Exercise: Find the Taylor series of the function $\log(z)$ on the disk $|z-1|<1$ of his principal branch. Then, continue analytically the function along the curve $\gamma:z(t)=e^{it},\, 0\leq t \leq 2\pi$.
My attempt: I've already find the Taylor series of $\log(z)$, which is
$$ \log(z) = \sum_{n=1}^{\infty} \frac{f^{(n)}(1)}{n!}(z-1)^n = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{(z-1)^n}{n}, \, |z-1|<1. $$
Now, i've been looking all the theory about analytic continuation along curves but still don't understand, i've never seen any example at all. I know that if
$$D_1=\{z\in \mathbb{C}:|z-1|=1\}, \quad D_0=\{z\in \mathbb{C}:|z|=1\} $$
the curve $\gamma$ has the intersection point $a=(1/2,\sqrt{3}/2)$ of $D_0,D_1$. Now, i can try to find an analytic function $f_1$ on the disk $D_{a}$ centered at $a$ of radius $1/2$ such that $f_1$ is analytic and $f_1 \equiv log(z)$ on $D_0 \cap D_a$. Should i do this with every circle centered on a point of $\gamma$? I still can't understand the process of this type of exercises (then, i've to do the same with the function $\sqrt{z}$).
Thank you for your time