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!



Monodromy of "complex Schwarz-Christoffel maps
Monodromy is affine in the general case. It is generated by finitely many elliptic transformations of the form $az+b$. Proof: your function satisfies the differential
equation $u^{\prime\prime}=Ru',$ where $R$ is rational. The general solution of this equation
is obtained from a particular solution by the formula $au+b$.


Computing this monodromy explicitly is as difficult as in
the classical case: you need to compute the integrals of your $u$ from $x\_j$ to $x\_k$.


