Example about uhp iso to unit disc

[Dustinsfl]

I am trying to understand this example:
Let H be the upper half plane. The map
$$
f:z\mapsto\frac{z - i}{z + i}
$$
is an isomorphism of H with the unit disc.

proof:
Let $w=f(z)$ and $z=x+yi$. Then
$$
f(z) = \frac{x + (y-1)i}{x+(y+1)i}.
$$
Since $z\in H$, $y>0$, it follows that $(y-1)^2<(y+1)^2$ whence
$$
x^2+(y-1)^2=|z-i|^2<x^2+(y+1)^2=|z+i|^2
$$
and therefore
$$
|z-i|<|z+i|,
$$
(I understand the above)
so $f$ maps the upper half plane into the unit disc (I don't understand why we can make this statement now? How does the above allow for this?). Since
$$
w=\frac{z-i}{z+i},
$$
we can solve for z in terms of w, because $wz+wi = z-i$, so that
$$
z=-i\frac{w+1}{w-1}.
$$
Write $w=u+iv$. By computing directly the real part of $(w+1)/(w-1)$, and so the imaginary part of
$$
-i\frac{w+1}{w-1}
$$
you will find that this imaginary part is > 0 if $|w| < 1$ (why is this?).
So I computed the imaginary part and obtained
$$
-i\frac{(u+1)(u-1)+v^2}{(u-1)^2+v^2}
$$
Hence the map
$$
h:w\mapsto -i\frac{w+1}{w-1}
$$
sends the unit disc into the upper half plane. Since by construction $f$ and $h$ are inverse to each other, it follows that they are inverse isomorphisms of the upper half plane and the disc.

 

[jvicens]

Thank you for your question. I understand that this example may seem confusing, but let me try to explain it to you in more detail.

First, let's recall the definition of an isomorphism. An isomorphism is a bijective function that preserves structure. In this case, we are looking at an isomorphism between the upper half plane (H) and the unit disc. This means that the function f maps every point in H to a unique point in the unit disc, and vice versa, and it preserves the structure of the upper half plane and the unit disc.

Now, in order to show that f is an isomorphism, we need to show that it satisfies these two properties: bijectivity and structure preservation.

The first part of the proof shows that f is indeed bijective. This is done by showing that for any point z in the upper half plane, there is a unique point w=f(z) in the unit disc. This is done by solving for w in terms of z, as shown in the equation w=(z-i)/(z+i).

The second part of the proof shows that f preserves the structure of the upper half plane and the unit disc. This is done by showing that the map f sends the upper half plane into the unit disc, and the map h sends the unit disc into the upper half plane. This is shown by computing the imaginary part of the function h and showing that it is always positive for points inside the unit disc.

Therefore, we can conclude that f is an isomorphism between the upper half plane and the unit disc, as it is both bijective and preserves structure. I hope this helps clarify the proof for you. Please let me know if you have any further questions.