Difference between revisions of "Bohner logarithm"
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(Created page with "Define the logarithm to be $$L_p(t,t_0) = \displaystyle\int_{t_0}^t \dfrac{p^{\Delta}(\tau)}{p(\tau)} \Delta \tau$$ Note while $$\dfrac{(pq)^{\Delta}}{pq} = \dfrac{p^{\Delta}...") |
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the following "bad" formula holds | the following "bad" formula holds | ||
$$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.$$ | $$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.$$ | ||
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| + | =References= | ||
| + | [http://web.mst.edu/~bohner/papers/tlots.pdf Bohner, Martin. The logarithm on time scales. J. Difference Equ. Appl. 11 (2005), no. 15, 1305--1306.] | ||
Revision as of 00:22, 22 May 2015
Define the logarithm to be $$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},$$ $$\dfrac{(\frac{p}{q})^{\Delta}}{\frac{p}{q}} = \dfrac{p^{\Delta}}{p} \ominus \dfrac{q^{\Delta}}{q},$$ and $$\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.$$
