# Two Complex Variable Problems

See the attachment for the problems.

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- the image of the closed rectangular region...

- principal branch of the square root...

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(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.