What is the image of y>0 under the map $w= e^{z}$

Question

What is the image of y>0 under the map $w= e^{z}$

It is and entire circunference of radius 1? or only half circunference of radius 1?

I see this video https://www.youtube.com/watch?v=d18M390rMaI

And, in minute 4, he only graphics half of the circunference... And my question is why??? If y>0 is not to supossed to be an entire circunference of radius 1?

Thanks you for help

Answer 1

They only drew the half circumference because they stoped at drawing on the left at $y = \pi$. If there were to keep going upward on the $y$ axis, the point in the right picture would go round and around the unit circle (making a full trip for each interval $y = k 2\pi i$ and $y = (k + 1) 2\pi i$, $k$ an integer)