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  "body": "\n## Summary\n\nA holomorphic function on a domain has the boundary as natural boundary if it cannot be continued across any boundary point.\nSee the [[definition|WER1M071EK]].\nIt is the extreme case of the obstruction to [[analytic continuation|0M3YQJ81NB]].\nThe typical example is a [[power series|0D6KXBHKC0]] with large gaps, whose circle of convergence is a natural boundary by the [[Hadamard gap theorem|DA8M8VZZHC]].\n\n## Explanations\n\n### L1 human\n\nMany holomorphic functions given on a disc can be extended past the circle, except at a few bad points.\nFor $1/(1 - z)$, only the point $1$ is bad.\nFor some functions, every point of the circle is bad, so the disc is the end of the world for the function.\nThen the circle is called a natural boundary.\nAn example is $z + z^2 + z^4 + z^8 + \\cdots$, which blows up near each root of unity whose order is a power of $2$, and these points are dense on the circle.\n",
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