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Mobius Transformations

So it says by Lemma D2D which is disk to disk but all we have in notes for this lemma is a graph of what is happening (very helpful) I don't know what the equation is??

I tried to compare the solutions to 26 and 27 to get an answer but the denominator is confusing.
From 26 it looks like w=k((z-z0)/(conjugate(z0)*z-1)) but in 27 it looks like -z0 so I have no clue what the right equation is.
If anyone knows that would be great.
complex.png101.1KB
Original post by CammieInfinity
So it says by Lemma D2D which is disk to disk but all we have in notes for this lemma is a graph of what is happening (very helpful) I don't know what the equation is??

I tried to compare the solutions to 26 and 27 to get an answer but the denominator is confusing.
From 26 it looks like w=k((z-z0)/(conjugate(z0)*z-1)) but in 27 it looks like -z0 so I have no clue what the right equation is.
If anyone knows that would be great.

Could you state Lemma D2D please?
In the notes lemma D2D is basically the unit disc with z0 being some random point being transformed to unit disc with zo at (0,0)
complex.jpg56.4KB
Original post by CammieInfinity
In the notes lemma D2D is basically the unit disc with z0 being some random point being transformed to unit disc with zo at (0,0)

The form of the Moebius transformation you've called D2D, sending w to 0, is x↦x−w−iwx+(1−(1−i)w)x \mapsto \dfrac{x-w}{- i w x+ (1-(1-i)w)}. I got that by knowing that the map is determined by the image of three points, so I worked out the map that would take w to 0, 1 to 1 and i to i.

Does that help?
It's ok I've got it now but many thanks anyway! Much obliged.
Original post by CammieInfinity
It's ok I've got it now but many thanks anyway! Much obliged.

It has been pointed out to me that my answer is nonsense. While it does take the specified three points to the specified three points, it doesn't actually take the disc to the disc.

In fact, the true answer is (w∗−1)(w−z)(w−1)(zw∗−1)\displaystyle \frac{\left(w^*-1\right) (w-z)}{(w-1) \left(z w^*-1\right)} (that's the answer that also sends 1 -> 1; rotate by multiplying by an arbitrary modulus-1 complex number).

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