Hey I am doing a basic undergraduate course in complex analysis and need some help on Möbius transformations.
When determining the Möbius transformation does it really matter what 3 points I'm choosing to use the for the specifying transformation by 3 point property? The particular question I am trying to do is:
Q. Use part (b) to find the Möbius transformation mapping the crecented-shaped region that lies between $\|z-2\|=1$ and $\|z-1\|=4$ onto the strip $0
$(partb)$ Specialize the cross ratio to the limiting case $w_3=\infty$
From part b my answer was $\frac{(z-z_1)(z_2-z_3)}{(z_2-z_3)(z_2-z_1)}=\frac{w-w_1}{w_2-w_1}$
So my plan was to pick any 3 points of the boundary in the intersection and have them map to $0$, $i$ and $\infty$ and solve the equation above. Is there anything wrong with this method?
Thanks
EDIT: It seems there was a mistake in the set questions =luckily I found the answer online on the fullerton website.