{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $f \\colon [0, \\infty) \\to \\C$ be integrable on each interval $[0, T]$.\nAssume that the Laplace integral of $f$ converges at some real number $s_0$, and let $F$ be the [[Laplace transform|05XT5AVJBK]] of $f$.\nAssume that the one-sided [[limit|0TXTSEMHR5]] $f(0^+) = \\lim_{t \\to 0^+} f(t)$ exists and is finite.\nThen\n$$\\lim_{s \\to +\\infty} s F(s) = f(0^+),$$\nwhere $s$ tends to $+\\infty$ through real values.\n\n## Remarks\n\nSince $s \\int_0^\\infty e^{-s t} \\, dt = 1$, the difference $s F(s) - f(0^+)$ is $s \\int_0^\\infty e^{-s t} (f(t) - f(0^+)) \\, dt$.\nThe weight $s e^{-s t}$ has total mass $1$ and concentrates near $t = 0$ as $s \\to \\infty$.\nOn $[0, \\delta]$ the integrand is small by the existence of $f(0^+)$; on $[\\delta, \\infty)$ the contribution tends to $0$ by the convergence at $s_0$.\nExample: for $f(t) = \\cos t$, $s F(s) = s^2 / (s^2 + 1) \\to 1 = f(0)$.\nThe finiteness of $f(0^+)$ matters: for $f(t) = t^{-1/2}$ one has $F(s) = \\sqrt{\\pi / s}$, so $s F(s) = \\sqrt{\\pi s} \\to +\\infty$.\n\n## Explanations\n\n### L1 human\n\nThe behaviour of the transform for very large $s$ tells the starting value of the function.\nOne can read $f(0^+)$ from $F$ without inverting the transform.\n\n### L2 human\n\nIn control and circuit theory one uses it to check the value of a response just after a switch is closed.\nIt is the simplest Abelian theorem for the Laplace transform: behaviour of $f$ near $0$ gives behaviour of $F$ near $\\infty$.\nIts companion is the [[final value theorem|BDHADVMJ8G]], which links $f$ near $\\infty$ with $F$ near $0$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "0V49JEZW06"
    ],
    "kind": "statement",
    "msc": [
      "44A10",
      "40E05"
    ],
    "name": "Initial value theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q4272300",
      "wikipedia": "Initial value theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Initial_value_theorem"
      }
    ],
    "tag": "KZPVHDKV9E",
    "uses": [
      "05XT5AVJBK",
      "0TXTSEMHR5"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/KZPVHDKV9E/",
  "links": [
    {
      "field": "about",
      "target": "0V49JEZW06"
    },
    {
      "field": "uses",
      "target": "05XT5AVJBK"
    },
    {
      "field": "uses",
      "target": "0TXTSEMHR5"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/KZPVHDKV9E.md",
  "msc": [
    "44A10",
    "40E05"
  ],
  "name": "Initial value theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/KZP/KZPVHDKV9E.md",
  "status": null,
  "tag": "KZPVHDKV9E",
  "uri": "m2://KZPVHDKV9E",
  "url": "https://tamnd.github.io/math2/wiki/initial-value-theorem/",
  "writing": {
    "tag": "KZPVHDKV9E",
    "kind": "statement",
    "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
      }
    ]
  }
}