Four vertices of rectangle domain are as follows:
$(0,0)$, $(0,k)$, $(k,i)$, $(0,i)$, where $k>0$.
Any assistance that you can provide is greatly appreciated.
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Closely related is this previous Question, Schwarz-Christoffel mapping of the upper half-plane, which describes an inverse mapping, from upper half-plane to rectangle. – hardmath Aug 23 '16 at 03:15
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Seems to me that you may be able to use the Weierstrass $\wp$-function for an appropriate rectangular lattice. Any $\wp$ will map a fundamental domain two-to-one over the whole plane, and I think that with a rectangular lattice you should have $\wp(\bar z)=\overline{\wp(z)}$, meaning that the real line gets mapped to the reals. At this point, I get lost — this is definitely not my meat. – Lubin Aug 23 '16 at 04:47