# Hilger pure imaginary

From timescalewiki

Let $h>0$ be fixed. The Hilger pure imaginary numbers, $\mathring{\iota} \omega$, where $-\dfrac{\pi}{h} < \omega \leq \dfrac{\pi}{h}$ is defined by the formula $$\mathring{\iota} \omega = \dfrac{e^{2\pi i \omega}-1}{h},$$ where $i=\sqrt{-1}$.

# Properties

**Proposition:** If $z \in \mathbb{C}_h$, the Hilger complex plane, then $\mathring{\iota} \mathrm{Im}_h(z) \in \mathbb{I}_h$, the Hilger circle.

**Proof:** █

**Theorem:** Let $h>0$ be fixed. If $-\dfrac{\pi}{h} < \omega \leq \dfrac{\pi}{h}$, then
$$\left| \mathring{\iota} \omega \right|=\dfrac{4}{h^2} \sin^2 \left( \dfrac{\omega h}{2} \right).$$

**Proof:** █