1. ## principal value mapping

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

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

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

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

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