\begin{definition}[Definition:Analytic Function/Complex Plane]
Let $U \subset \C$ be an open set.
Let $f : U \to \C$ be a complex function.
Then $f$ is '''analytic''' in $U$ {{iff}} for every $z_0 \in U$ there exists a sequence $\sequence {a_n}: \N \to \C$ such that the series:
:$\ds \sum_{n \mathop = 0}^\infty a_n \paren {z - z_0}^n$
converges to $\map f z$ in a neighborhood of $z_0$ in $U$.
\end{definition}