Homework Help: Show that function maps unit cirle onto a line

1. May 29, 2012

zezima1

1. The problem statement, all variables and given/known data
Show that the function f(z) = z/(1-z) maps the unit circle to an infinite line.

2. Relevant equations
Polar form z = rexp(iθ)

3. The attempt at a solution
I've tried to see what happens, when we let f(z) = f(e) and then get:

f(e) = e/(1-e)

But I need some help on making this expression more illuminating. I want something for which I can take the real and imaginary part - what tricks can I use?

2. May 29, 2012

SammyS

Staff Emeritus
Multiply the numerator & denominator of $\displaystyle \frac{e^{i\theta}}{1-e^{i\theta}}$ by the complex conjugate of $\displaystyle 1-e^{i\theta}$ which is $\displaystyle 1-e^{-i\theta}\ .$

3. May 29, 2012

zezima1

Thanks, I'm still unsure how to do it though. Multiplying by the conjugate you get:

f(e) = (e-1)/(2-2cos(θ))

How do I show that this is the equation for a straight line?

4. May 29, 2012

SammyS

Staff Emeritus
Write (e-1) in terms of sin(θ) & cos(θ) .