Difference between revisions of "Euler-Cauchy logarithm"
From timescalewiki
(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 $$t\sigma(t)y^{\Delta \Delta}(t) - 3ty^{\Del...") |
|||
| (14 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
| − | Let $\mathbb{T}$ be a [[time scale]]. | + | Let $\mathbb{T}$ be a [[time scale]] and let $s \in \mathbb{T}$. The Euler-Cauchy logarithm is defined by the formula |
| − | $$ | + | $$L(t,s)=\displaystyle\int_{s}^t \dfrac{1}{\tau + 2\mu(\tau)} \Delta \tau.$$ |
| − | + | ||
| − | + | =Properties= | |
| − | + | ||
| − | + | =See also= | |
| − | \ | + | [[Euler-Cauchy dynamic equation]]<br /> |
| − | + | [[Jackson logarithm]]<br /> | |
| − | + | [[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
