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    Two Complex Variable Problems

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    See the attachment for the problems.

    - the image of the closed rectangular region...
    - principal branch of the square root...

    (See attached file for full problem description).

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    Solution Preview

    See attachment.

    Question 4.

    Proof. WE prove the following two things.

    (1) Verify that A->A', B->B', C->C' and D->D'
    In z plane, A<-> z=0, B<->z=1-i, C<->z=1, D<->z=1+i

    So, by mapping w(z)=z^2, we have
    w(A)=0^2=0, w(B)=(1-i)^2=-2i, w(C)=1^2=1 and w(D)=(1+i)^2=2i
    which correspond to A', B', C' and D' in the w-plane, respectively.

    We are done.

    (2) Triangle ABC is mapped to a parabola

    To prove this case, we want to show
    (a) the segment BD: x=1, y ...

    Solution Summary

    This solution is comprised of a detailed explanation to solve the image of the closed rectangular region and the principal branch of the square root.