Show that function maps unit cirle onto a line

[zezima1]

Homework Statement

Show that the function f(z) = z/(1-z) maps the unit circle to an infinite line.

Homework Equations

Polar form z = rexp(iθ)

The Attempt at a Solution

I've tried to see what happens, when we let f(z) = f(e^iθ) and then get:

f(e^iθ) = e^iθ/(1-e^iθ)

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?

 

[SammyS]

Homework Statement

Show that the function f(z) = z/(1-z) maps the unit circle to an infinite line.

Homework Equations

Polar form z = rexp(iθ)

The Attempt at a Solution

I've tried to see what happens, when we let f(z) = f(e^iθ) and then get:

f(e^iθ) = e^iθ/(1-e^iθ)

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?

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}\ .

 

[zezima1]

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

f(e^iθ) = (e^iθ-1)/(2-2cos(θ))

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

 

[SammyS]

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

f(e^iθ) = (e^iθ-1)/(2-2cos(θ))

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

Write (e^iθ-1) in terms of sin(θ) & cos(θ) .