I am trying to find the value of:
$$\int_\Gamma \frac{1}{z} dz.$$
where $\Gamma$ is the semi circle in the right hand plane, traversed from $-i$ to $i$ my plan was to use the parameterization $\gamma(t)=\exp(it)$. So $\gamma(t)'=i\exp(it)$. Now using the formula:
$$\int_\Gamma f(z) \, dz= \int_0^1 f(\gamma(t))\gamma(t)' \, dt$$
$$\int_\Gamma \frac{1}{z} dz= \int_0^1 \frac{i\exp(it)}{\exp(it)} \, dt$$ $$\int_\Gamma \frac{1}{z} dz= \int_0^1 i \, dt$$ $$\int_\Gamma \frac{1}{z} \, dz= i.$$
However this doesn't seem right and I am not sure how to verify it! Can anyone confirm? Also does anyone have any tips to check these things like online calculators or intuition because I can't see how complex integration works compared to real integration!
It is almost correct. However, the domain of $\gamma$ should be $\left[-\frac\pi2,\frac\pi2\right]$, since you want to go from $-i$ to $i$. It follows that the value of the integral is $\pi i$.