Difference between revisions of "Hilger imaginary part"
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Revision as of 15:45, 22 September 2016
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
