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Math Help - Complex Numbers Mapping (DUE TOMORROW)

  1. #1
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    Complex Numbers Mapping (DUE TOMORROW)

    From your knowledge of the map z to z^2, explain how the map z to e^z works. Discuss how to construct an inverse map. Make the domains and ranges clear in your discussion and use simple diagrams to clarify your answer.

    and

    Consider the wedge 0<=|z|<=r, and 0<=arg(z)<=(pie/4) in the z-plane. Find a 1-1 map that takes any point inside (but not on the boundary of) this wedge to a point inside (but not on the boundary of) the infinite horizontal strip in another copy of the z-plane that lies between the horizontal lines y=0 and y=2pie.

    Help!!! For the second question, I am not sure how to parameterise....HELP!
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  2. #2
    Super Member wingless's Avatar
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    Do you know complex mapping? Where are you stuck?
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  3. #3
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    I know the z to z^2 map, but I am not sure how to parameterise e^z to continue. Also, how about the angle?
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  4. #4
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    The function f(z) = z^2 shall map the quater-disk to the semi-disk.
    The function f(z) = (1-z)/(1+z) shall map the the semi-disk to the IV quadrant.
    The function f(z) = iz shall map the IV quadrant to I quadrant.
    The function f(z) = z^2 shall map the I quadrant to the upper-half plane.
    The function f(z) = \log z shall map to the symettric 2\pi-strip.
    The function f(z) = z+\pi i shall map the symettric 2\pi-strip to the desired region.

    Now your job is to compose them.
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