Evaluate $\int_{C} log z~ dz$ where C is $|z|=y.$

Solution:

$I = \int_{C} \log z dz$ -------------------------(1)

Put $z=e^{j \theta}, \gamma = y$,

Differentiating z with respect to $\theta$.

$dz=j e^{j \theta} d \theta$ $\left[ \cfrac{d}{dx} e^{ax}=e^{ax} \cfrac{d}{dx}ax \right]$

and since it is unit circle, $\theta$ varies from $0 \ to \ 2\pi$.

Put z and dz value in …