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Since linear fractional transformations are compositions of translations, mutliplications by a constant and inversion, I tried proving that an LTF would transform circles into circles by writing the equation of a circle as $z_0 + r\cdot\exp ( i \cdot \theta )$ , $0 \le \theta \le 2\pi$ and then looking at the effect of translations, multiplications by a constant and inversion.

The first two clearly leave me with a circle, as does inversion of a circle centered around the origin, but I can't get this to work for a circle centered at an arbitrary point $z_0$.

On the other hand, if I write down the general equation of a circle: $$A(x^2 + y^2) + Bx +Cy +D = 0$$

...and write $w = u+iv = \frac{1}{z} = \frac{1}{(x+iy)}$ and make the appropriate substitutions, I get the equation of circle in terms of $u$ and $v$. Is it possible to write $$\frac{1}{z_0 + re^{i\theta}}$$ where $0 \le \theta \le 2\pi$, in the form $$w_0 + r'e^{i\phi}$$ with $0 \le \phi \le 2\pi$?

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    Please use LaTeX to write mathematics in this site. You can go to the FAQ section and search there.2012-12-14
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    I realize LaTeX can be intimidating when you need to use it the first couple of times. Thus, I've just suggested an edit on your post that added the necessary LaTeX. If you look over it, you may find some commands you find useful.2012-12-14
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    @anorton: I suspect that the 2nd last equation should be $\frac{1}{z_0 + r e^{i \theta}}$.2012-12-14
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    @copper.hat I really have no idea (I haven't had complex analysis). I'll suggest the change... EDIT: Oops. I don't have the rep to suggest that trivial of a change. Never mind.2012-12-14

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