1

Could you please explain how to get the poles of the complex function $f(z) =\frac{z}{1+z^n}$. I am computing the integral $\int_0^\infty \frac{dx}{1+x^n}$ by considering the complex integral $\int_C \frac{dz}{1+z^n}$ around an arc segment enclosing a single pole. In order to complete my answer, I need to find the residue, for which I need to know the location of the poles.

Thank you in advanced

Ryan Maguire
  • 13

1 Answers1

1

The poles of $f$ are the $z$ such that $z^n+1=0$. As $z^n+1=0$ iff $(z e^{i \frac{\pi}{n}})^n =1$ you obtain a rotation of the roots of the unity: $$ z = e^{-i \frac{\pi}{n}+\frac{2 i k \pi}{n}}, k \in \{0,\ldots,n-1\}$$

Delta-u
  • 6,038