Difference between revisions of "Hilger pure imaginary"
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$$\mathring{\iota} \omega = \dfrac{e^{2\pi i \omega}-1}{h},$$ | $$\mathring{\iota} \omega = \dfrac{e^{2\pi i \omega}-1}{h},$$ | ||
where $i=\sqrt{-1}$. | where $i=\sqrt{-1}$. | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Proposition:</strong> If $z \in \mathbb{C}_h$, the [[Hilger complex plane]], then $\mathring{\iota} \mathrm{Im}_h(z) \in \mathbb{I}_h$, the [[Hilger circle]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> Let $h>0$ be fixed. If $-\dfrac{\pi}{h} < \omega \leq \dfrac{\pi}{h}$, then | ||
| + | $$\left| \mathring{\iota} \omega \right|=\dfrac{4}{h^2} \sin^2 \left( \dfrac{\omega h}{2} \right).$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | [[Hilger real part oplus Hilger imaginary part equals z]]<br /> | ||
| + | |||
| + | =References= | ||
| + | * {{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=}}: $(2.3)$ | ||
| + | |||
| + | [[Category:Definition]] | ||
| + | |||
| + | <center>{{:Hilger complex plane footer}}</center> | ||
Latest revision as of 15:40, 21 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
References
- Martin Bohner and Allan Peterson: Dynamic Equations on Time Scales (2001)... (previous): Definition 2.4
- Robert J. Marks II, Ian A. Gravagne and John M. Davis: A generalized Fourier transform and convolution on time scales (2008)... (previous): $(2.3)$
