# Through the Looking-Glass II

Just a few illustrations that did not fit in the previous post. A brief recap: given a holomorphic map $f$ of the upper half-plane, we can extend it to the bottom half-plane by setting $f(\bar z)=f'(z)(\bar z-z)$. This extension is not holomorphic unless $f$ is linear; however, it extends $f$ to a global homeomorphism provided that $|f''(z)|\cdot |\bar z-z|<|f'(z)|$. A calculation shows that the power map $f(z)=z^p$ satisfies this inequality whenever $|p-1|<1/2$.

Here are two examples where this inequality is seriously violated: $p=1.8$ and $p=2$. The images of concentric circles $|z|=r$ are shown, the upper half (power map) in green and the lower half (extended map) in red.

When $p=2$ the extension is $f(\bar z)=2|z|^2-2z^2$, so the images of concentric circles are again circles. The plot reminded me of this:

However, here the angle is $2\arcsin (1/2)=60^{\circ}$. Oh well. Maybe I should blog about the Kelvin wake pattern instead of holomorphic maps. The Wikipedia article does not offer much in the way of explanation.

By the way, nothing forces $p$ to be a real number. The Ahlfors extension works for any complex exponent with $|p-1|<1/2$. For example, $1+i/3$:

The plot shows also images of radial segments. In the green territory, the radial segments and circles are mapped to logarithmic spirals. The red part is more complicated. Here is what happens when $|p-1|$ exceeds $1/2$:

Finally, a flower with $p=2+3i$: