\begin{definition}[Definition:Residue (Complex Analysis)]
Let $f: \C \to \C$ be a complex function.
Let $z_0 \in U \subset \C$ such that $f$ is analytic in $U \setminus \set {z_0}$.
Then by Existence of Laurent Series, there is a Laurent series:
:$\ds \sum_{j \mathop = -\infty}^\infty a_j \paren {z - z_0}^j$
such that the sum converges to $f$ in $U - \set {z_0}$.  
The '''residue''' at a point $z = z_0$ of a function $f: \C \to \C$ is defined as $a_{-1}$ in that Laurent series.
It is denoted $\Res f {z_0}$ or just $\map {\mathrm {Res} } {z_0}$ when $f$ is understood.
Category:Definitions/Complex Analysis
\end{definition}