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?

  • 2
    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

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