# Conformal properties of complex Schwarz-Christoffel tranformations

Let $a_1< a_2<\dots < a_n$ be $n$ real numbers and assume that
$\beta_1,\dots,\beta_n\in \mathbb R$ are such that $\sum_i \beta_i=n-2$.

In this case, it is well-knonw that the Schwarz-Christoffel map $$U: z\mapsto U(z): =\int^z\frac{dx}{\prod_{i=1}^n (x-a_i)^{1-{\beta_i}}}$$ realizes a conformal isomorphism from the upper half-plane $\lbrace z\in \mathbb C \lvert {\rm Im}(z)>0 \rbrace$ onto a closed $n$-gon in the plane, with interior angles $\beta_1 \pi, \beta_2\pi,\ldots,\beta_n\pi$.

QUESTION: what is known when the $a_i$ are complex numbers?

In this case, the formula above for $U$ still defines a multivalued function on $\mathbb C\setminus \lbrace a_1,\dots,a_n \rbrace$ (ie., defines a holomorphic function on the universal covering of $\mathbb C\setminus \lbrace a_1,\ldots,a_n \rbrace$).

The question is if it is know that $U$ induces a conformal isomorphism between some nice domains.

Thanks for any help!

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