I am doing some revision of complex analysis, and am stuck on this question. I am looking for A mobius mapping sending the set {z: |z+1|<$\sqrt{2}$}, |z-1|<$\sqrt2$} onto the sector {z:3pi/4< argz< 5pi<4}.
I thought that a Mobius mapping is uniquely determined by three points, so I was going to consider the points i,0 and -i in the first set, and send those to -1+i, 0, -1-i respectively? But if this is right (and I'm not sure if it is), I can't find values for a Mobius transformation to work.
Any help appreciated.
Hint: Try to find a Möbius transform which maps
$$ i \mapsto 0, ~-i \mapsto \infty, ~0 \mapsto -1$$
Afterwards, check that that this Möbius transform is indeed what you want (e.g. compute the values of $1+\sqrt{2}$ and $1-\sqrt{2}$ and remember that it maps circles and lines to circles and lines)