Difference between revisions of "Jackson logarithm"
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=Properties= | =Properties= | ||
| − | + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | |
| − | + | <strong>Theorem:</strong> The following formula holds: | |
| − | + | $$\log_{\mathbb{T}} e_p(t,s) = \dfrac{(e_p(t,s))^{\Delta}}{e_p(t,s)} = p(t).$$ | |
| − | + | <div class="mw-collapsible-content"> | |
| − | + | <strong>Proof:</strong> █ | |
| − | $\log_{\mathbb{T}} | + | </div> |
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> For nonvanishing $\Delta$-differentiable functions $f,g$, | ||
| + | $$\log_{\mathbb{T}} \dfrac{f(t)}{g(t)} = \log_{\mathbb{T}} f(t) \ominus \log_{\mathbb{T}} g(t).$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> If $f$ $\Delta$-differentiable nonvanishing function then | ||
| + | $$e_{\log_{\mathbb{T}}f}(t,s)=\dfrac{f(t)}{f(s)}.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> For nonvanishing $\Delta$-differentiable functions $f,g$, | ||
| + | $$\log_{\mathbb{T}} f(t)g(t) = \log_{\mathbb{T}} f(t) \oplus \log_{\mathbb{T}} g(t).$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
=References= | =References= | ||
[http://www.sciencedirect.com/science/article/pii/S0893965907001309 Jackson, Billy. The time scale logarithm. Appl. Math. Lett. 21 (2008), no. 3, 215--221.] | [http://www.sciencedirect.com/science/article/pii/S0893965907001309 Jackson, Billy. The time scale logarithm. Appl. Math. Lett. 21 (2008), no. 3, 215--221.] | ||
Revision as of 22:00, 19 February 2016
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
Jackson, Billy. The time scale logarithm. Appl. Math. Lett. 21 (2008), no. 3, 215--221.
