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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 $0<Im(w)<1$

$(partb)$ 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?


EDIT: It seems there was a mistake in the set questions =luckily I found the answer online on the fullerton website.

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it appears that in your answer to part (b) you still have $w_3$ finite, whereas it should really go out of the ratio – Valentin Jun 4 '12 at 11:04
Oops Thanks fixed! – Pk.yd Jun 4 '12 at 11:29

A Möbius transformation is determined by any three points, so if you know where three points should go, you can determine the transformation!

The trick is knowing where three points should go.

Möbius transformations map circles to circles (a line is just a circle through $\infty$), so that tells you a lot about where points could possibly go.

For example, the point where your two circles intersect must map to the point where the edges of the strip intersect.

Any Möbius transformation that sends the point where the circles touch to $\infty$ will map your pair of circles that intersect once to a pair of lines that intersect once (at $\infty$) -- i.e. to a pair of parallel lines.

So, you have to use your two remaining choices to ensure those parallel lines go where you want them....

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Hmm so am I correct when I say the boundary of the $\|z-2\|=1$ will map to $\infty$ so I can say for example $1->\infty$ and the boundary of the outer circle say I can make $5->i$ and $-3->0$ (map them to some boundary of the strip) to get a formula for $w$ in terms of $z$? – Pk.yd Jun 4 '12 at 13:18

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