{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\nu \u003e -1$ be real and let $J_\\nu$ be the [[Bessel function of the first kind|CZCMRDWDYW]] of order $\\nu$.\nThe positive zeros of $J_\\nu$ form an infinite set with no finite accumulation point.\nFor $k \\ge 1$, the $k$-th *zero* $j_{\\nu,k}$ of $J_\\nu$ is the $k$-th smallest element of this set, so that\n$$0 \u003c j_{\\nu,1} \u003c j_{\\nu,2} \u003c j_{\\nu,3} \u003c \\cdots, \\qquad J_\\nu(j_{\\nu,k}) = 0.$$\n\n## Remarks\n\nFor $\\nu \u003e -1$, every zero of $z^{-\\nu} J_\\nu(z)$ in $\\C$ is real and simple, so the $j_{\\nu,k}$ and their negatives are all of them.\nExample: $J_{1/2}(x) = \\sqrt{\\frac{2}{\\pi x}} \\sin x$, so $j_{1/2,k} = k\\pi$; the check is $\\sin(k\\pi) = 0$ and $\\sin x \\ne 0$ on $(0, \\pi)$.\nNumerical values: $j_{0,1} \\approx 2.4048$ and $j_{1,1} \\approx 3.8317$.\nThe zeros of $J_\\nu$ and $J_{\\nu+1}$ interlace: $j_{\\nu,1} \u003c j_{\\nu+1,1} \u003c j_{\\nu,2} \u003c j_{\\nu+1,2} \u003c \\cdots$.\nFor fixed $\\nu$, $j_{\\nu,k+1} - j_{\\nu,k} \\to \\pi$ as $k \\to \\infty$, and $j_{\\nu,k} = \\left( k + \\frac{\\nu}{2} - \\frac{1}{4} \\right)\\pi + O\\left( \\frac{1}{k} \\right)$ (McMahon's expansion).\nA non-example: $0$ is a zero of $J_\\nu$ for $\\nu \u003e 0$, but it is not one of the $j_{\\nu,k}$, which are positive by definition.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "M324R146XH",
    "kind": "definition",
    "msc": [
      "33C10"
    ],
    "name": "Zeros of Bessel functions",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Bessel_function"
      }
    ],
    "tag": "MSXT31P944",
    "uses": [
      "CZCMRDWDYW"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/MSXT31P944/",
  "links": [
    {
      "field": "defines",
      "target": "M324R146XH"
    },
    {
      "field": "uses",
      "target": "CZCMRDWDYW"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/MSXT31P944.md",
  "msc": [
    "33C10"
  ],
  "name": "Zeros of Bessel functions",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/MSX/MSXT31P944.md",
  "status": null,
  "tag": "MSXT31P944",
  "uri": "m2://MSXT31P944",
  "url": "https://tamnd.github.io/math2/wiki/zeros-of-bessel-functions-definition/",
  "writing": {
    "tag": "MSXT31P944",
    "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
      }
    ]
  }
}