{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\sin$ and $\\cos$ be the [[sine and cosine|08WFZP6P89]] functions.\n1. The *sine integral* is the function $\\operatorname{Si} \\colon \\R \\to \\R$ given by\n$$\\operatorname{Si}(x) = \\int_0^x \\frac{\\sin t}{t}\\, dt,$$\nwhere the integrand is given the value $1$ at $t = 0$.\n2. The *cosine integral* is the function on $x \u003e 0$ given by the [[improper integral|05KQ3F99BQ]]\n$$\\operatorname{Ci}(x) = -\\int_x^\\infty \\frac{\\cos t}{t}\\, dt.$$\n3. The *hyperbolic sine integral* and *hyperbolic cosine integral* are $\\operatorname{Shi}(x) = \\int_0^x \\frac{\\sinh t}{t}\\, dt$ for real $x$ and $\\operatorname{Chi}(x) = \\gamma + \\ln x + \\int_0^x \\frac{\\cosh t - 1}{t}\\, dt$ for $x \u003e 0$, where $\\gamma$ is the [[Euler–Mascheroni constant|00C5P9JZBK]].\n\n## Remarks\n\nEquivalently, $\\operatorname{Ci}(x) = \\gamma + \\ln x + \\int_0^x \\frac{\\cos t - 1}{t}\\, dt$ for $x \u003e 0$.\n\nExample: $\\operatorname{Si}$ is odd, $\\operatorname{Si}(0) = 0$, and $\\lim_{x \\to \\infty} \\operatorname{Si}(x) = \\frac{\\pi}{2}$ by the [[Dirichlet integral|0GZNSHMG4Q]].\nThe largest value of $\\operatorname{Si}$ on $[0, \\infty)$ is $\\operatorname{Si}(\\pi) \\approx 1.8519$.\n\nNon-example: $\\int_0^x \\frac{\\cos t}{t}\\, dt$ is not a valid definition of $\\operatorname{Ci}$; it diverges at $0$ because $\\frac{\\cos t}{t} \\sim \\frac{1}{t}$ there.\nThis is why $\\operatorname{Ci}$ is defined from $\\infty$, and it has a logarithmic singularity at $0$.\n\nThe shifted function $\\operatorname{si}(x) = \\operatorname{Si}(x) - \\frac{\\pi}{2} = -\\int_x^\\infty \\frac{\\sin t}{t}\\, dt$ is also used.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "VB37Q6FY1Y",
    "kind": "definition",
    "msc": [
      "33E20",
      "33B10"
    ],
    "name": "Trigonometric integrals",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Trigonometric_integral"
      }
    ],
    "tag": "T761KPKPV6",
    "uses": [
      "08WFZP6P89",
      "05KQ3F99BQ",
      "00C5P9JZBK"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/T761KPKPV6/",
  "links": [
    {
      "field": "defines",
      "target": "VB37Q6FY1Y"
    },
    {
      "field": "uses",
      "target": "08WFZP6P89"
    },
    {
      "field": "uses",
      "target": "05KQ3F99BQ"
    },
    {
      "field": "uses",
      "target": "00C5P9JZBK"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/T761KPKPV6.md",
  "msc": [
    "33E20",
    "33B10"
  ],
  "name": "Trigonometric integrals",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/T76/T761KPKPV6.md",
  "status": null,
  "tag": "T761KPKPV6",
  "uri": "m2://T761KPKPV6",
  "url": "https://tamnd.github.io/math2/wiki/trigonometric-integrals-definition/",
  "writing": {
    "tag": "T761KPKPV6",
    "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
      }
    ]
  }
}