{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $f \\colon [0, \\infty) \\to \\C$ be integrable on each interval $[0, T]$ with $T \u003e 0$.\nThe *abscissa of convergence* of the [[Laplace transform|05XT5AVJBK]] of $f$ is\n$$\\sigma_c = \\inf \\set{ \\sigma \\in \\R : \\int_0^\\infty f(t) e^{-\\sigma t}\\, dt \\text{ converges} } \\in [-\\infty, +\\infty],$$\nwhere convergence is that of the [[improper integral|0VCTBJMEJF]] $\\lim_{T \\to \\infty} \\int_0^T f(t) e^{-\\sigma t}\\, dt$, and $\\inf \\emptyset = +\\infty$.\n\n## Remarks\n\nIf the Laplace integral converges at a complex number $s_0$, it converges at every $s$ with $\\operatorname{Re} s \u003e \\operatorname{Re} s_0$.\nHence the integral converges for $\\operatorname{Re} s \u003e \\sigma_c$ and diverges for $\\operatorname{Re} s \u003c \\sigma_c$.\nExample: for $f(t) = e^{at}$ with $a \\in \\R$, $\\int_0^T e^{(a - \\sigma)t}\\, dt$ has a finite limit exactly when $\\sigma \u003e a$, so $\\sigma_c = a$.\nNon-example of a finite value: for $f(t) = e^{t^2}$ the integral diverges for every real $\\sigma$, so $\\sigma_c = +\\infty$; for $f(t) = e^{-t^2}$ it converges for every real $\\sigma$, so $\\sigma_c = -\\infty$.\nFor $f = 1$, $\\sigma_c = 0$, and on the line $\\operatorname{Re} s = 0$ the integral diverges at every point.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "FTHFSJ30WJ",
    "kind": "definition",
    "msc": [
      "44A10"
    ],
    "name": "Abscissa of convergence",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Laplace_transform"
      }
    ],
    "tag": "HEMXNPQBSV",
    "uses": [
      "05XT5AVJBK",
      "0VCTBJMEJF"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/HEMXNPQBSV/",
  "links": [
    {
      "field": "defines",
      "target": "FTHFSJ30WJ"
    },
    {
      "field": "uses",
      "target": "05XT5AVJBK"
    },
    {
      "field": "uses",
      "target": "0VCTBJMEJF"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/HEMXNPQBSV.md",
  "msc": [
    "44A10"
  ],
  "name": "Abscissa of convergence",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HEM/HEMXNPQBSV.md",
  "status": null,
  "tag": "HEMXNPQBSV",
  "uri": "m2://HEMXNPQBSV",
  "url": "https://tamnd.github.io/math2/wiki/abscissa-of-convergence-definition/",
  "writing": {
    "tag": "HEMXNPQBSV",
    "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
      }
    ]
  }
}