{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "7KA78K00QJ"
    },
    {
      "field": "uses",
      "target": "D3JWE193EM"
    },
    {
      "field": "uses",
      "target": "JEXQ0MFJ0Q"
    },
    {
      "field": "uses",
      "target": "ZNC09JXRC5"
    },
    {
      "field": "defines",
      "target": "VSD1D6ASVH"
    }
  ],
  "body": "\n## Summary\n\nThe modified Bessel functions $I_\\nu$ and $K_\\nu$ solve the modified Bessel equation $z^2 y'' + z y' - (z^2 + \\nu^2) y = 0$.\nThey are the [[Bessel functions|CZCMRDWDYW]] evaluated on the imaginary axis, made real for real arguments.\nOn the positive real axis $I_\\nu$ grows like $e^x$ and $K_\\nu$ decays like $e^{-x}$.\nThey appear in heat flow, diffusion and electrostatics in cylindrical geometry.\n\n## Explanations\n\n### L1 human\n\nBessel's equation behaves like $y'' + y = 0$ for large $x$, so its solutions oscillate.\nChanging one sign gives an equation that behaves like $y'' - y = 0$, whose solutions grow or decay like $e^{x}$ and $e^{-x}$.\nThe modified Bessel functions are the solutions of this changed equation.\n$I_\\nu$ is the growing one, finite at the origin, and $K_\\nu$ is the decaying one, infinite at the origin.\nFor example, $I_{1/2}(x) = \\sqrt{\\frac{2}{\\pi x}} \\sinh x$ for $x \u003e 0$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "33C10",
      "34B30"
    ],
    "name": "Modified Bessel functions",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Bessel_function"
      }
    ],
    "tag": "2ET068BZZH"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/2ET068BZZH/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/2ET068BZZH.md",
  "msc": [
    "33C10",
    "34B30"
  ],
  "name": "Modified Bessel functions",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/2ET/2ET068BZZH.md",
  "status": null,
  "tag": "2ET068BZZH",
  "uri": "m2://2ET068BZZH",
  "url": "https://tamnd.github.io/math2/wiki/modified-bessel-functions/",
  "writing": {
    "tag": "2ET068BZZH",
    "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
      }
    ]
  }
}