Suppose that $f : \mathbb{C} \setminus \{0, ±i\} → \mathbb{C}$ is the rational function $$f(z) = {a_{−3}\over z^3} −{a_{−2}\over z^2}+{a_{−1}\over z}+ a_0 + za_1 +{z\over (z^2+1)}.$$ Let $Γ$ denote the positively oriented circle of radius $1/2$ about the origin. Determine the integral of $f(z)$ and express your answer in terms of $\mathrm{Res}(f; 0)$, the residue for $f$ at the pole $z_0 = 0$.
So far: I found that $Γ(t)=(1/2)e^{it}$. I also found that the partial fractions of $z\over (z^2+1)$ are $1\over 2(z+i)$ + $1\over 2(z-i)$ and therefore there are poles at $0,i,-i$.
Because only the pole $0$ is in or on the contour, the integral of $z\over(z^2+1)$ = $0$. Furthermore because $a_0 + za_1$ is entire, this integral is also zero. I'm now stuck with how to finish the problem. Any help greatly appreciated.
If $n \ne -1$, then \begin{align} \int_\Gamma z^n dz &= \int_0^{2\pi} \left(\frac{1}{2}\right)^n e^{in t} \frac{1}{2}ie^{it}dt\\ &=\left(\frac{1}{2}\right)^{n+1}i\left[\frac{1}{i(n+1)}e^{i(n+1)t}\right]_0^{2\pi}\\ &=0. \end{align} Then only $\frac{a_{-1}}{z}$ term remains.