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 $z_{0}$:

$\lim _{z\to z_{0}}(z-z_{0})f(z)$

(the limit is zero iff the function is holomorphic at $z_{0}$, and is finite nonzero iff it has a simple pole at $z_{0}$).