1. ## principal value mapping

Sketch an image under $\displaystyle w = \text{Log} (z)$ of the half disk $\displaystyle |z| < 1, \text{Re} (z) > 0$.

To this point, I do not have much experience with mappings under the principal value mapping of the logarithm. I know that

$\displaystyle \text{Log} (z) := \ln | z | + i \cdot \text{Arg} (z)$.

However, I do not see what this image would look like.

2. If you plot $\displaystyle \ln |r|$ in the x-direction and the principal argument $\displaystyle \text{Arg}(z)$ in the y-direction, then for the right half unit circle, won't the argument go from $\displaystyle -\pi/2$ to $\displaystyle \pi/2$? So that's a range in the y direction from $\displaystyle \pi/2$ to $\displaystyle -\pi/2$. Now, the x-direction: when r is very small, then $\displaystyle \ln |r|$ is very big and when r=1, then $\displaystyle \ln |r|=0$. Isn't that then a block from zero to infinity along the x-axis with the upper boundary at $\displaystyle \pi/2$ and lower one at $\displaystyle -\pi/2$? And since $\displaystyle |z|<1$, then it's that block except for the left side at x=0.