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>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> | ||
Revision as of 19:38, 29 December 2015
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
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: █
