Calculating $\oint_{\gamma}z^ndz$ for a triangle

Question

I wish to calculate the following integral:

$$\oint_{\gamma}z^ndz$$

$\gamma$ is a triangle with vertices at $0,1,i$, in the positive direction and $n\in \mathbb Z$.

Here is what I think, please correct me if I'm wrong

For $n\ge 0$ $z^n$ is analytic on and inside the triangle, and $\gamma$ is a closed path, so $\oint_{\gamma}z^ndz=0$

For $n<0$ the function is not defined at $z=0$, so I think can't use the same theorem.

I tried calculating it using the following parameterization:

$$\gamma(t)=\begin{cases} \gamma_1 && t\in[0,1] && t \\ \gamma_2 && t\in[0,1] && (1-t) + i\cdot t \\ \gamma_3 && t\in[0,1] && (1-t)i \end{cases}$$

But since I don't know $n$, I don't think I can solve it.

Edit: I think I can use Green theorem, Is that right?

Answer 1

For $n\ge 0$, I was correct.

For $n<0$, the function is not defined at $0$, so this integral is unsolvable.