Residue of function at point

Definition

Suppose $U\subset \mathbb{C}$ is an open subset and $f:U\to \mathbb{C}$ be a holomorphic function. Suppose $z_0$ is a point in $\mathbb{C}\setminus U$ such that there exists an open neighborhood $V\ni z_0$ such that $V\setminus z_0\subset U$ (in other words, $z_0$ is an isolated singularity of $f$). The residue of $f$ at $z_0$ is defined in the following equivalent ways:

• It is the coefficient of $1/(z-z_0)$ in the Laurent series expansion of $f$ about $z_0$
• It is given by the following formula, where $\gamma$ is a small counter-clockwise circular loop about $z_0$ that lies completely inside $U$:

$res(f;z_0):=\frac{1}{2\pi i}{\oint }_{\gamma }f(z)dz$

If the following limit is a finite complex number, then that complex number equals the residue at $z_0$:

$\underset{z\to z_0}{lim}(z-z_0)f(z)$

(the limit is zero iff the function is holomorphic at $z_0$, and is finite nonzero iff it has a simple pole at $z_0$).