I am learning a bit about integration on one-dimensional complex tori. It's exciting stuff, but I have some trouble making things a bit explicit.
Let's consider the elliptic curve $E = \mathbb C/\mathbb Z[i]$. Consider the loop $$\gamma: [0,1]\to E, \quad t\mapsto \frac{1}{2}\exp(2\pi i t) + \mathbb Z[i].$$
I want to integrate some differential forms along this loop.
Let $\omega_n = z^n dz$ for $n=0,1,\ldots$. How to easily compute $$ \int_\gamma \omega_n?$$ I was thinking of simply substituting everything. This would give $$ \int_\gamma \omega_n = \int_{0}^1 z^n dz = \frac{1}{n+1}.$$ Is that correct?
I highly doubt it, because I think I should write $\omega = (x+iy)^n d(x+iy)$ and then proceed. Or not?