So, I am given the following task:
Compute:
$$\int_{+\gamma} \frac{2z}{(z^2 - i)^3} dz$$
when $+ \gamma$ is any curve $z = z(t)$ in $\mathbb{C}$ with $|z(t)| > 2$ for every $t$, with start point at $3$ and endpoint at $2i$.
The only way I can think of evaluating such an integral is to simply use the fundamental theorem of calculus in Complex Variable.
In other words, I can notice that $$\frac{d}{dz}\frac{-1}{2(z^2-i)^2} = \frac{2z}{(z^2 - i)^3}$$
However, I am unsure if $\frac{-1}{2(z^2-i)^2}$ is holomorphic in the set on which $+\gamma$ is defined. Is there another way to tackle this problem that perhaps I'm not thinking of?