I've got a problem to do something. We consider $\alpha(t) = \sin(5t) +i\sin(4t)$, for $t \in [0;2\pi]$. We consider as well : $f_s(z) = \frac{s^4}{z^4-s^4}$. The first question I struggle with is that I don't even succeed to find the set where $f_s$ is continuous. Actually, I know that it's the case for $s^4 \notin \alpha([0;2\pi])$, but the point is to determine $s \in \mathbb{C}$ which verify this property.
So we know that : $z^4=s^4 \iff \exists t\in [0;2\pi] \; (\sin(5t)+i\sin(4t))^4 = s^4 $, but the fact is that I don't know how to transform $\sin(5t)+i\sin(4t)$... I've tried to use trigonometric identities but without success...
Then, I've to determine $\int_{\alpha} f_s(z)dz$, but at the moment I just want to find out where $f_s$ is continuous.
If someone could help me, thank you very much !