{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "X6GEZGZ8HQ"
    }
  ],
  "body": "\n## Summary\n\nA Fourier–Bessel series expands a function on an interval $[0, b]$ in the functions $J_\\nu(j_{\\nu,k} x / b)$.\nHere $J_\\nu$ is the Bessel function of the first kind and $j_{\\nu,k}$ are its positive zeros.\nThese functions are orthogonal for the weight $x$, so the series is a [[generalized Fourier series|SNJC87TBHD]].\nThey are the eigenfunctions of a singular [[Sturm–Liouville problem|7WV6B2502K]] coming from [[Bessel's equation|EAV97QH9DF]].\nFourier–Bessel series solve problems with circular or cylindrical symmetry, such as the vibrations of a circular drum or heat flow in a cylinder.\n\n## Explanations\n\n### L1 human\n\nA vibrating string has sine waves as its basic shapes, and any shape of the string is a sum of them.\nA circular drum skin has different basic shapes: in the radial direction they are Bessel functions, scaled so that they vanish at the rim.\nA Fourier–Bessel series writes a radial profile on the drum as a sum of these basic shapes.\nFor example, the constant profile $1$ on $[0, 1)$ is $\\sum_k \\frac{2}{j_k J_1(j_k)} J_0(j_k x)$, where $j_k$ are the zeros of $J_0$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "42C10",
      "33C10"
    ],
    "name": "Fourier–Bessel series",
    "same_as": {
      "wikidata": "Q5475809",
      "wikipedia": "Fourier–Bessel series"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Fourier%E2%80%93Bessel_series"
      }
    ],
    "tag": "34VJ4T7ZSB"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/34VJ4T7ZSB/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/34VJ4T7ZSB.md",
  "msc": [
    "42C10",
    "33C10"
  ],
  "name": "Fourier–Bessel series",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/34V/34VJ4T7ZSB.md",
  "status": null,
  "tag": "34VJ4T7ZSB",
  "uri": "m2://34VJ4T7ZSB",
  "url": "https://tamnd.github.io/math2/wiki/fourier-bessel-series/",
  "writing": {
    "tag": "34VJ4T7ZSB",
    "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
      }
    ]
  }
}