In the Schwarz-Christoffel mapping, we write the integral as
$$F(z) = \int_{z_0}^z (s-x_1)^{-k_1}\dotsb(s-x_n)^{-k_n} ds$$
How do we choose $z_0$ in this case?
On the wikipedia article, it writes the integral as
$$F(\zeta) = \int^\zeta \left( \frac{K}{(w-a)^{1-\alpha/\pi}(w-b)^{1-\beta/\pi}\dotsb} \right),$$
omitting reference to $z_0$ altogether. What can this mean?
The proof of the Schwarz-Christoffel formula focuses on $F'$, especially its argument. Once we figure out what $F'$ is, the map $F$ is determined up to the infamous additive constant $+C$. The additive constant can be found, alongside other constants involved in the formula ($K$, $x_1,\dots,x_n$) from the requirement that $F(x_j)$ is the $j$th vertex of the polygon. The example given in the linked Wikipedia article illustrates this process.
The notation with $\int_{z_0}^z$ may be convenient when the polygon contains $0$ (possibly on the boundary), because we will choose $z_0$ to be the point that should be mapped to $0$. However, if the target polygon stays away from $0$, writing the map as $\int_{z_0}^z$ is incorrect.