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  "body": "\n## Summary\n\nLet $U \\subseteq V$ be open subsets of $\\mathbb{C}$, with $U$ nonempty and $V$ a [[domain|097KNJFJ7V]], and let $f$ be [[holomorphic|0DG0Z72CQD]] on $U$.\nAn analytic continuation of $f$ to $V$ is a function $F$ that is holomorphic on $V$ with $F(z) = f(z)$ for each $z \\in U$.\nBy the [[identity theorem|0YBVKMYX0W]], $f$ has at most one analytic continuation to $V$.\n\n## Remarks\n\nThe geometric series $\\sum_{n \\ge 0} z^n$ is holomorphic on the unit disc, and $1/(1 - z)$ is its analytic continuation to $\\mathbb{C} \\setminus \\{1\\}$.\nThe Riemann zeta function $\\sum_{n \\ge 1} n^{-s}$ on $\\operatorname{Re} s \u003e 1$ has an analytic continuation to $\\mathbb{C} \\setminus \\{1\\}$.\nContinuation along a path is a more general process. Its result can depend on the path: the [[logarithm|0X2FY491PF]] continued once around $0$ changes by $2\\pi i$.\n\n## Explanations\n\n### L1 human\n\nA formula such as a power series often defines a function only on a small region.\nAnalytic continuation extends the function to a larger region in the only possible holomorphic way.\n",
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