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I have seen many examples of mapping the vertical half-strip, ie $-\pi/2 \lt x < \pi/2$, $y \gt 0$ to $H$(the upper half-plane) in $\mathbb{C}$ using the transformation $f = \sin z$. Would the corresponding mapping from the horizontal half-strip $0 \lt y \lt \pi$, $x \gt 0$ to $H$ be $f = \cos z$?

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Try working it out. The answer should be no. Notice, in particular, that $\cos{z} = \sin(z + \pi/2)$. –  Christopher A. Wong Nov 9 '12 at 3:24
    
Shift $\pi/2$ to the right then rotate by $-\pi/2$ and then shift $\pi$ upward. Make an equation for that and combine it with your initial half-strip. –  James S. Cook Nov 9 '12 at 3:34
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