Mobius transformation :
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Properties:

      1 . The composition of two Mobius transformations is again a Mobius transformation.
      2 . A Mobius tranfromation can have atmost two fixed points unless $T(z) =z $ for all $z$.

      3 . Let MATH be a Mobius tranformation. Rewrite $T(z)$ as a sequence of transformations in the following: MATH            when $c \neq 0$. If $c = 0$ then MATH. Then, the Mobius transformation $T$ is the composition of            translations, dilations, and the inversion.
           Every Mobius transformation is the composition of translations, dilations, and the inversion.

      4 . Every Mobius transformation is a conformal mapping of the extended complex plane onto itself. (See Pages            62-63, ``Introductory Complex Anlaysis" by R. A. Silverman, Dover, 1972).

      5. Given any three distinct points $z_{1}$, $z_{2}$, $z_{3}$ in the extended complex plane, and any given three distinct values           (points) $w_{1}$, $w_{2}$, $w_{3}$ in the extended complex plane, there is a unique Mobius transformation $w=T(z)$ such            that $w_{1}=T(z_{1})$, $w_{2} = T(z_{2})$, $w_{3} = T(z_{3})$. This unique Mobius tranformation is given by solving for $w$ the            following equation MATH
 
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