Mobius Transform :
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Definition: A Mobius transformation is a mapping of the form MATH where $a$, $b$, $c$, $d$ are complex constants satisfying $ad - bc \neq 0$.

Mobius transforations are also called fractional linear transformations (or linear fractional transforations) or bilinear transformations or homographic transformations. Since MATH the condition $ad - bc \neq 0$ guarantees that $T(z)$ is not constant.

Examples:

  1. MATH is a Mobius transformation.

  2. Mobius transformations of the form MATH where $b \neq 0$ are called translations . Under this mapping every point is shifted by the vector corresponding to $b$.

  3. Mobius transformations of the form MATH where MATH are called rotations . Every point is rotated about the origin through the angle $\theta$ under this transformation.
 
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