I'm taking a look at Ahlfors's Complex Analysis, Third Edition. In Section 2.3 "Mapping on a Rectangle", the author talks about how to extend a conformal map from a rectangle to the upper half plane to a meromorphic function on the entire plane.
This is what Ahlfors does there: The original conformal map $f$ from a fixed rectangle $R$ from the upper half plane $\mathbf H$ is given by a Schwarz-Christoffel formula. Ahlfors points out that $f$ has one singularity (corresponding to $\infty$) somewhere on the boundary of $R$. He then extends the map to the whole plane by applying Schwarz reflection principle on each edge of $R$. Lastly he claims to have a meromorphic map on the whole plane.
But I believe there are some subtleties here. First, it is not obvious that the resulted function on the plane does not have essential isolated singularities. Since the singularity mentioned above is on the boundary of a line segment where the functions are "glued" together, I don't see how the big function behaves around this singularity.
Second, there is a problem of where to put the singularity. Ahlfors put it in the middle of one of the edges, but it seems to me that it complicates the gluing process along the edge. One can put it at one of the vertices, but I fear it might complicate the reasoning about the behavior of the singularity, because the vertices are where the integrand of the S-C formula can go wild, i.e., in the domain of the integrand there should be some slit from the image of the vertices downwards to pick some branches of the integrand.
There exist a related question Schwarz-Christoffel mapping of the upper half-plane here, but I am not sure whether or not my question is identical to the question asked or answered there.
How can I address these subtleties?
Best Answer
If the singularity were essential after the reflecting and gluing, then the image of a (punctured) neighborhood of it would be dense in $\mathbb C$. But the images of the restrictions of this neighborhood to the adjacent rectangle copies are all bounded away from a suitable finite point.