Riemann Mapping question

I'm having trouble getting my teeth into this problem.

Let f be the riemann mapping from the square with vertices 1+i, 1-i,
-1+i, -1-i
to the unit disk, determined by f(0)=0 and f ' (0)>0.

I would like to prove that this is a meromorphic function, and would
also like to find the poles and zeros of this map.

1) I can see that using the Riemann mapping theorem I can get a unique
map from the square to the circle, and that by applying schwarz
reflections (and showing they agree on the overlap) that I can get the
squares to cover the complex plane, which I think obviously (well
visually I can see this!) shows that this set of circles also cover the
complex plane.
The problem is, I'm uncertain whether this is showing that f DOES
actually extend to a meromorphic function

2) With the aid of a diagram I can see that the zeros of the map are
images of 0.
im pretty sure that not all the maps of 0 (2a+2b*i where a,b element of
the integers) are zeros, but im confused to how to progress with this
question.

any help would be much appreciated

.

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