Finding the Equation of the Image of the Unit Circle Under an Extended Mobius Transformation Using the Inverse Transformation
Given an extended Mobius transformationwe can find the image of the unit circleunderby considering the effect of the inverse transformation on the image of the unit circle.
The inverse transformation isThe inverse transformation will send the image of the unit circle underback to the unit circle. If we denote a point on the image curve by thenso that
Clearing the fractions gives
This is the equation of a generalized circle (a circle or a line (which we can think of as a circle with infinite radius) depending on the values of a, b, c and d.
Example: find the equation of the image of the unit circle under the generalized mobius transformation
The inverse transformation is
Replace z by w and put
Now clear the fractions to giveor
This is the equation of a circle in the complex plane.