Evaluate $\oint_{|z|=2}\frac{\cos(z)}{z^5 -32} dz$
Honestly, I have no idea how to proceed. My professor talked about integral with poles in the interior of the curves so that case was simple. I have never seen this type of integral. Can someone show me how can I proceed? Thanks before!
$$\int_{|z|=2}\frac{\cos(z)}{z^5 -32} dz$$ diverges.
What is well-defined, because $2 e^{2i\pi n/5} $ are simple poles, is $$PV(\int_{|z|=2}\frac{\cos(z)}{z^5 -32} dz)$$ (principal value)
Then the theorem is that it is $$=\frac12(\int_{|z|=2+\epsilon}\frac{\cos(z)}{z^5 -32} dz+\int_{|z|=2-\epsilon}\frac{\cos(z)}{z^5 -32} dz)$$ to which we can apply the residue theorem.