{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "B5ZWZHMD12"
    },
    {
      "field": "uses",
      "target": "DQRXS8PJ15"
    },
    {
      "field": "uses",
      "target": "QBRJN93K0P"
    }
  ],
  "body": "\n## Summary\n\nIn [[operational calculus|8050R3Z6HA]], $p$ denotes the operator $d/dt$ that sends a function of time to its [[derivative|0V1EJCT8G6]].\nIt is treated as an algebraic symbol: one forms polynomials and fractions in $p$, and $1/p$ is integration from $0$.\nBecause $p$ undoes integration but integration does not quite undo $p$, initial values appear in the rules, as in $p^{-1} p f = f - f(0)$.\n\n## Explanations\n\n### L1 human\n\nWriting $p$ for \"take the derivative\" lets one write $y'' + 3y' + 2y$ as $(p^2 + 3p + 2) y$, and even factor it as $(p + 1)(p + 2) y$.\nIts partner $1/p$ means \"integrate from $0$\".\nDifferentiating an integral gives back the function, but integrating a derivative from $0$ loses the starting value.\nFor example, for $f(t) = t + 5$, integrating $f' = 1$ from $0$ gives $t$, not $t + 5$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "44A45",
      "44A40"
    ],
    "name": "Differentiation operator p",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Operational_calculus"
      }
    ],
    "tag": "BB15T3J891"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/BB15T3J891/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/BB15T3J891.md",
  "msc": [
    "44A45",
    "44A40"
  ],
  "name": "Differentiation operator p",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/BB1/BB15T3J891.md",
  "status": null,
  "tag": "BB15T3J891",
  "uri": "m2://BB15T3J891",
  "url": "https://tamnd.github.io/math2/wiki/differentiation-operator-p/",
  "writing": {
    "tag": "BB15T3J891",
    "kind": "concept",
    "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
      }
    ]
  }
}