# Inversion mapping of z

1. Jan 18, 2012

1. The problem statement, all variables and given/known data
Show that the inversion mapping w = f(z) = 1/z maps the circle |z - 1| = 1 onto the vertical line x = 1/2.

2. Relevant equations

3. The attempt at a solution
z = a + ib
w = x + yi = a^2/(a^2 + b^2) + ib^2/(a^2 + b^2)
|z - 1| = |(a -1) + ib | = 1
(a - 1)^2 + b^2 = 1
a^2 + b^2 = 2ab

Not sure what to do afterwards.

2. Jan 18, 2012

### A. Bahat

Hint: The last equation is wrong. It should read $a^2+b^2=2a$, which makes your job a whole lot easier.

3. Jan 18, 2012