I would love some help with this question please:
Sketch the image under the function $w = \log z$ of the following set $\{z : Im z > 0\}$
What I have done so far is as follows:
$R= \{x+iy: y>0, x \in {\bf R}\}$
$R= \{re^{iθ}: 0<θ\le \pi,r>0\}$
$f(R)= \{~\log re^{iθ}: 0<θ\le \pi,r>0,\}$
$=\{\log r + iθ: 0<θ\le \pi,r>0\}$
I am unsure if this is correct and if it correct would the sketch be all the area above the x axis from $0$ to $\pi$?
Also would the sketch exclude the origin since $\log(0)$ is not defined and would it also exclude the line of the x-axis since $θ>0$?
I hope that my question is clear and I would really love any help on this.
Many thanks.