$$\int_\gamma \frac1z \, dz$$
where $\gamma = \{z \in \mathbb C, z= e^{i\theta} , \theta \in [-\pi/2, \pi/2]\}$. I know how to calculate the counter integrals (Last process). However, in this question, I just could not start somewhere.
I could not parameterize it and clarify the boundaries for it. If you can help me I would be very appreciated it.
Just using the definition:
$$\int_{\gamma} \frac{1}{z} dz=\int_{\frac{-\pi}{2}}^\frac{\pi}{2} f(\gamma(\theta))\gamma'(\theta)d\theta=\int_{\frac{-\pi}{2}}^\frac{\pi}{2}\frac{ie^{i\theta}}{e^{i\theta}} d\theta=\int_{\frac{-\pi}{2}}^\frac{\pi}{2}i d\theta=i\pi$$