{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "PKRGY8HD0A"
    }
  ],
  "body": "\n## Summary\n\nThe Bromwich justification gives the formal expressions $\\varphi(p) H$ of the [[Heaviside operational calculus|QQ98C2AXZP]] a precise meaning as contour integrals.\nIt defines $\\varphi(p) H$ by the [[Bromwich integral|5KNAXNHRY6]] of $\\varphi(s) e^{st} / s$ along a vertical line in the complex plane.\nWith this definition, Heaviside's rules become theorems of complex analysis, and his expansion theorem becomes the residue theorem.\n\n## Explanations\n\n### L1 human\n\nHeaviside's symbolic tricks worked but nobody could say why.\nThe answer was to read each symbolic expression as an integral in the complex plane.\nEvaluating that integral by residues reproduces exactly the answers that Heaviside's rules give.\nFor example, $p^{-1} H$ becomes an integral whose only residue gives the function $t$, the integral of the unit step.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "44A45",
      "44A10"
    ],
    "name": "Bromwich justification of the Heaviside calculus",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Operational_calculus"
      }
    ],
    "tag": "GD8B883592"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/GD8B883592/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/GD8B883592.md",
  "msc": [
    "44A45",
    "44A10"
  ],
  "name": "Bromwich justification of the Heaviside calculus",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GD8/GD8B883592.md",
  "status": null,
  "tag": "GD8B883592",
  "uri": "m2://GD8B883592",
  "url": "https://tamnd.github.io/math2/wiki/bromwich-justification-of-the-heaviside-calculus/",
  "writing": {
    "tag": "GD8B883592",
    "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
      }
    ]
  }
}