{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $a \u003c x \u003c b$ and let $\\varphi \\colon [a, b] \\to \\R$ be integrable, differentiable near $x$, with $\\varphi'$ [[Hölder continuous|012PBA0FMD]] near $x$.\nThe *Hadamard finite part* integral of $\\varphi(t)/(t - x)^2$ is\n$$\\mathrm{f.p.}\\!\\int_a^b \\frac{\\varphi(t)}{(t - x)^2}\\, dt = \\lim_{\\varepsilon \\to 0^+} \\left( \\int_{\\set{t \\in [a, b] : \\abs{t - x} \u003e \\varepsilon}} \\frac{\\varphi(t)}{(t - x)^2}\\, dt - \\frac{2\\varphi(x)}{\\varepsilon} \\right).$$\nUnder these hypotheses the limit exists.\n\n## Remarks\n\nThe general rule is the same.\nExpand the integral over the region at distance more than $\\varepsilon$ from the singularity in powers of $\\varepsilon$ and $\\log \\varepsilon$.\nThen drop the terms that diverge as $\\varepsilon \\to 0$ and keep the limit of the rest.\nFor example, $\\int_\\varepsilon^1 t^{-3/2}\\, dt = 2\\varepsilon^{-1/2} - 2$, so $\\mathrm{f.p.}\\!\\int_0^1 t^{-3/2}\\, dt = -2$.\n\nExample: $\\varphi = 1$ on $[-1, 1]$ and $x = 0$.\nThen $\\int_{\\varepsilon \u003c \\abs{t} \\le 1} t^{-2}\\, dt = \\frac{2}{\\varepsilon} - 2$, so the finite part is $-2$.\n\nNon-example: for $\\varphi(t) = \\abs{t}^{1/2}$ and $x = 0$, the derivative condition fails at $0$.\nThe integral over $\\varepsilon \u003c \\abs{t} \\le 1$ equals $4\\varepsilon^{-1/2} - 4$, and subtracting $2\\varphi(0)/\\varepsilon = 0$ leaves a divergent quantity, so this limit does not exist.\n\nFor $\\varphi$ as above, the finite part is the derivative of a principal value:\n$$\\mathrm{f.p.}\\!\\int_a^b \\frac{\\varphi(t)}{(t - x)^2}\\, dt = \\frac{d}{dx}\\, \\mathrm{PV}\\!\\int_a^b \\frac{\\varphi(t)}{t - x}\\, dt,$$\nwhere $\\mathrm{PV}$ is the [[Cauchy principal value|0SYR10HD4R]].\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "G7VE41JHHH",
    "kind": "definition",
    "msc": [
      "45E99",
      "26A42"
    ],
    "name": "Hadamard finite part",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hadamard_regularization"
      }
    ],
    "tag": "R519NKG3V6",
    "uses": [
      "0SYR10HD4R",
      "012PBA0FMD"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/R519NKG3V6/",
  "links": [
    {
      "field": "defines",
      "target": "G7VE41JHHH"
    },
    {
      "field": "uses",
      "target": "0SYR10HD4R"
    },
    {
      "field": "uses",
      "target": "012PBA0FMD"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/R519NKG3V6.md",
  "msc": [
    "45E99",
    "26A42"
  ],
  "name": "Hadamard finite part",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/R51/R519NKG3V6.md",
  "status": null,
  "tag": "R519NKG3V6",
  "uri": "m2://R519NKG3V6",
  "url": "https://tamnd.github.io/math2/wiki/hadamard-finite-part-definition/",
  "writing": {
    "tag": "R519NKG3V6",
    "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
      }
    ]
  }
}