Hilger imaginary part

Let $h>0$ and let $z \in \mathbb{C}_h$, the Hilger complex plane. The Hilger imaginary part of $z$ is defined by $$\mathrm{Im}_h(z)=\dfrac{\mathrm{Arg}(zh+1)}{h},$$ where $\mathrm{Arg}$ denotes the principal argument of $z$ (i.e. $-\pi < \mathrm{Arg(z)} \leq \pi$).

Properties

Theorem: The following inequality holds for $z \in \mathbb{C}_h$: $$-\dfrac{\pi}{h} < \mathrm{Im}_h(z) \leq \dfrac{\pi}{h}.$$

Proof: █

Theorem

The following formula holds: $$\displaystyle\lim_{h \rightarrow 0^+} \left[ \mathrm{Re}_h(z) + \mathring{\iota} \mathrm{Im}_h(z) \right] = \mathrm{Re}(z) + i \mathrm{Im}(z),$$ where $\mathrm{Re}_h$ denotes the Hilger real part of $z$ and $\mathrm{Im}_h$ denotes the Hilger imaginary part of $z$.

Theorem

The following formula holds: $$z = \mathrm{Re}_h(z) \oplus_h \mathring{\iota} \mathrm{Im}_h(z),$$ where $\mathrm{Re}_h$ denotes the Hilger real part of $z$, $\mathrm{Im}_h$ denotes the Hilger imaginary part of $z$, $\oplus_h$ denotes the circle plus operation, and $\mathring{\iota}$ denotes the Hilger pure imaginary.