identity theorem of holomorphic functionsIdentityTheoremOfHolomorphicFunctions2007-03-04 14:07:302008-05-01 04:10:33Theoremholomorphic% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{thmplain}{Theorem}
If the functions $f$ and $g$ are holomorphic in a domain $D$ of
the complex plane and the equation
\begin{align}
f(z) = g(z)
\end{align}
is true in an infinite subset $S$ of $D$ having an accumulation point $z_0$ in $D$, then (1) is true in the whole $D$.
\textbf{Remark.}\, The subset $S$ may be e.g. some neighbourhood of $z_0$ or some arc containing $z_0$.