{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "ZNC09JXRC5"
    }
  ],
  "body": "\n## Summary\n\nThe Kontorovich–Lebedev transform integrates a function $f$ on $(0, \\infty)$ against the [[modified Bessel function|2ET068BZZH]] $K_{iy}(x)$ of imaginary order $iy$.\nUnlike most transforms, the transform variable $y$ enters through the order of the Bessel function, not through its argument.\nIt has an explicit inversion formula with the weight $y \\sinh(\\pi y)$.\nIt is used to solve boundary value problems in wedge-shaped and conical regions, where the separation of variables leads to Bessel functions of imaginary order.\n\n## Explanations\n\n### L1 human\n\nMany transforms take a family of functions, such as waves of every frequency, and measure how much of each one a given function contains.\nHere the family is $x \\mapsto K_{iy}(x)$, indexed by a real number $y$.\nThese functions oscillate faster and faster near $x = 0$ as $y$ grows and decay exponentially for large $x$.\nThe transform records the overlap of $f$ with each of them.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "44A20",
      "33C10"
    ],
    "name": "Kontorovich–Lebedev transform",
    "same_as": {
      "wikidata": "Q6430373",
      "wikipedia": "Kontorovich–Lebedev transform"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Kontorovich%E2%80%93Lebedev_transform"
      }
    ],
    "specializes": [
      "YCK709QB06"
    ],
    "tag": "XFJDYMB91J"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/XFJDYMB91J/",
  "links": [
    {
      "field": "specializes",
      "target": "YCK709QB06"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/XFJDYMB91J.md",
  "msc": [
    "44A20",
    "33C10"
  ],
  "name": "Kontorovich–Lebedev transform",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/XFJ/XFJDYMB91J.md",
  "status": null,
  "tag": "XFJDYMB91J",
  "uri": "m2://XFJDYMB91J",
  "url": "https://tamnd.github.io/math2/wiki/kontorovich-lebedev-transform/",
  "writing": {
    "tag": "XFJDYMB91J",
    "kind": "concept",
    "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
      }
    ]
  }
}