Difference between revisions of "Bohner logarithm"

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Define the logarithm to be
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Let $\mathbb{T}$ be a [[time scale]] and let $p \colon \mathbb{T} \rightarrow \mathbb{C}$ [[delta derivative|delta differentiable]]. The Bohner logarithm is defined by
$$L_p(t,t_0) = \displaystyle\int_{t_0}^t \dfrac{p^{\Delta}(\tau)}{p(\tau)} \Delta \tau$$
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$$L_p(t,t_0) = \displaystyle\int_{t_0}^t \dfrac{p^{\Delta}(\tau)}{p(\tau)} \Delta \tau.$$
  
Note while
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=Properties=
$$\dfrac{(pq)^{\Delta}}{pq} = \dfrac{p^{\Delta}}{p} \oplus \dfrac{q^{\Delta}}{q},$$
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[[Bohner logarithm sub a product]]<br />
$$\dfrac{(\frac{p}{q})^{\Delta}}{\frac{p}{q}} = \dfrac{p^{\Delta}}{p} \ominus \dfrac{q^{\Delta}}{q},$$
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and
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=See also=
$$\alpha \odot \dfrac{p^{\Delta}}{p} = \dfrac{(p^{\alpha})^{\Delta}}{p^{\alpha}}$$
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[[Euler-Cauchy logarithm]]<br />
the following "bad" formula holds
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[[Jackson logarithm]]<br />
$$L_{pq}(t,t_0)=L_p(t,t_0)+L_q(t,t_0)+\displaystyle\int_{t_0}^t \dfrac{\mu(\tau)p^{\Delta}(\tau)q^{\Delta}(\tau)}{p(\tau)q(\tau)} \Delta \tau.$$
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[[Mozyrska-Torres logarithm]]<br />
  
 
=References=
 
=References=
[http://web.mst.edu/~bohner/papers/tlots.pdf Bohner, Martin. The logarithm on time scales. J. Difference Equ. Appl. 11 (2005), no. 15, 1305--1306.]
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*{{PaperReference|The logarithm on time scales|2005|Martin Bohner|prev=Euler-Cauchy logarithm}}: $(3)$

Latest revision as of 17:02, 11 February 2017

Let $\mathbb{T}$ be a time scale and let $p \colon \mathbb{T} \rightarrow \mathbb{C}$ delta differentiable. The Bohner logarithm is defined by $$L_p(t,t_0) = \displaystyle\int_{t_0}^t \dfrac{p^{\Delta}(\tau)}{p(\tau)} \Delta \tau.$$

Properties

Bohner logarithm sub a product

See also

Euler-Cauchy logarithm
Jackson logarithm
Mozyrska-Torres logarithm

References