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Properties of the modulus - complex variables question

  1. Sep 21, 2009 #1
    f(z) = (z+1) / (z-1) for z not equal to 1

    My teacher wrote
    |f(z)| = |x+1+iy| / |x-1+iy| = sqrt((x+1)^2 +1) / sqrt((x-1)^2 + 1)

    How do the values within the modulus work out to the right hand side? I can't figure it out.
  2. jcsd
  3. Sep 21, 2009 #2
    That's the definition of the modulus. You square the real part and then you square the imaginary part and add the two together and finally you take the square root. But I think there is an error since the imaginary part of z = x + iy is y, not 1.
  4. Sep 21, 2009 #3
    [tex]f(z) = \frac{z+1}{z-1}[/tex]

    [tex]|f(x+iy)| = \frac{|x+iy+1|}{|x+iy-1|}[/tex]

    [tex] = \frac{|x+1+iy|}{|x-1+iy|}[/tex]
    [tex] = \frac{\sqrt{(x+1)^2+(iy)^2}}{\sqrt{(x-1)^2+(iy)^2}}[/tex]
    [tex] = \frac{\sqrt{(x^2+2x+1)+(-y^2)}}{\sqrt{(x^2-2x+1)+(-y^2)}}[/tex]
    [tex] = \frac{\sqrt{(x^2+2x+1)+(-y^2)}}{\sqrt{(x^2-2x+1)+(-y^2)}}[/tex]

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