# Hilger imaginary part

From timescalewiki

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:** █

Limit of Hilger real and imag parts yields classical

Hilger real part oplus Hilger imaginary part equals z