Difference between revisions of "Euler-Cauchy logarithm"

From timescalewiki
Jump to: navigation, search
 
(13 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=
 
=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.]
+
*{{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

Retrieved from "http://timescalewiki.org/index.php?title=Euler-Cauchy_logarithm&oldid=2218"