I am reading through a text on time scale calculus and I came upon a certain transformation whose follow up question I cannot solve. Its defined as follows
For $h >0$, let $\mathbb{Z}_h$ be the strip $$ \mathbb{Z}_h := \biggl\{ z \in \mathbb{C}: \frac{-\pi}{h} < Im(z) \leq \frac{\pi}{h}\biggr\} $$ and for $h = 0$, let $\mathbb{Z}_0 := \mathbb{C}$
Definition: For $h > 0$, we define the cylinder transformation $\xi_h : \mathbb{C}_h \rightarrow \mathbb{Z}_h $ by $$ \xi_h (z) = \frac{1}{h} \log (1 + zh), $$ whew log is the principal logarithm function. For $h = 0$, we define $\xi_0 (z) = z$ for all $z \in \mathbb{C}.$ Where $\mathbb{C}_h := \{z \in \mathbb{C}: z \neq -\frac{1}{h} \}$
We call $\xi _h$ the cylinder transformation because when $h > 0$ we can view $\mathbb{Z}_h$ as a cylinder if we glue the bordering lines $Im(z) = \frac{-\pi}{h}$ and $Im(z) = \frac{\pi}{h}$ of $\mathbb{Z}_h$ together to form a cylinder. We define addition on $\mathbb{Z}_h$ by $$ z+w:=z+w \;\bigl(\text{mod}\frac{2\pi i}{h}\bigr)\; for \; z,w \in \mathbb{Z}_h. $$
The question is: Show that the cylinder transformation $\xi_h$ when $h > 0$ maps open rays emanating from the point -$\frac{1}{h}$ in $\mathbb{C}$ onto horizontal lines on the cylinder $\mathbb{Z}_h$. Also show that circles with center at -$\frac{1}{h}$ are mapped onto vertical lines on the strip $\mathbb{Z}_h$ (circles on the cylinder $\mathbb{Z}_h$).
My attempt was taking an example ray, eg $\{z: zh + 1 > 0 \}$ and try to plug it in the transformation $\xi_h$, but I get stuck and fail to show it maps to horizontal lines on the cylinder. Perhaps a fresh set of eyes might help
Any ray $R_{\theta}, -\pi < \theta \le \pi$ as in the OP is defined by $\arg(z+\frac{1}{h})=\theta$ so we have that $\arg (zh+1)=\arg (h(z+\frac{1}{h}))=\theta$ is constant too as $h>0$.
But this precisely means that $\Im \xi_h (z)=\frac{\theta}{h}, z \in R_{\theta}$ is constant so indeed the image of $R_{\theta}$ is a horizontal line in the strip.
For the circle $C_r$ we have that $|z+\frac{1}{h}|=r$ so $|zh+1|=rh$ or $\Re \xi_h (z)=\frac{\log hr}{h}$ is constant on $C_r$ which means that its image is a vertical line (hence a circle on the cylinder)