# How do I evaluate the following integral for a complex-valued function?

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?

• the path integral depends only on the endpoints, so you evaluate $\frac{-1}{2(z^2-i)^2}$ between $2i$ and $3$. About holomorphicity: the points where the demoninator is zero are in the circle of radious $1$ centered in the origin. So if your curve is prescribed with $|z(t)|>2$ you are ok. – Baol Oct 11 at 12:24