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Sincethe critical points are exactlyThe holomorphic inverse function theorem shows that $f$ is conformal locally everywhere else.

For $z=x+iy$,Its imaginary part is positive throughout $S$. On the three boundary pieces,These intervals traverse the boundary of the upper half-plane. The inverse branchexists for $\Im \zeta >0$, proving that the image is preciselyThis is the hyperbolic-cosine half-strip map.