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?
 A: This only happens if the edges of the square are parallel to the axes.
Let $\alpha = {(1+i) \over \sqrt{2}}$. Then multiplying by $\alpha$ corresponds to rotating by ${\pi \over 4}$. Let $S$ be the square formed from the points $\alpha \{ 0,1,1+i,i \}$, and let $T= \{ e^z | z \in S\}$. This is not a sector of an annulus. 
Here is the evidence 'by picture' obtained by plotting $z \mapsto e^{\alpha z}$ (thanks to Wolfram Alpha):

A: A rectangle in the complex plane with sides parallel to the real and imaginary axes will be mapped under the complex exponential function to a partial sector of an annulus if the height is less than the circumference of a unit circle ($<2\pi$), or to a full annulus if the height is greater.
If the rectangle's sides are not parallel to the axes, the image looks much funkier. The image will always looked like a warped rectangle. However, if one puts a "Cartesian grid" on the inside of the rectangle, the grid will also appear in the image and the curves will remain intersecting at right angles like they did originally. This is because holomorphic maps are conformal (preserve angles).
The exponential map is essentially taking the point $(x,y)$ as interpreted by Cartesian coordinates and mapping it to the point $(r,\theta)=(e^x,y)$ as interpreted by polar coordinates. Therefore, any vertical line segment will be mapped to a circular arc, and any horizontal line segment will be mapped to a ray segment pointing out from the origin. When you consider all four sides of a rectangle at once, they are mapped to the edges of an annular sector.
A: 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!
