{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\N^n$ be the set of [[multi-indices|0PJXBR6QA3]] $\\alpha = (\\alpha_1, \\dots, \\alpha_n)$ with $\\alpha_j \\ge 0$ integers.\nFor $w \\in \\C^n$, write $w^\\alpha = w_1^{\\alpha_1} \\cdots w_n^{\\alpha_n}$, with $w_j^0 = 1$.\nLet $a \\in \\C^n$ and let $(c_\\alpha)_{\\alpha \\in \\N^n}$ be complex numbers.\nThe *power series* with centre $a$ and coefficients $c_\\alpha$ is the formal sum\n$$\\sum_{\\alpha \\in \\N^n} c_\\alpha (z - a)^\\alpha.$$\nIt *converges* at $z \\in \\C^n$ if $\\sum_{\\alpha \\in \\N^n} \\abs{c_\\alpha} \\abs{(z - a)^\\alpha} \u003c \\infty$.\nIts *sum* at such a point is the sum of the family $(c_\\alpha (z - a)^\\alpha)_\\alpha$, which does not depend on the order of the terms.\n\n## Remarks\n\nConvergence here is [[absolute convergence|0YBDSJARWN]], since the index set $\\N^n$ has no preferred order when $n \\ge 2$.\nExample: for $n = 2$, $a = 0$ and $c_\\alpha = 1$ for all $\\alpha$, at $z = (1/2, 1/3)$ the sum of absolute values is $\\sum_j 2^{-j} \\sum_k 3^{-k} = 2 \\cdot \\tfrac32 = 3 \u003c \\infty$, so the series converges, with sum $3$.\nNon-example: the same series at $z = (1/2, 1)$ has the terms $2^{-j}$ with $k$ arbitrary, so the sum of absolute values is infinite and the series does not converge there.\nGrouping by total degree $\\abs{\\alpha} = \\alpha_1 + \\cdots + \\alpha_n$ writes the series as $\\sum_{m \\ge 0} P_m(z - a)$, with $P_m$ a homogeneous polynomial of degree $m$.\nFor $n = 1$ this is the ordinary [[power series|0MZB06ST2B]], and in that case absolute convergence holds at every point inside the disc of convergence.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "R3QTCPTJK7",
    "kind": "definition",
    "msc": [
      "32A05"
    ],
    "name": "Power series in several variables",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Power_series"
      }
    ],
    "tag": "ZBNQEJ09QE",
    "uses": [
      "BH57D2FBNR",
      "0PJXBR6QA3",
      "0YBDSJARWN"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/ZBNQEJ09QE/",
  "links": [
    {
      "field": "defines",
      "target": "R3QTCPTJK7"
    },
    {
      "field": "uses",
      "target": "BH57D2FBNR"
    },
    {
      "field": "uses",
      "target": "0PJXBR6QA3"
    },
    {
      "field": "uses",
      "target": "0YBDSJARWN"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/ZBNQEJ09QE.md",
  "msc": [
    "32A05"
  ],
  "name": "Power series in several variables",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZBN/ZBNQEJ09QE.md",
  "status": null,
  "tag": "ZBNQEJ09QE",
  "uri": "m2://ZBNQEJ09QE",
  "url": "https://tamnd.github.io/math2/wiki/power-series-in-several-variables-definition/",
  "writing": {
    "tag": "ZBNQEJ09QE",
    "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
      }
    ]
  }
}