I wish to calculate the following integral:
$$\oint_{\gamma}\bar z^ndz$$
$\gamma$ is a triangle with vertices at $0,1,i$, in the positive direction and $n\in \mathbb Z$.
Since $\bar z^n$ is not analytic, I can't say that the integral is 0, so it has to be calculated manually. I tried to use that $\bar z=x-iy$, but I still could not solve this.
Let $\eta(t)=t$, with $t\in[0,1]$. Then\begin{align}\int_\eta\overline z^n\,\mathrm dz&=\int_0^1\overline{\eta(t)}^n\eta't\,\mathrm dt\\&=\int_0^1t^n\,\mathrm dt\\&=\frac1{n+1}.\end{align}Now, do the same thing with the other two sides of the triangle (the side that gois from $1$ to $i$ and the side that goes from $i$ to $0$).