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Show that function maps unit cirle onto a line

  1. May 29, 2012 #1
    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. jcsd
  3. May 29, 2012 #2

    SammyS

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    Multiply the numerator & denominator of [itex]\displaystyle \frac{e^{i\theta}}{1-e^{i\theta}}[/itex] by the complex conjugate of [itex]\displaystyle 1-e^{i\theta}[/itex] which is [itex]\displaystyle 1-e^{-i\theta}\ .[/itex]
     
  4. May 29, 2012 #3
    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?
     
  5. May 29, 2012 #4

    SammyS

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