Parametrization of half circle in complex plane I'm looking for a parametrization of the half-circle in the upper plane, going from $-1$ to $1$. Is $$\gamma(t):\begin{cases}[0,\pi]\to\mathbb C\\\t\mapsto-e^{-it}\end{cases}$$ correct?

That is correct. $$ \gamma(t) = -e^{-it} = e^{i(\pi - t)} \quad (0 \le t \le \pi) $$ is an arc on the unit circle with the argument of $\gamma(t)$ decreasing from $\pi$ to zero.