Difference between revisions of "Euler-Cauchy logarithm"

From timescalewiki
Jump to: navigation, search
(Created page with "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^{\Del...")
 
 
(14 intermediate revisions by the same user not shown)
Line 1: Line 1:
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
+
 
$$\left\{\begin{array}{ll}
+
=Properties=
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 />
 +
 
 +
=References=
 +
*{{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 15:15, 21 January 2023

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

Properties

See also

Euler-Cauchy dynamic equation
Jackson logarithm
Mozyrska-Torres logarithm

References