{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nFor $y \\in \\R$ and $x \u003e 0$ let $K_{iy}(x) = \\int_0^\\infty e^{-x\\cosh t} \\cos(yt)\\, dt$ be the [[modified Bessel function of the second kind|2ET068BZZH]] of order $iy$.\nLet $f \\colon (0, \\infty) \\to \\C$ be measurable with $\\int_0^\\infty \\abs{f(x)}\\, K_0(x)\\, dx \u003c \\infty$.\nThe *Kontorovich–Lebedev transform* of $f$ is the function $g \\colon \\R \\to \\C$ given by the [[Lebesgue integral|DWJF7ABPR1]]\n$$g(y) = \\int_0^\\infty f(x)\\, K_{iy}(x)\\, dx.$$\n\n## Remarks\n\nThe integral converges absolutely because $\\abs{K_{iy}(x)} \\le \\int_0^\\infty e^{-x\\cosh t}\\, dt = K_0(x)$.\nThe function $K_{iy}(x)$ is real and even in $y$, so $g$ is even and is usually considered on $(0, \\infty)$.\nUnder suitable conditions on $f$, the inversion formula is\n$$f(x) = \\frac{2}{\\pi^2 x} \\int_0^\\infty g(y)\\, K_{iy}(x)\\, \\sinh(\\pi y)\\, y\\, dy.$$\n\nExample: $f(x) = 1$.\nSince $K_0(x)$ behaves like $-\\log x$ near $0$ and like $\\sqrt{\\pi/(2x)}\\, e^{-x}$ at infinity, $K_0$ is integrable and the condition holds.\nThe formula $\\int_0^\\infty K_\\mu(x)\\, dx = \\frac{\\pi}{2\\cos(\\pi\\mu/2)}$ for $\\abs{\\operatorname{Re} \\mu} \u003c 1$, with $\\mu = iy$, gives $g(y) = \\frac{\\pi}{2\\cosh(\\pi y / 2)}$.\n\nNon-example: $f(x) = e^x$.\nFor fixed $y$, $K_{iy}(x) \\sim \\sqrt{\\pi/(2x)}\\, e^{-x}$ as $x \\to \\infty$, so $f(x) K_{iy}(x) \\sim \\sqrt{\\pi/(2x)}$ and the integral diverges at infinity.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "XFJDYMB91J",
    "kind": "definition",
    "msc": [
      "44A20",
      "33C10"
    ],
    "name": "Kontorovich–Lebedev transform",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Kontorovich%E2%80%93Lebedev_transform"
      }
    ],
    "tag": "ZNC09JXRC5",
    "uses": [
      "2ET068BZZH",
      "DWJF7ABPR1"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/ZNC09JXRC5/",
  "links": [
    {
      "field": "defines",
      "target": "XFJDYMB91J"
    },
    {
      "field": "uses",
      "target": "2ET068BZZH"
    },
    {
      "field": "uses",
      "target": "DWJF7ABPR1"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/ZNC09JXRC5.md",
  "msc": [
    "44A20",
    "33C10"
  ],
  "name": "Kontorovich–Lebedev transform",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZNC/ZNC09JXRC5.md",
  "status": null,
  "tag": "ZNC09JXRC5",
  "uri": "m2://ZNC09JXRC5",
  "url": "https://tamnd.github.io/math2/wiki/kontorovich-lebedev-transform-definition/",
  "writing": {
    "tag": "ZNC09JXRC5",
    "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
      }
    ]
  }
}