri": "m2://CG7BFJ04YE",
  "url": "https://tamnd.github.io/math2/wiki/mellin-transform-under-a-power-substitution/",
  "writing": {
    "tag": "CG7BFJ04YE",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}