Difference between revisions of "Hilger pure imaginary"
From timescalewiki
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=References= | =References= | ||
* {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Hilger imaginary part|next=}}: Definition 2.4 | * {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Hilger imaginary part|next=}}: Definition 2.4 | ||
| + | *{{PaperReference|A generalized Fourier transform and convolution on time scales|2008|Robert J. Marks II|author2=Ian A. Gravagne|author3=John M. Davis|prev=Inverse cylinder transformation|next=}}: $(2.3)$ | ||
Revision as of 16:00, 15 January 2023
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: █
Hilger real part oplus Hilger imaginary part equals z
