Question

Describe the image of the circle |z − 3| = 1 under the M¨obius transformation w = f(z) = (z − i)/(z − 4). Be sure to explain why your description is correct.

could someone please help me on this problem?

Answer 1

First lets write .

Now consider the mappings

.

It is easy to see that . Let us call the circle centered at with radius . Note that is not in the domain of the function, so we ignore that point (or define ).

Since is just a shift of units to the right, we must have the circle centered at with radius .

Now we find the image of under the reciprocal function. We can write . Therfore, for a point ,






for .
So the image is the line .

We next want to find the image . Remember that is the multiplication by . Therefore, for we have



And therefore is the line trough and parallel to .

Finally, since is just a horizontal translation of we get that is the line passing trough and parallel to .

The image of under is . In other words, .