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- Jan 26, 2012

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**Problem**: A

*fixed point*of a function $f(z)$ is a point $z_0$ satisfying $f(z_0)=z_0$. Show that a Möbius transformation $f(z)$ can have at most two fixed points in the complex plane unless $f(z)\equiv z$.

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**Recall**: A

*Möbius transformation*(also called a

*linear fractional transformation*) is any function of the form $f(z)=\dfrac{az+b}{cz+d}$ with the restriction that $ad\neq bc$ (so that $f(z)$ is not a constant function).

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