Difference between revisions of "Euler-Cauchy logarithm"

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Let $\mathbb{T}$ be a [[time scale]]. Define the Euler-Cauchy logarithm to be part of a solution of the [[Cauchy-Euler equation]]
Let $\mathbb{T}$ be a [[time scale]] and let $s \in \mathbb{T}$. The Euler-Cauchy logarithm is defined by the formula
$$t\sigma(t)y^{\Delta \Delta}(t) - 3ty^{\Delta}(t)+4y(t)=0$$
$$L(t,s)=\displaystyle\int_{s}^t \dfrac{1}{\tau + 2\mu(\tau)} \Delta \tau.$$
whose linearly independent solutions are
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)}.
=See also=
\end{array} \right.$$
[[Euler-Cauchy dynamic equation]]<br />
This suggests that an analogue to the logarithm could be given by
[[Jackson logarithm]]<br />
$$L(t,t_0)=\displaystyle\int_{t_0}^t \dfrac{\Delta \tau}{\tau + 2\mu(\tau)}.$$
[[Mozyrska-Torres logarithm]]<br />
[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=Delta exponential dynamic equation|next=Bohner logarithm}}: $(2)$
{{PaperReference|The Natural Logarithm on Time Scales|2008|Dorota Mozyrska|author2 = Delfim F. M. Torres|prev=Mozyrska-Torres logarithm composed with forward jump|next=findme}}

Latest revision as of 15:30, 21 October 2017

Let $\mathbb{T}$ be a time scale and let $s \in \mathbb{T}$. The Euler-Cauchy logarithm is defined by the formula $$L(t,s)=\displaystyle\int_{s}^t \dfrac{1}{\tau + 2\mu(\tau)} \Delta \tau.$$


See also

Euler-Cauchy dynamic equation
Jackson logarithm
Mozyrska-Torres logarithm


Dorota Mozyrska and Delfim F. M. Torres: The Natural Logarithm on Time Scales (2008)... (previous)... (next)