Evaluate $\int_C{\frac{\sin z}{z^2(z-2)}}dz$ where $C$ is given by $x^2+xy+y^2=1$.
Okay so I know that you can do this one of two ways, either using General Cauchy integration and or Residue Theorem. I also know that this integral has two discontinuities at $z=0$ and $z=2$ however 2 falls outside the contour. After which, I'm not really familiar on how to apply either theorem.
We only care about $z=0$ since $z=2$ falls outside this contour. This is a first-order (simple) pole since $\frac{\sin z}{z}$ has a removable singularity. Hence, the residue is
$$ \operatorname*{Res}_{z=0}\big(f(z)\big) = \lim_{z\to 0} z\ f(z) = \lim_{z\to 0} \frac{\sin z}{z(z-2)} = -\frac12 $$