Evaluate the integral $$ \int_{\gamma} z^{\large \frac{1}{2}} dz \ $$
for the principal branch of $ \ \large z^{\frac{1}{2}} \ $ along the the contour indicated below:
Answer:
Take the principal branch:
$f(z)=\sqrt z=\sqrt e^{\frac{i \theta}{2}} , \ 0 \leq \theta \leq 2 \pi \ $
But how to evaluate the integral?