Latest revision as of 15:41, 21 January 2023

Let $h>0$ and let $z \in \mathbb{C}_h$, the Hilger complex plane. The Hilger real part of $z$ is defined by $$\mathrm{Re}_h(z)=\dfrac{|zh+1|-1}{h}.$$

Properties

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

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 =Properties=
-<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
+[[Inequality for Hilger real part]]<br />
-<strong>Theorem:</strong> The following inequality holds for $z \in \mathbb{C}_h$:
+[[Limit of Hilger real and imag parts yields classical]]<br />
-$$-\dfrac{1}{h} < \mathrm{Re}_h(z) < \infty.$$
+[[Hilger real part oplus Hilger imaginary part equals z]]<br />
-<div class="mw-collapsible-content">
-<strong>Proof:</strong> █
-</div>
-</div>
-{{:Limit of Hilger real and imag parts yields classical}}
+=References=
-{{:Hilger real part oplus Hilger imaginary part equals z}}
+* {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Hilger circle|next=Hilger imaginary part}}: Definition 2.3
+[[Category:Definition]]
+<center>{{:Hilger complex plane footer}}</center>