Evaluate $I = \frac {1}{2\pi i} \int_{\gamma_ \rho} \frac {1}{z^n} \, dz$

Question

I'm having some difficulty calculating this integral :

For $\rho > 0$ and $n \in \Bbb N$ calculate the value of the integral

$$I = \frac {1}{2\pi i} \int_{\gamma_ \rho} \frac {1}{z^n} \, dz$$

where $\gamma_ \rho$ is the circle centered at $0$ and radius $\rho$.

In evaluating this integral I tried to use the circle $\lvert z \rvert = \rho$ parametrized by $z(t) = \rho e^{it}$, with $t \in [0, 2\pi]$, then $dz/dt = i\rho e^{it}$ but I'm confused by the power $n$ in the integrand when I have to make the substitution.

Thanks in advance for your help.

Answer 1

$$\int_C \frac 1{z^n} = \int_0^{2\pi} \frac{i\rho e^{it}}{\rho^n e^{itn}}=\frac i{\rho^{n-1}} \int_0^{2\pi} \frac 1{e^{it(n-1)}}$$ Can you take it from there?