Let:

-- $x_1,\ldots,x_n$ be $n$ distinct points on the complex plane $\mathbb C$.

-- $r_1,\ldots,r_n$ be $n$ real numbers.

Consider the map $$ z\mapsto u(z)=\int^z \frac{1}{(x-x_1)^{r_1}\cdots (x-x_n)^{r_n}} d\xi $$

It defines a multivalued holomorphic function on $\mathbb C\setminus (x_1,\ldots,x_n)$

**Question: what is the monodromy of this map?**

Remark: when the $x_i$'s are real and if the $r_i$'s verify $\sum_i r_i=2$, the map $u$ is nothing but a classical Schwarz-Chritoffel map that maps conformally the upper half-plane onto a closed $n$-gon.

In this classical case, it is well known that the monodromy is affine. But I have been unable to find any reference where the monodromy of $u$ is explicited.

Any help would be welcome!