find this integral $\int_{|z|=10} \frac{dz}{z^5+z+1}$

Question

Good time of day

I try to compute this integral $\displaystyle \int_{|z|=10} \frac{dz}{z^5+z+1}$

Can you help me please

My attempt is the following

I try to compute using residues. There are no poles. And consider residue at $\infty$. $\operatorname{Res}_{z=\infty}f(z)=0$

And as a result this integral is equal zero

I'm correct? or it's not true. I'm not sure about it.

Answer 1

It is false that there are no poles. There are always poles when you have a rational function which is not polynomial. However, it is true that if $|z|\geqslant2$, then $|z^5+z+1|\geqslant29>0$. Therefore, by Cauchy's theorem,$$\int_{|z|=R}\frac{\mathrm dz}{z^5+z+1}$$is independent of the choice of $R$, as long as $R\geqslant2$. So, since$$\lim_{R\to\infty}\int_{|z|=R}\frac{\mathrm dz}{z^5+z+1}=0,$$your integral is indeed $0$.