What is the conformal map of a rectangle into a band?

Question

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.

Answer 1

Perhaps $x \mapsto x$ and $y \mapsto \tan \left( \frac{\pi y}{2w} \right) $. This should send $y$ to $\pm \infty$ as $y$ tends to $\pm w$. Is this conformal? Here $w=f(z)$ with $f = u+iv$ has $u = x$ and $v = \tan \left( \frac{\pi y}{2w} \right)$ so $u_x=1$, $u_y=0$ and $v_x=0$ whereas $v_y = \frac{\pi}{2w}\sec^2\left( \frac{\pi y}{2w} \right)$ so apparently this map is not conformal. That said, it might still be interesting for your application.

Answer 2

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 $\ddot\frown$.