{
  "aliases": [
    "termwise-differentiation-of-power-series"
  ],
  "backlinks": [
    {
      "field": "uses",
      "target": "0QEBN8DGVE"
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  "body": "\n## Statement\n\nLet $\\sum_{n \\ge 0} a_n (z - c)^n$ be a [[power series|0VARVQ9ZW1]] with radius of convergence $R \u003e 0$, and let $D = \\{z : |z - c| \u003c R\\}$ be its [[disc of convergence|045EWAXY9P]].\nLet $f(z) = \\sum_{n=0}^{\\infty} a_n (z - c)^n$ for $z \\in D$.\nThen $f$ is [[holomorphic|0DG0Z72CQD]] on $D$, and for each $z \\in D$,\n$$f'(z) = \\sum_{n=1}^{\\infty} n a_n (z - c)^{n-1}.$$\nThe series for $f'$ has the same radius of convergence $R$.\n\n## Remarks\n\nBy induction, $f$ has complex derivatives of all orders on $D$, and $a_n = f^{(n)}(c) / n!$. So the power series is the Taylor series of $f$ at $c$.\nFor example, the [[complex exponential|054VXHMCA3]] has $R = \\infty$, so it is [[entire|0G2YJN0H5S]].\nThe converse is also true: [[holomorphic functions are analytic|0826SGV3R0]].\n\n## Explanations\n\n### L1 human\n\nInside its disc of convergence, a power series can be differentiated term by term, like a polynomial.\nThe result is again a power series with the same disc, so you can do it again and again.\n",
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    "aliases": [
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    "name": "Power series are holomorphic in the disc of convergence",
    "role": "theorem",
    "sources": [
      {
        "locator": "Differentiation and integration",
        "url": "https://en.wikipedia.org/wiki/Power_series"
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      {
        "locator": "Analytic functions",
        "url": "https://en.wikipedia.org/wiki/Power_series"
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    "tag": "0YMP521ZR5",
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  "markdown": "https://math2.tamnd.com/node/0YMP521ZR5.md",
  "msc": [
    "30B10"
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  "name": "Power series are holomorphic in the disc of convergence",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0YM/0YMP521ZR5.md",
  "status": null,
  "tag": "0YMP521ZR5",
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  "url": "https://math2.tamnd.com/wiki/power-series-are-holomorphic-in-the-disc-of-convergence/",
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