I have several concentric circles with center (0,0) and one eccentric circle which are inside smallest concentric circles with center (d,0), I need to transform all the circles to concentric circles. Joe Manlove had answered similar question: Explicit formula of Mobius Transformation that maps non intersecting circles to concentric circles
But I don't know why we should do like this. Is there any recommended reference book to read?
Thanks, Tang Laoya
updated @ 04/19/2017: suppose two circles： C1: $|z-d|=r_1$, C2: $|z|=r_2$, where $ 0<d<1$, according to Joe Manlove's answer, we have the following transformation:
$$f(z) = \frac{\frac{z - d}{r_1}+d}{1+d\frac{z-d}{r_1}}=\frac{z-d+r_1d}{r_1+dz-d^2}$$
Now we'll check whether C1 and C2 are concentric circles with center (0,0). For C1, we have $|z-d|=r_1$, $$|f(z)|^2=\frac{(z-d+r_1d)(\overline{z-d}+r_1d)}{(d(z-d)+r_1)(d(\overline{z-d})+r_1)}=\frac{|z-d|^2+2r_1dRe(z-d))+(r_1d)^2}{d^2|z-d|^2+2r_1dRe(z-d)+r_1^2}=1$$
But for C2: $|z|=r_2$, how to prove that $$|f(z)|^2=const$$
Thanks,
Tang Laoya