Difference between revisions of "Bohner logarithm"

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Define the logarithm to be
Let $\mathbb{T}$ be a [[time scale]] and let $p \mathbb{T} \rightarrow \mathbb{C}$ [[delta derivative|delta differentiable]]. The Bohner logarithm is defined by
$$L_p(t,t_0) = \displaystyle\int_{t_0}^t \dfrac{p^{\Delta}(\tau)}{p(\tau)} \Delta \tau$$
$$L_p(t,t_0) = \displaystyle\int_{t_0}^t \dfrac{p^{\Delta}(\tau)}{p(\tau)} \Delta \tau.$$
Note while
$$\dfrac{(pq)^{\Delta}}{pq} = \dfrac{p^{\Delta}}{p} \oplus \dfrac{q^{\Delta}}{q},$$
[[Bohner logarithm sub a product]]<br />
$$\dfrac{(\frac{p}{q})^{\Delta}}{\frac{p}{q}} = \dfrac{p^{\Delta}}{p} \ominus \dfrac{q^{\Delta}}{q},$$
$$\alpha \odot \dfrac{p^{\Delta}}{p} = \dfrac{(p^{\alpha})^{\Delta}}{p^{\alpha}}$$
the following "bad" formula holds
$$L_{pq}(t,t_0)=L_p(t,t_0)+L_q(t,t_0)+\displaystyle\int_{t_0}^t \dfrac{\mu(\tau)p^{\Delta}(\tau)q^{\Delta}(\tau)}{p(\tau)q(\tau)} \Delta \tau.$$
[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.]
{{PaperReference|The logarithm on time scales|2005|Martin Bohner|prev=findme|next=findme}}: (3)

Revision as of 22:49, 10 February 2017

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


Bohner logarithm sub a product


Martin Bohner: The logarithm on time scales (2005)... (previous)... (next): (3)