## 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.