I'm really struggling to visualise why certain conformal mappings give me these new geometries. In particular, I'm struggling to understand the figure I attached above: the author of the figure claims that the following mapping was used: $$w=\log z$$
I tried to understand this by letting $z=re^{i\theta}$ and so $w=\log r + j\theta$. This gives me for $w=u+jv$ that $u=\log r$ and $\theta = v$. Now how do I go from the $w$-plane to the $z$-plane?
For the left boundary on the $w$ plane, I have $u=0$, which gives me $r=1$, or a unit circle. But since $v\in[0,\pi]$, I only get a half unit circle centred at the origin. I'm guessing this corresponds to the "Insulation" label in the $z$-diagram.
For the bottom boundary, I have $v=0$ and so $\theta=0$, meaning I'm somewhere on the positive axis. Since $u\in[0,-\infty]$, I'm not sure what to do now since $u=\log r$. What am I missing?