In Ahlfors' Complex Analysis, there is one section about multiply connected domain. Suppose $\Omega$ is a domain with $n$ holes $A_1, A_2, \cdots, A_n$. $\gamma_i$ is a circle around each $A_i$. Then $\gamma_1, \gamma_2, ..., \gamma_n$ is a homology basis for the region $\Omega$. Let $\gamma$ be some cycle (a sum of closed curves) in $\Omega$. Then $$\int_\gamma fdz = \sum_{i=1}^n c_i\int_{\gamma_i}fdz, $$ where $c_i = n(\gamma_i, a)$ for some $a \in A_i$. The numbers $\int_{\gamma_i} fdz$ are called the peridods of the indefinite integral.
It is said that the integral along an arc from $z_o$ to $z$ is determined up to additive multiples of the periods. What is the meaning of that? Why is that true?
And why is it true that the vanishing of the periods is a necessary and sufficient condition for the existence of a single-valued indefinite integral?