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Please help me draw sketches in the complex plane

  1. Feb 17, 2012 #1
    1. The problem statement, all variables and given/known data
    I do not have specific problem, I am struggling in my complex variables class and I think a large part of it is because I struggle at sketching regions in ℂ.

    For instance let z=x+

    I full understand what |z|< 1 looks like and all that (punctured disk, things in that form)
    but when I am asked to sketch a set such as {z = x+iy : x = y, |z| < 1} then I completely fall apart.

    here is a problem from a textbook,
    Let Ω be the unit disk |z| < 1. Sketch sets Ω and f(Ω) where w=f(z)=z-i. Likewise where

    I understand the unit disk but then after that I fall apart completely.

    Thank you very much
  2. jcsd
  3. Feb 17, 2012 #2
    Ω:|z|<1 is the open unit circular disc, center at the origin.
    f(Ω) for f(z)=z-i, translate Ω to -i. The unit circular disc, center at -i.
    f(Ω) for f(z)=2z+3i, dilation by 2, then translate to 3i. The circular disc with radius 2, center at 3i.
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