{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Counterexample\n\nThe function $f(t) = e^{t^2}$ on $[0, \\infty)$ is continuous, but its [[Laplace integral|05XT5AVJBK]] $\\int_0^\\infty e^{t^2} e^{-st}\\, dt$ converges for no $s \\in \\C$.\nCheck: suppose it converged at some $s_0 \\in \\C$.\nBy [[the half-plane theorem|6X05N89XDP]] it would then converge at every real $s \u003e \\operatorname{Re} s_0$.\nBut for real $s$ and $t \\ge \\max(s, 0)$, the integrand satisfies $e^{t^2 - st} = e^{t(t - s)} \\ge 1$, so $\\int_0^T e^{t^2 - st}\\, dt \\to \\infty$ as $T \\to \\infty$.\nThis is a contradiction, so the abscissa of convergence of $f$ is $+\\infty$.\nThe function is not of [[exponential order|TKZAV3KR86]]: $e^{t^2} / e^{at} \\to \\infty$ for every real $a$.\n\n## Remarks\n\nExponential order is sufficient, not necessary, for a Laplace transform to exist.\nFor example, $f(t) = 2t e^{t^2} \\cos(e^{t^2})$ is not of exponential order, yet $\\int_0^T f(t)\\, dt = \\sin(e^{T^2}) - \\sin 1$ stays bounded, and integration by parts shows that its Laplace integral converges for every $s$ with $\\operatorname{Re} s \u003e 0$.\nA different obstruction is local: $f(t) = 1/t$ has no Laplace transform because $\\int_0^1 dt/t$ diverges, whatever $s$ is.\n\n## Explanations\n\n### L1 human\n\nThe Laplace transform weighs $f(t)$ by $e^{-st}$, which shrinks at most exponentially fast.\nThe function $e^{t^2}$ grows faster than any exponential, so no choice of $s$ can tame it and the integral always blows up.\nFast growth is not the only way to fail: a function such as $1/t$ is too large near $0$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "counterexample",
    "msc": [
      "44A10"
    ],
    "name": "Function with no Laplace transform",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Laplace_transform"
      }
    ],
    "tag": "105G7PWKS1",
    "uses": [
      "05XT5AVJBK",
      "6X05N89XDP",
      "TKZAV3KR86"
    ]
  },
  "kind": "counterexample",
  "link": "https://tamnd.github.io/math2/node/105G7PWKS1/",
  "links": [
    {
      "field": "uses",
      "target": "05XT5AVJBK"
    },
    {
      "field": "uses",
      "target": "6X05N89XDP"
    },
    {
      "field": "uses",
      "target": "TKZAV3KR86"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/105G7PWKS1.md",
  "msc": [
    "44A10"
  ],
  "name": "Function with no Laplace transform",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/105/105G7PWKS1.md",
  "status": null,
  "tag": "105G7PWKS1",
  "uri": "m2://105G7PWKS1",
  "url": "https://tamnd.github.io/math2/wiki/function-with-no-laplace-transform/",
  "writing": {
    "tag": "105G7PWKS1",
    "kind": "counterexample",
    "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
      },
      {
        "what": "edge refutes or not_instance_of out",
        "have": 0,
        "want": 1
      }
    ]
  }
}