Difference between revisions of "Cylinder transformation"
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Revision as of 02:14, 2 December 2016
Let $\mathbb{T}$ be a time scale. We define the cylinder transformation $\xi_h \colon \mathbb{C}_h \rightarrow \mathbb{Z}_h$, where $\mathbb{C}_h$ denotes the Hilger complex plane and $\mathbb{Z}_h=\left\{ z \in \mathbb{C} \colon -\dfrac{\pi}{h} < \mathrm{Im}(z) \leq \dfrac{\pi}{h} \right\}$, and is defined by the formula $$\xi_h(z)=\dfrac{1}{h} \mathrm{Log}(1+zh),$$ where $\mathrm{Log}$ denotes the principal logarithm.
