{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nThe Joukowsky transform is the map $J(z) = z + \\dfrac{1}{z}$ on $\\C \\setminus \\set{0}$.\nIt is [[holomorphic|0DG0Z72CQD]] with $J'(z) = 1 - z^{-2}$, which vanishes only at $z = \\pm 1$.\nSo $J$ is a [[conformal map|0VHNKP1N3K]] near each point other than $0$ and $\\pm 1$.\n\nCircles go to ellipses.\nFor $z = r e^{i\\theta}$ with $r \u003e 1$,\n$$J(z) = \\left(r + \\frac{1}{r}\\right) \\cos \\theta + i \\left(r - \\frac{1}{r}\\right) \\sin \\theta,$$\nso the circle $\\abs{z} = r$ goes onto the [[ellipse|0FF6CAQT2Y]] with semi-axes $r + 1/r$ and $r - 1/r$ and foci $\\pm 2$.\nFor $r = 1$ one gets $J(e^{i\\theta}) = 2 \\cos \\theta$, so the unit circle is folded onto the segment $[-2, 2]$.\n\nThe exterior of the unit disc is mapped bijectively onto $\\C \\setminus [-2, 2]$.\nCheck of injectivity: $J(z) - J(w) = (z - w)(1 - \\frac{1}{zw})$, and $zw \\neq 1$ when $\\abs{z} \u003e 1$ and $\\abs{w} \u003e 1$.\nCheck of surjectivity: for $c \\in \\C$, the two roots of $z^2 - cz + 1 = 0$ have product $1$.\nIf neither root has modulus greater than $1$, both have modulus $1$, and then $c = 2\\cos\\theta$ lies in $[-2, 2]$.\n\n## Remarks\n\nCircles that pass through $-1$ or $1$ and contain the other point go to curves with one sharp edge, the Joukowsky airfoils.\nThis is why the map is used to model the flow of air around a wing.\nSome authors use $\\frac{1}{2}(z + 1/z)$ instead, which only rescales the picture by $\\frac{1}{2}$.\n\n## Explanations\n\n### L1 human\n\nThe Joukowsky map adds a number and its reciprocal.\nFar from the origin $1/z$ is tiny, so the map is almost the identity there.\nNear the unit circle the two terms are of the same size, and the circle is squashed flat onto a segment.\nCircles a bit larger than the unit circle become ellipses, and suitably shifted circles become wing shapes.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "illustrates": [
      "0YYT36R4YD"
    ],
    "kind": "example",
    "msc": [
      "30C20",
      "76B10"
    ],
    "name": "Joukowsky transform",
    "same_as": {
      "wikidata": "Q1794703",
      "wikipedia": "Joukowsky transform"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Joukowsky_transform"
      }
    ],
    "tag": "3DJ46V21KH",
    "uses": [
      "0DG0Z72CQD",
      "0VHNKP1N3K",
      "0FF6CAQT2Y"
    ]
  },
  "kind": "example",
  "link": "https://math2.tamnd.com/node/3DJ46V21KH/",
  "links": [
    {
      "field": "illustrates",
      "target": "0YYT36R4YD"
    },
    {
      "field": "uses",
      "target": "0DG0Z72CQD"
    },
    {
      "field": "uses",
      "target": "0VHNKP1N3K"
    },
    {
      "field": "uses",
      "target": "0FF6CAQT2Y"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/3DJ46V21KH.md",
  "msc": [
    "30C20",
    "76B10"
  ],
  "name": "Joukowsky transform",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/3DJ/3DJ46V21KH.md",
  "status": null,
  "tag": "3DJ46V21KH",
  "uri": "m2://3DJ46V21KH",
  "url": "https://math2.tamnd.com/wiki/joukowsky-transform/",
  "writing": {
    "tag": "3DJ46V21KH",
    "kind": "example",
    "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
      }
    ]
  }
}