{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nThe inverse hyperbolic sine $\\operatorname{arsinh} \\colon \\mathbb{R} \\to \\mathbb{R}$ is the inverse of [[$\\sinh$|0GWD21EWRW]].\nThe inverse hyperbolic cosine $\\operatorname{arcosh} \\colon [1, \\infty) \\to [0, \\infty)$ is the inverse of the restriction of [[$\\cosh$|0EQFWKJ37J]] to $[0, \\infty)$.\nThe inverse hyperbolic tangent $\\operatorname{artanh} \\colon (-1, 1) \\to \\mathbb{R}$ is the inverse of [[$\\tanh$|0MWYQTZ07Z]].\n\n## Remarks\n\nSolving a quadratic equation in $e^y$ gives the formulas $\\operatorname{arsinh} x = \\ln(x + \\sqrt{x^2 + 1})$ and $\\operatorname{arcosh} x = \\ln(x + \\sqrt{x^2 - 1})$.\nSimilarly, $\\operatorname{artanh} x = \\frac{1}{2} \\ln \\frac{1 + x}{1 - x}$ for $|x| \u003c 1$.\nTheir derivatives are $\\frac{1}{\\sqrt{x^2 + 1}}$, $\\frac{1}{\\sqrt{x^2 - 1}}$ for $x \u003e 1$, and $\\frac{1}{1 - x^2}$. Mathlib defines the first one as `Real.arsinh`.\n\n## Explanations\n\n### L1 human\n\nThe inverse hyperbolic functions undo the hyperbolic functions.\nSince the hyperbolic functions come from $e^x$, their inverses come from the logarithm.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0WFZX06698",
    "kind": "definition",
    "msc": [
      "33B10"
    ],
    "name": "Inverse hyperbolic functions",
    "sources": [
      {
        "locator": "Definitions in terms of logarithms",
        "url": "https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions"
      }
    ],
    "tag": "04D7CHT5V1",
    "uses": [
      "0GWD21EWRW",
      "0EQFWKJ37J",
      "0MWYQTZ07Z",
      "0044CPHJBV",
      "0V7E9KP6Q2"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/04D7CHT5V1/",
  "links": [
    {
      "field": "defines",
      "target": "0WFZX06698"
    },
    {
      "field": "uses",
      "target": "0GWD21EWRW"
    },
    {
      "field": "uses",
      "target": "0EQFWKJ37J"
    },
    {
      "field": "uses",
      "target": "0MWYQTZ07Z"
    },
    {
      "field": "uses",
      "target": "0044CPHJBV"
    },
    {
      "field": "uses",
      "target": "0V7E9KP6Q2"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/04D7CHT5V1.md",
  "msc": [
    "33B10"
  ],
  "name": "Inverse hyperbolic functions",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/04D/04D7CHT5V1.md",
  "status": null,
  "tag": "04D7CHT5V1",
  "uri": "m2://04D7CHT5V1",
  "url": "https://math2.tamnd.com/wiki/inverse-hyperbolic-functions-definition/",
  "writing": {
    "tag": "04D7CHT5V1",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}