{
  "aliases": [
    "periodicity-of-the-exponential"
  ],
  "backlinks": null,
  "body": "\n## Statement\n\nFor each $z \\in \\mathbb{C}$,\n$$e^{z + 2 \\pi i} = e^{z},$$\nwhere $e^z$ is the [[complex exponential|054VXHMCA3]] of $z$.\n\n## Remarks\n\nProof: by [[Euler's formula|04R3CY41VJ]], $e^{2 \\pi i} = \\cos 2\\pi + i \\sin 2\\pi = 1$. By the [[product of exponentials|0WM7DJCM6G]], $e^{z + 2\\pi i} = e^z e^{2\\pi i} = e^z$.\nThe periods of $\\exp$ are exactly the numbers $2 \\pi i k$ with $k \\in \\mathbb{Z}$, by [[exp z = 1 iff z is in 2πiZ|013AHC4DRT]].\nSo $\\exp$ is not injective on $\\mathbb{C}$. It is injective on each horizontal strip $\\{z : a \u003c \\operatorname{Im} z \\le a + 2\\pi\\}$ with $a \\in \\mathbb{R}$.\n\n## Explanations\n\n### L1 human\n\nAdding $2 \\pi i$ to $z$ adds a full turn to the direction of $e^z$ and does not change its size.\nSo $e^z$ takes the same value again: the exponential repeats itself along each vertical line, with step $2\\pi$.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "aliases": [
      "periodicity-of-the-exponential"
    ],
    "kind": "statement",
    "msc": [
      "33B10"
    ],
    "name": "The complex exponential is 2πi-periodic",
    "role": "proposition",
    "sources": [
      {
        "locator": "Complex exponential",
        "url": "https://en.wikipedia.org/wiki/Exponential_function"
      }
    ],
    "tag": "02P2PYH3X3",
    "uses": [
      "054VXHMCA3",
      "0WM7DJCM6G",
      "04R3CY41VJ"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/02P2PYH3X3/",
  "links": [
    {
      "field": "uses",
      "target": "054VXHMCA3"
    },
    {
      "field": "uses",
      "target": "0WM7DJCM6G"
    },
    {
      "field": "uses",
      "target": "04R3CY41VJ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/02P2PYH3X3.md",
  "msc": [
    "33B10"
  ],
  "name": "The complex exponential is 2πi-periodic",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/02P/02P2PYH3X3.md",
  "status": null,
  "tag": "02P2PYH3X3",
  "uri": "m2://02P2PYH3X3",
  "url": "https://math2.tamnd.com/wiki/the-complex-exponential-is-2-i-periodic/",
  "writing": {
    "tag": "02P2PYH3X3",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}