Difference between revisions of "Euler-Cauchy logarithm"
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This suggests that an analogue to the logarithm could be given by | This suggests that an analogue to the logarithm could be given by | ||
$$L(t,t_0)=\displaystyle\int_{t_0}^t \dfrac{\Delta \tau}{\tau + 2\mu(\tau)}.$$ | $$L(t,t_0)=\displaystyle\int_{t_0}^t \dfrac{\Delta \tau}{\tau + 2\mu(\tau)}.$$ | ||
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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
Let $\mathbb{T}$ be a time scale. Define the Euler-Cauchy logarithm to be part of a solution of the Cauchy-Euler equation $$t\sigma(t)y^{\Delta \Delta}(t) - 3ty^{\Delta}(t)+4y(t)=0$$ whose linearly independent solutions are $$\left\{\begin{array}{ll} y_1(t)&=e_{\frac{2}{t}}(t,t_0) \\ y_2(t)&=e_{\frac{2}{t}}(t,t_0) \displaystyle\int_{t_0}^t \dfrac{\Delta \tau}{\tau + 2\mu(\tau)}. \end{array} \right.$$ This suggests that an analogue to the logarithm could be given by $$L(t,t_0)=\displaystyle\int_{t_0}^t \dfrac{\Delta \tau}{\tau + 2\mu(\tau)}.$$
