There are a few nice "recipes" for calculating Residues

1. Let $f:\mathbb{C}-\{0\}\to \mathbb{C}$ be given by $f(z)=\frac{e^z-1}{z^{10}}$. Then $\underset{z=0}{Res}f(z)=\frac{1}{9!}$
2. Let $f:U\to \mathbb{C}$ be holomorphic, and $z_0$ a pole of degree $m$. Suppose there exists $\varphi :B_r(z_0)\to \mathbb{C}$ holomorphic such that on $B_r(z_0)-\{z_0\}$, we have that $f(z)=\frac{\varphi (z)}{(z-z_0{)}^m}$. Then we have that $\underset{z=z_0}{res}(f(z))=\frac{{\varphi }^{(m-1)}(z_0)}{(m-1)!}$
3. Suppose $f:U\to \mathbb{C}$ is holomorphic, and $f(z)=\frac{1}{g(z)}$ for $z_0$ asimple zero of $g$ ($g$ is also holomorphic). Then we have that $\underset{z=z_0}{Res}f(z)=\frac{1}{g^{\prime }(z_0)}$.