Does the $e^z$ function always map a square to an annulus on the plane? Does the $e^z$ function always map a square to an annulus on the plane?
I was doing a few examples recently on my paper where I would take a map and then apply the $e^z$ function to it and I would always get some sort of annulus or a sector of an annulus.
Would there be a time when this would not occur?

this always occurs. e to the power of the imaginary part of z gives you an angle and the real part contributes the distance from zero. Applying this to the endpoints of a square/rectangle gives you what you observe.
edit: I'm not so sure what you mean by "take a map" mean (and I cannot comment).
edit2: Oops, I forgot about rectangles not parallel to the axes!Thx to the others for the better answers!
