{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "JCNCD8PSGF"
    }
  ],
  "body": "\n## Example\n\nLet $a \\in \\C$ and $f(t) = e^{at}$ for $t \\ge 0$, an [[exponential function|0RMVS7RXZX]].\nIts [[Laplace transform|05XT5AVJBK]] is\n$$\\mathcal{L}\\{e^{at}\\}(s) = \\frac{1}{s - a} \\quad (\\operatorname{Re} s \u003e \\operatorname{Re} a).$$\nCheck: for $s \\ne a$ and $T \u003e 0$,\n$$\\int_0^T e^{at} e^{-st}\\, dt = \\frac{1 - e^{-(s - a)T}}{s - a}.$$\nSince $\\abs{e^{-(s - a)T}} = e^{-\\operatorname{Re}(s - a) T}$, this tends to $0$ as $T \\to \\infty$ exactly when $\\operatorname{Re} s \u003e \\operatorname{Re} a$, which gives the formula.\nWhen $\\operatorname{Re} s \u003c \\operatorname{Re} a$ the term $e^{-(s-a)T}$ grows without bound.\nWhen $\\operatorname{Re} s = \\operatorname{Re} a$ it stays on the unit circle without a limit (or the integrand is $1$ if $s = a$), so the integral diverges.\nSo the abscissa of convergence is $\\operatorname{Re} a$.\nThe case $a = 0$ gives $\\mathcal{L}\\{1\\}(s) = 1/s$.\n\n## Explanations\n\n### L1 human\n\nThe exponential $e^{at}$ is the basic building block of Laplace transform tables.\nIts transform $1/(s - a)$ has a single pole at $s = a$, and the location of the pole records the growth rate of $e^{at}$.\nMany other entries, such as those for sine, cosine and hyperbolic functions, follow from this one by linearity.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "illustrates": [
      "0V49JEZW06"
    ],
    "kind": "example",
    "msc": [
      "44A10"
    ],
    "name": "Laplace transform of the exponential function",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Laplace_transform"
      }
    ],
    "tag": "X5JYH26J3M",
    "uses": [
      "05XT5AVJBK",
      "0RMVS7RXZX"
    ]
  },
  "kind": "example",
  "link": "https://tamnd.github.io/math2/node/X5JYH26J3M/",
  "links": [
    {
      "field": "illustrates",
      "target": "0V49JEZW06"
    },
    {
      "field": "uses",
      "target": "05XT5AVJBK"
    },
    {
      "field": "uses",
      "target": "0RMVS7RXZX"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/X5JYH26J3M.md",
  "msc": [
    "44A10"
  ],
  "name": "Laplace transform of the exponential function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/X5J/X5JYH26J3M.md",
  "status": null,
  "tag": "X5JYH26J3M",
  "uri": "m2://X5JYH26J3M",
  "url": "https://tamnd.github.io/math2/wiki/laplace-transform-of-the-exponential-function/",
  "writing": {
    "tag": "X5JYH26J3M",
    "kind": "example",
    "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
      }
    ]
  }
}