{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $n$ be an integer and let $J_n$ be the [[Bessel function of the first kind|CZCMRDWDYW]] of order $n$.\nThen for every $z \\in \\C$,\n$$J_n(z) = \\frac{1}{\\pi} \\int_0^\\pi \\cos(n\\tau - z \\sin \\tau)\\, d\\tau.$$\n\n## Remarks\n\nThe generating function $\\exp\\left( \\frac{z}{2} (t - t^{-1}) \\right) = \\sum_{k \\in \\Z} J_k(z)\\, t^k$ with $t = e^{i\\tau}$ gives $e^{iz \\sin \\tau} = \\sum_{k} J_k(z)\\, e^{ik\\tau}$, the [[Jacobi–Anger expansion|KFHYQM8T7E]].\nSo $J_n(z)$ is the $n$-th Fourier coefficient $\\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} e^{i(z \\sin \\tau - n\\tau)}\\, d\\tau$.\nWriting $e^{i(z \\sin \\tau - n\\tau)} = \\cos(n\\tau - z\\sin\\tau) - i \\sin(n\\tau - z\\sin\\tau)$, the sine term is odd in $\\tau$ and the cosine term is even, which gives the formula on $[0, \\pi]$.\nExample: for $n = 0$ and $z = 0$ the integral is $\\frac{1}{\\pi} \\int_0^\\pi 1\\, d\\tau = 1 = J_0(0)$.\nThe hypothesis that $n$ is an integer is needed: for non-integer $\\nu$ the same integral defines the Anger function, which differs from $J_\\nu$.\nFor $\\Re z \u003e 0$ and any $\\nu$, Schläfli's extension reads $J_\\nu(z) = \\frac{1}{\\pi} \\int_0^\\pi \\cos(\\nu\\tau - z \\sin\\tau)\\, d\\tau - \\frac{\\sin(\\nu\\pi)}{\\pi} \\int_0^\\infty e^{-z \\sinh t - \\nu t}\\, dt$.\n\n## Explanations\n\n### L1 human\n\nThis formula writes a Bessel function as an average of a cosine over half a turn.\nIt was Bessel's own definition, coming from a problem in planetary motion.\nIt shows at once that $\\abs{J_n(x)} \\le 1$ for real $x$, since the integrand lies between $-1$ and $1$.\n\n### L2 human\n\nThe integral turns many questions about $J_n$ into questions about an oscillatory integral.\nApplying stationary phase to it gives the large-$x$ asymptotics of $J_n(x)$.\nDifferentiating under the integral sign checks Bessel's equation directly.\nIt is also the bridge between Bessel functions and Fourier series of $e^{iz\\sin\\tau}$, which is how they enter Kepler's equation and frequency modulation.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "CZCMRDWDYW"
    ],
    "kind": "statement",
    "msc": [
      "33C10"
    ],
    "name": "Bessel's integral",
    "role": "proposition",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Bessel_function"
      }
    ],
    "tag": "PCE6XQTT6R",
    "uses": [
      "CZCMRDWDYW"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/PCE6XQTT6R/",
  "links": [
    {
      "field": "about",
      "target": "CZCMRDWDYW"
    },
    {
      "field": "uses",
      "target": "CZCMRDWDYW"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/PCE6XQTT6R.md",
  "msc": [
    "33C10"
  ],
  "name": "Bessel's integral",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PCE/PCE6XQTT6R.md",
  "status": null,
  "tag": "PCE6XQTT6R",
  "uri": "m2://PCE6XQTT6R",
  "url": "https://tamnd.github.io/math2/wiki/bessels-integral/",
  "writing": {
    "tag": "PCE6XQTT6R",
    "kind": "statement",
    "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
      }
    ]
  }
}