# Monodromy of "complex Schwarz-Christoffel maps

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!

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+1. However, the function is not well defined on $\mathbb{C}$ minus a set of points. You have to choose branches of logarithms at each point or only consider it as a function on the upper halfplane. Can you elaborate what you mean by "the monodromy is affine" and how this conclusion arises (only for the sake of my education;)? – Marc Palm Apr 19 '13 at 15:52

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