Compute the integral $\int_\gamma log z \,dz$ where you need to select a continuous branch of the logarithm based on the given value of the logarithm at one of the points on $\gamma$:
1- $\gamma$ is the unit circle |$z$|$=1$ oriented counterclockwise and started at $z=1$, and $ log( 1)=0$.
2- $\gamma$ is the unit circle |$z$|$=1$ oriented counterclockwise and started at $z=i$, and $ log( i)=\pi i/2$.
I doubt that my solution is correct because I didn't use the given information of the logarithmic 1 and i.
My attempt:
I use for both cases a cut at the origin point.
Part 1:
$z=e^{i \theta}$
$\int_0^{2\pi} log (e^{i \theta}) i e^{i \theta} \,d\theta$ = $\int_0^{2\pi} {i \theta} i e^{i \theta} \,d\theta$ = - $\int_0^{2\pi} {\theta} e^{i \theta} \,d\theta$ = $2\pi i$
Part 2:
$\int_{\pi/2}^{2\pi} log (e^{i \theta}) i e^{i \theta} \,d\theta$ + $\int_0^{\pi/2} log e^{i \theta + 2i\pi} i e^{i \theta} \,d\theta$