Calculate $\int_{\gamma} \frac{z^2}{\sin(z^3)} dz $ where $\gamma=\{z \in \mathbb{C} : |z|=\frac{3}{2}\}$

Question

Calculate $$\int_{\gamma} \frac{z^2}{\sin(z^3)} dz $$ where $\gamma=\{z \in \mathbb{C} : |z|=\frac{3}{2}\}$

Can I use the Laurent expansion?

Answer 1

As the tree poles are simple one, we get that

$$Res_{z=0}(f)=\lim_{z\to0}\;zf(z)=\lim_{z\to0}\frac{z^3}{\sin z^3}=1$$

Residue at $\;z=\pm\sqrt[3]\pi\;$:

$$Res_{z=\pm\sqrt[3]\pi}(f)=\lim_{z\to\pm\sqrt[3]\pi}\frac{(z\pm\sqrt[3]\pi)z^2}{\sin z^3}=\lim_{z\to\pm\sqrt[3]\pi}\frac{z^2+2(z\pm\sqrt[3]\pi)z}{3z^2\cos z^3}=\frac{\pi^{2/3}}{3\pi^{2/3}\cos(\pm\pi)}=\pm\frac13$$

I think the above is the easiest way to get the residues, but if you want to do Laurent series go ahead...