# Hilger pure imaginary

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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:

# References

## Hilger complex plane and friends

$\Huge\mathbb{A}_h$
Hilger alternating axis
$\Huge\mathbb{I}_h$
Hilger circle
$\Huge\mathbb{C}_h$
Hilger complex plane
$\Huge\mathrm{Im}_h$
Hilger imaginary part
$\Huge\mathring{\iota}$
Hilger pure imaginary
$\Huge\mathbb{R}_h$
Hilger real axis
$\Huge\mathrm{Re}_h$
Hilger real part