Difference between revisions of "Jackson logarithm"

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This definition attempts to define the logarithm as the inverse of an [[exponential_functions | exponential function]]. Let $\mathbb{T}$ be a time scale. Let $p \in \mathcal{R}(\mathbb{T},\mathbb{R})$ be [[regressive_function | regressive]]. Define $F \colon \mathcal{R}(\mathbb{T},\mathbb{R}) \rightarrow C_n^1(\mathbb{T},\mathbb{R})$ by $F(p)=e_p(t,s)$, where $C_n^1$ denotes nonvanishing continuously $\Delta$-differentible functions. Let $g \in C_n^1(\mathbb{T},\mathbb{R})$. Define
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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
 
$$\log_{\mathbb{T}}g(t)=\dfrac{g^{\Delta}(t)}{g(t)}.$$
 
$$\log_{\mathbb{T}}g(t)=\dfrac{g^{\Delta}(t)}{g(t)}.$$
  
 
=Properties=
 
=Properties=
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*$\log_{\mathbb{T}} e_p(t,s) = \dfrac{(e_p(t,s))^{\Delta}}{e_p(t,s)} = p(t)$
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[[Jackson logarithm of delta exponential]]<br />
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*If $f$ $\Delta$-differentiable nonvanishing function then $e_{\log_{\mathbb{T}}f}(t,s)=\dfrac{f(t)}{f(s)}.$
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[[Delta exponential of Jackson logarithm]]<br />
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*For nonvanishing $\Delta$-differentiable functions $f,g$,
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[[Jackson logarithm of a product]]<br />
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$\log_{\mathbb{T}} f(t)g(t) = \log_{\mathbb{T}} f(t) \oplus \log_{\mathbb{T}} g(t).$
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*For nonvanishing $\Delta$-differentiable functions $f,g$,
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=See also=
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$\log_{\mathbb{T}} \dfrac{f(t)}{g(t)} = \log_{\mathbb{T}} f(t) \ominus \log_{\mathbb{T}} g(t).$
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[[Bohner logarithm]]<br />
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[[Euler-Cauchy logarithm]]<br />
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[[Mozyrska-Torres logarithm]]<br />
  
 
=References=
 
=References=
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[http://www.sciencedirect.com/science/article/pii/S0893965907001309 Jackson, Billy. The time scale logarithm. Appl. Math. Lett. 21 (2008), no. 3, 215--221.]
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*{{PaperReference|The time scale logarithm|2008|Billy Jackson|next=Jackson logarithm of delta exponential}}: Definition $1.1$, $(1.1)$

Latest revision as of 17:43, 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

Jackson logarithm of delta exponential
Delta exponential of Jackson logarithm
Jackson logarithm of a product

See also

Bohner logarithm
Euler-Cauchy logarithm
Mozyrska-Torres logarithm

References

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