Let $D_0$ be the disc centered at $1$ of radius $1$. Let $f_0$ be the restriction of the principal branch of $\sqrt z$ to $D_0$. Let $\gamma(t)=e^{2\pi i t}$ and $\sigma(t)=e^{4\pi i t}$ for $0 \leq t \leq 1$. Find an analytic continuation $\{(f_t, D_t) : 0 \leq t \leq 1\}$ of $(f_0, D_0)$ along $\gamma$ and show that $f_1(1)=-f_0(1)$.
By definition we have $f_0 (z)=e^{\frac{1}{2} {\rm Log}(z)}$ with ${\rm Log}$ the principal branch of $\log$. $\gamma$ goes around the origin once and $\sigma$ goes around the origin twice. I don't understand the first question of finding the analytic continuation. As we go along $\gamma$ what happens to the principal branch of the square root? What is the method for solving such problems? Thx.
EDIT:thank you to all of you for these amazing answers. I never knew how to quite approach this problem. Now you gave me an idea how to do it.