Saturday, February 23, 2013

Let's Get Mobius Up In Here

Here is an applet to explore the function $$\phi(z) = \frac{z - \lambda}{1 - z \bar{\lambda}}$$ for $| \lambda | \leq 1$. In the bottom left is the unit disc with Darth Vader super imposed on it. On the top is also the unit disc. For each point $|z| < 1$, the applet determines the color of $\phi(z)$ and and then colors $z$ that color.

The unit disc on the bottom right allows you to adjust $\lambda$. Just click anywhere in the unit disc and $\lambda$ will change appropriately.

Note that this demonstrates that $\phi$ maps the unit disc to itself. Click here for a proof.


  1. The proof I'm a little confused on what your first claim is. I see you've built the inverse and demonstrated it's a right inverse (technically we'd need to show it's also a left inverse, but it's pretty trivial so we can grant it's a true inverse) but then your claim is that it shows the original function only maps into the disc is that right? I believe your argument is sound but that leap in logic is losing me. Maybe I'm just tired.
    The rest seems sound though.

    PS: The app doesn't seem to load for me for some reason. Just shows a P in the upper left corner, which maybe means I should click something but not familiar with it.

    1. The key idea is that $\phi_{-\lambda}:D \rightarrow C$, so $\phi_{-\lambda}^{-1}(D) \subseteq D$. (I'm not sure if LaTeX will render in the comments.)

      The P in the corner just indicates that it's loading, but that shouldn't take more than a few seconds. Is your Java up to date?

    2. Curious. Apparently Firefox doesn't like it either, getting the same loading screen perpetually.

    3. Oh well, thanks for trying.

      I updated the pdf to include your proof. Thanks!