Difference between revisions of "Jackson logarithm"
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| − | + | Let $\mathbb{T}$ be a time scale. Let $p \in \mathcal{R}(\mathbb{T},\mathbb{R})$ be [[regressive_function | regressive]]. Let $g \colon \mathbb{T} \rightarrow \mathbb{R}$ be nonvanishing. Define the Jackson logarithm of $g$ by | |
$$\log_{\mathbb{T}}g(t)=\dfrac{g^{\Delta}(t)}{g(t)}.$$ | $$\log_{\mathbb{T}}g(t)=\dfrac{g^{\Delta}(t)}{g(t)}.$$ | ||
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</div> | </div> | ||
</div> | </div> | ||
| + | |||
| + | =See also= | ||
| + | [[Bohner logarithm]]<br /> | ||
| + | [[Euler-Cauchy logarithm]]<br /> | ||
| + | [[Mozyrska-Torres logarithm]]<br /> | ||
=References= | =References= | ||
*{{PaperReference|The time scale logarithm|2008|Billy Jackson|prev=findme|next=findme}}: $(1.1)$ | *{{PaperReference|The time scale logarithm|2008|Billy Jackson|prev=findme|next=findme}}: $(1.1)$ | ||
Revision as of 17:26, 11 February 2017
Let $\mathbb{T}$ be a time scale. Let $p \in \mathcal{R}(\mathbb{T},\mathbb{R})$ be regressive. Let $g \colon \mathbb{T} \rightarrow \mathbb{R}$ be nonvanishing. Define the Jackson logarithm of $g$ by $$\log_{\mathbb{T}}g(t)=\dfrac{g^{\Delta}(t)}{g(t)}.$$
Properties
Theorem: The following formula holds: $$\log_{\mathbb{T}} e_p(t,s) = \dfrac{(e_p(t,s))^{\Delta}}{e_p(t,s)} = p(t).$$
Proof: █
Theorem: For nonvanishing $\Delta$-differentiable functions $f,g$, $$\log_{\mathbb{T}} \dfrac{f(t)}{g(t)} = \log_{\mathbb{T}} f(t) \ominus \log_{\mathbb{T}} g(t).$$
Proof: █
Theorem: If $f$ $\Delta$-differentiable nonvanishing function then $$e_{\log_{\mathbb{T}}f}(t,s)=\dfrac{f(t)}{f(s)}.$$
Proof: █
Theorem: For nonvanishing $\Delta$-differentiable functions $f,g$, $$\log_{\mathbb{T}} f(t)g(t) = \log_{\mathbb{T}} f(t) \oplus \log_{\mathbb{T}} g(t).$$
Proof: █
See also
Bohner logarithm
Euler-Cauchy logarithm
Mozyrska-Torres logarithm
References
- Billy Jackson: The time scale logarithm (2008)... (previous)... (next): $(1.1)$
