# Find image under map of complex numbers

I need to find the image of the square $S={z; 0\le Re(z)\le1; 1\le Im(z)\le2}$under the function $f(z)=e^{ipz}$ where z is a complex number. I'm not sure what the variable p represents. Could it be the argument of z? Or some other convention i'm unaware of?

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My best guess would be that $p$ stands for $\pi$, but of course it's hard to know for sure without context. In any event, welcome to Math SE! It's usually better to explain what you know about a given problem and what means you've employed to solve it, so that we both know how to best assist and that you've given the problem due consideration (you'll find that you get better responses that way). Also, just in case this is homework, we've got a special tag for that. – Jonathan Y. Sep 22 '13 at 18:45

If $p=\pi$, then $f(S)$ is the upper half of the annulus between the circles of radius $\mathrm e^{-2\pi}$ and $\mathrm e^{-\pi}$ centered at $(0,0)$. Thus, $f(S)$ is delimited by the half-circles $[x^2+y^2=\mathrm e^{-2\pi},y\geqslant0]$ and $[x^2+y^2=\mathrm e^{-4\pi},y\geqslant0]$ and the horizontal segments $[\mathrm e^{-2\pi},\mathrm e^{-\pi}]\times\{0\}$ and $[-\mathrm e^{-\pi},-\mathrm e^{-2\pi}]\times\{0\}$.
I presume $p$ is some positive constant. Hint: note that the images of horizontal line segments are line segments, and the images of vertical line segments are circular arcs.