# Thread: Complex Numbers Mapping (DUE TOMORROW)

1. ## 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!

2. Do you know complex mapping? Where are you stuck?

3. 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?

4. 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.