# Conformal properties of complex Schwarz-Christoffel maps

Let $a_1< a_2 < \cdots < a_n$ be $n$ real numbers assume that $\beta_1,\ldots,\beta_n\in \mathbb R$ 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 $\{ z\in \mathbb C \lvert {\rm Im}(z)>0 \}$ onto a closed $n$-gon in the plane, with interior angles $\beta_1 \pi, \beta_2\pi,\ldots,\beta_n\pi$.

When the $a_i$ are complex numbers, the formula above for $U$ still defines a multivalued function on $\mathbb C\setminus \{a_1,\ldots,a_n \}$ (ie. defines a holomorphic function on the universal covering of $\mathbb C\setminus \{a_1,\ldots,a_n \}$).

Is it known if $U$ induces a conformal isomorphism between some `nice big domains'?

More generally, I'm unable to find any reference where such complex Schwarz-Christoffel map have been considered. Does anyone know some?

Thanks for any help!

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