Definition:Residue (Complex Analysis)

Definition

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

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

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 $f$ 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.

Also see

• Cauchy's Residue Theorem