Difference between revisions of "Jackson logarithm"
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Revision as of 17:24, 11 February 2017
This definition attempts to define the logarithm as the inverse of an exponential function. Let $\mathbb{T}$ be a time scale. Let $p \in \mathcal{R}(\mathbb{T},\mathbb{R})$ be 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 $$\log_{\mathbb{T}}g(t)=\dfrac{g^{\Delta}(t)}{g(t)}.$$
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: █
References
- Billy Jackson: The time scale logarithm (2008)... (previous)... (next): $(1.1)$
