{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "JWSNY03J2Y"
    }
  ],
  "body": "\n## Summary\n\nThe Bergman kernel of a domain is the reproducing kernel of its [[Bergman space|2D619WTPYC]] $A^2$ of square-integrable [[holomorphic functions|0NAHXR1FRH]].\nIt is a function $K(z, w)$, holomorphic in $z$ and antiholomorphic in $w$, such that every $f \\in A^2$ satisfies $f(w) = \\int f(z)\\overline{K(z, w)}\\, dA(z)$.\nFor the unit disc it is $K(z, w) = \\frac{1}{\\pi (1 - z\\bar w)^2}$.\nIt gives the orthogonal projection from $L^2$ onto $A^2$, it transforms simply under biholomorphic maps, and in several variables it defines the Bergman metric.\n\n## Explanations\n\n### L1 human\n\nIn the Bergman space, the value of a function at a point is an inner product with one special function attached to that point.\nThe Bergman kernel collects all these special functions into one formula.\nOn the unit disc the kernel at the point $w = 0$ is the constant $1/\\pi$, which says that $f(0)$ is the average of $f$ over the disc.\nThe kernel encodes the shape of the domain, which is why it is used to study conformal and biholomorphic maps.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "30C40",
      "46E22",
      "32A25"
    ],
    "name": "Bergman kernel",
    "same_as": {
      "wikidata": "Q3345664",
      "wikipedia": "Bergman kernel"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Bergman_kernel"
      }
    ],
    "tag": "8RWDHW0CXC"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/8RWDHW0CXC/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/8RWDHW0CXC.md",
  "msc": [
    "30C40",
    "46E22",
    "32A25"
  ],
  "name": "Bergman kernel",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/8RW/8RWDHW0CXC.md",
  "status": null,
  "tag": "8RWDHW0CXC",
  "uri": "m2://8RWDHW0CXC",
  "url": "https://math2.tamnd.com/wiki/bergman-kernel/",
  "writing": {
    "tag": "8RWDHW0CXC",
    "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
      }
    ]
  }
}