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Math Help - principal value mapping

  1. #1
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    Mar 2009
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    principal value mapping

    Sketch an image under w = \text{Log} (z) of the half disk |z| < 1, \text{Re} (z) > 0.

    To this point, I do not have much experience with mappings under the principal value mapping of the logarithm. I know that

    \text{Log} (z) := \ln | z | + i \cdot \text{Arg} (z).

    However, I do not see what this image would look like.
    Last edited by chloe561; December 14th 2009 at 04:07 PM.
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  2. #2
    Super Member
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    Aug 2008
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    If you plot \ln |r| in the x-direction and the principal argument \text{Arg}(z) in the y-direction, then for the right half unit circle, won't the argument go from -\pi/2 to \pi/2? So that's a range in the y direction from \pi/2 to -\pi/2. Now, the x-direction: when r is very small, then \ln |r| is very big and when r=1, then \ln |r|=0. Isn't that then a block from zero to infinity along the x-axis with the upper boundary at \pi/2 and lower one at -\pi/2? And since |z|<1, then it's that block except for the left side at x=0.
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