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  "body": "\n## Definition\n\nA *function element* in $\\C^n$ is a pair $(U, f)$ where $U \\subseteq \\C^n$ is a nonempty connected [[open set|02BH3MCBQB]] and $f$ is a [[holomorphic function|507C34W0ZT]] on $U$.\n\n1. A function element $(V, g)$ is a *direct analytic continuation* of $(U, f)$ if $U \\cap V$ is nonempty and $f = g$ on some [[connected component|0JG9W3KA6B]] of $U \\cap V$.\n2. A function element $(V, g)$ is an *analytic continuation* of $(U, f)$ along a chain if a finite sequence of function elements runs from $(U_0, f_0) = (U, f)$ to $(U_k, f_k) = (V, g)$. Each $(U_j, f_j)$ must be a direct analytic continuation of $(U_{j-1}, f_{j-1})$.\n\nIf $V \\supseteq U$ and $g = f$ on $U$, one says that $g$ is an analytic continuation of $f$ to $V$.\n\n## Remarks\n\nBy the identity theorem, if $f$ and $g$ agree on a nonempty open subset of a component of $U \\cap V$, they agree on that whole component.\n\nExample: let $U = \\set{z \\in \\C^2 : \\operatorname{Re} z_1 \u003e 0}$ and $f(z) = \\log z_1$ with the principal branch.\nMoving through half-spaces in the $z_1$-direction around the plane $z_1 = 0$ gives a chain of direct continuations that returns to $U$ with the function $\\log z_1 + 2\\pi i$.\nSo continuation along a closed chain need not give back the original function element.\n\nNon-example: if $U \\cap V$ is empty, then $(V, g)$ is never a direct analytic continuation of $(U, f)$, whatever $g$ is.\n\nFor $n = 1$ this is the usual [[analytic continuation|0M3YQJ81NB]] of a holomorphic function along a chain of discs or domains.\n",
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