{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $f \\colon [0, \\infty) \\to \\C$ be a Lebesgue [[integrable function|X7GTF3D98F]].\nThe *cosine transform* of $f$ is the function on $[0, \\infty)$ given by\n$$\\mathcal{F}_c f(\\xi) = \\int_0^\\infty f(x) \\cos(2\\pi x \\xi)\\, dx.$$\nThe *sine transform* of $f$ is the function on $[0, \\infty)$ given by\n$$\\mathcal{F}_s f(\\xi) = \\int_0^\\infty f(x) \\sin(2\\pi x \\xi)\\, dx.$$\n\n## Remarks\n\nBoth integrals converge absolutely because $\\abs{\\cos}$ and $\\abs{\\sin}$ are at most $1$.\nExample: for $f(x) = e^{-x}$, two integrations by parts give $\\mathcal{F}_c f(\\xi) = \\frac{1}{1 + a^2}$ and $\\mathcal{F}_s f(\\xi) = \\frac{a}{1 + a^2}$ with $a = 2\\pi \\xi$.\nNon-example: the constant function $f = 1$ is not integrable on $[0, \\infty)$, and $\\int_0^\\infty \\cos(2\\pi x \\xi)\\, dx$ diverges for every $\\xi$.\n\nLet $f_e$ and $f_o$ be the even and the odd extensions of $f$ to $\\R$, and let $\\hat g$ denote the [[Fourier transform|DPVM34ADZV]] $\\int_\\R g(x) e^{-2\\pi i x \\xi}\\, dx$.\nThen $\\hat f_e(\\xi) = 2 \\mathcal{F}_c f(\\xi)$ and $\\hat f_o(\\xi) = -2i\\, \\mathcal{F}_s f(\\xi)$.\nConsequently, if $f$ is continuous and $\\mathcal{F}_c f$ is integrable, then $f(x) = 4 \\int_0^\\infty \\mathcal{F}_c f(\\xi) \\cos(2\\pi x \\xi)\\, d\\xi$ for $x \\ge 0$.\n\nOther conventions use $\\cos(x \\xi)$ and $\\sin(x \\xi)$, often with a factor $\\sqrt{2/\\pi}$ that makes the transforms their own inverses on $L^2(0, \\infty)$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "C07WNRSW52",
    "kind": "definition",
    "msc": [
      "42A38",
      "44A15"
    ],
    "name": "Sine and cosine transforms",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Sine_and_cosine_transforms"
      }
    ],
    "tag": "Q4HR7AF2R9",
    "uses": [
      "X7GTF3D98F",
      "DPVM34ADZV"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/Q4HR7AF2R9/",
  "links": [
    {
      "field": "defines",
      "target": "C07WNRSW52"
    },
    {
      "field": "uses",
      "target": "X7GTF3D98F"
    },
    {
      "field": "uses",
      "target": "DPVM34ADZV"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/Q4HR7AF2R9.md",
  "msc": [
    "42A38",
    "44A15"
  ],
  "name": "Sine and cosine transforms",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Q4H/Q4HR7AF2R9.md",
  "status": null,
  "tag": "Q4HR7AF2R9",
  "uri": "m2://Q4HR7AF2R9",
  "url": "https://tamnd.github.io/math2/wiki/sine-and-cosine-transforms-definition/",
  "writing": {
    "tag": "Q4HR7AF2R9",
    "kind": "definition",
    "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
      }
    ]
  }
}