What is the conformal map of a rectangle into a band? What is the conformal map from the rectangle to the "band" (space between two parallel lines) in the complex plane? By the Riemann mapping theorem, such a map exists.
We can define a rectangle as $\{a+bi: a \in (-1,1),b\in(-w,w)\}$ for some $w \in \mathbb R_+$, and the "band" as $\{a+bi: a \in (-1,1), b \in (-\infty,\infty)\}$.
Note in particular that the function should map the rectangles vertical axis to the bands vertical axis (i.e., it should map $\{0\} \times (-w,w)$ to $\{0\} \times (-\infty,\infty)$).
(The reason I want such a map is that I want to transform the band model of Circle Limit III into a desktop wallpaper.)
EDIT: I found this article claiming (on page 4) that you can map from a rectangle to a disk of the form
$$w = \frac{1 + i\sqrt k sn(\frac 1 \alpha (z + ib))}{1 + \sqrt k sn(\frac 1 \alpha (z + ib))}$$
where $z$ is a point the rectangle $[-a,a] \times [0,2b]$ and $w$ is a point in the unit circle ($\alpha = \frac K a$, $K$ is a quarter of the real period of $sn$, and $k$ is parameter drawn from $[0,1]$). It is trivial then to map the disk into the band.
The problem is that I looked up the $sn$ function, and it has two arguments, but in the equation above, only one argument is being given.
EDIT: $sn$ does not preserve symmetry.

Here's just a sketch of an argument. A suitable elliptic function will map the interior of the rectangle conformally and bijectively to the upper half plane $H$. The principal logarithm will map $H$ conformally and bijectively to a strip $\{x+yi:0<y<\pi\}$.
OK, I am skimping on details of the elliptic function. Roughly speaking
you take a fundamental region built of four copies of the given rectangle.
Apply the Weierstrass $\wp$-function. Avoiding half-period points, and using symmetry should give you what you want. I don't have the time or
inclination to check this fully right now $\ddot\frown$.
