Let $\gamma$ be a path in $\mathbb{C}$ such that $\gamma(0)=\gamma(1)=a$ and consider an analytic continuation {$(f_t, D_t): 0\leq t \leq 1$} along the path $\gamma$ such that $[f_1]_a = [f'_0]_a$. We also require that $f'_0$ is non zero. What are the possible solutions for $f_0$?
For any $K \in \mathbb{C}$ then $Ke^z$ is a solution to the problem but are there other functions satisfying this problem? By going along $-\gamma$ are we somehow integrating and getting a primitive? Thx.