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I'm having hard time figuring out the following sentence in my textbook.

"Two reflections will evidently result in a (fractional) linear transformation"

I'm confused because I don't know if the two reflections are with respect to the same circle or not. If it is the same circle, the composite would be identity and it is a fractional linear transformation. But it sounds a little weird to call it "linear transformation" to me. However, if they are with respect to different circles, it's not "evident" to me so I need some help with it.

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Reflection with respect to the circle $|z-a|=\rho$ takes $a + r e^{i\theta}$ to $a + (\rho^2/r) e^{i\theta} = a + \dfrac{\rho^2}{\overline{r e^{i\theta}}}$, i.e. it is the mapping $R_{a,\rho}(z) = a + \dfrac{\rho^2}{\overline{z - a}}$. Thus with two reflections with two circles we get $$ \eqalign{R_{a_2,\rho_2}(R_{a_1,\rho_1}(z)) &= a_2 + \frac{\rho_2^2}{\overline{R_{a_1,\rho_1}(z)}-a_2}\cr &=a_2 + \dfrac{\rho_2^2}{\overline{a_1 + \frac{\rho_1^2}{\overline{z - a_1}} - a_2}}\cr &= a_2 + \dfrac{\rho_2^2(z - a_1)}{(\overline{a_1} - \overline{a_2})(z - a_1) + \rho_1^2}}$$ which can be written in the form $\dfrac{\alpha z + \beta}{\gamma z + \delta}$ for suitable constants $\alpha, \beta, \gamma, \delta$. That's what's called a fractional linear transformation. I'll let you work out what $\alpha$, $\beta$, $\gamma$, $\delta$ are.

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Thank you for your explanation! Now I know how it turns into a fractional linear transformation, but I'm wondering how I can derive the equation of the mapping $R$? – Tengu Dec 4 '12 at 0:35

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