Definition:Residue (Complex Analysis)

From ProofWiki
Jump to: navigation, search

Definition

Let $f: \C \to \C$ be a complex function.

Let $z_0 \in U \subset \C$ such that $f$ is analytic in $U \setminus \left\{{z_0}\right\}$.


Then by Existence of Laurent Series, there is a Laurent series:

$\displaystyle \sum_{j \mathop = -\infty}^\infty a_j \left({z - z_0}\right)^j$

such that the sum converges to $f$ in $U - \left\{{z_0}\right\}$.


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 $\operatorname{Res} \left({f, z_0}\right)$ or just $\operatorname{Res}\left({z_0}\right)$ when $f$ is understood.