Difference between revisions of "Hilger imaginary part"
From timescalewiki
(Created page with "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 ...") |
|||
| (7 intermediate revisions by the same user not shown) | |||
| Line 2: | Line 2: | ||
$$\mathrm{Im}_h(z)=\dfrac{\mathrm{Arg}(zh+1)}{h},$$ | $$\mathrm{Im}_h(z)=\dfrac{\mathrm{Arg}(zh+1)}{h},$$ | ||
where $\mathrm{Arg}$ denotes the principal argument of $z$ (i.e. $-\pi < \mathrm{Arg(z)} \leq \pi$). | where $\mathrm{Arg}$ denotes the principal argument of $z$ (i.e. $-\pi < \mathrm{Arg(z)} \leq \pi$). | ||
| + | |||
| + | =Properties= | ||
| + | [[Range of Hilger imaginary part]]<br /> | ||
| + | [[Limit of Hilger real and imag parts yields classical]]<br /> | ||
| + | [[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 real part|next=Hilger pure imaginary}}: Definition 2.3 | ||
| + | |||
| + | [[Category:Definition]] | ||
| + | |||
| + | <center>{{:Hilger complex plane footer}}</center> | ||
Latest revision as of 15:40, 21 January 2023
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
Range of Hilger imaginary part
Limit of Hilger real and imag parts yields classical
Hilger real part oplus Hilger imaginary part equals z
References
- Martin Bohner and Allan Peterson: Dynamic Equations on Time Scales (2001)... (previous)... (next): Definition 2.3
