So, I am given the following task:


$$\int_{+\gamma} \frac{2z}{(z^2 - i)^3} dz$$

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
    $\begingroup$ 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. $\endgroup$ – Baol Oct 11 '18 at 12:24

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.