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Math Help - Elementary Transformations from z to w plane

  1. #1
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    Elementary Transformations from z to w plane

    I'm stuck on this question:
    For the transformation:
    w=(z+i)/(z-i)
    show that as z moves along the real axis, w moves along a circle centre O and radius 1.

    I know that since z only moves on the real axis, it has only an x component, so I can replace z with x, but I'm stuck after that. Do I rewrite w as u+iv? If so I get a very complicated equation which doesn't look like the equation of a circle. I'd be grateful of any help.
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  2. #2
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    free_to_fly,

    Plug in x for z and multiply (numerator and denominator) by the conjugate of the denominator, which is x + i). Then separate into real and imaginary parts, and take the modulus of the expression. It turns out the modulus of w is 1, (i.e. ||w|| = 1). Thus the image (in the w plane) of the real axis under the mapping defined by w = (z+i)/(z-i) is the unit circle.

    Best,

    m_s_d
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  3. #3
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    Quote Originally Posted by free_to_fly View Post
    I'm stuck on this question:
    For the transformation:
    w=(z+i)/(z-i)
    show that as z moves along the real axis, w moves along a circle centre O and radius 1.
    Let z=t then w=(t+i)/(t-i) = (t^2-1)/(t^2+1)+i(2t)/(t^2+1) this is a parametric equation for almost a circle. Because we never reach 1+0i.
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