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]]
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Let $\mathbb{T}$ be a [[time scale]] and let $s \in \mathbb{T}$. The Euler-Cauchy logarithm is defined by the formula
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$$t\sigma(t)y^{\Delta \Delta}(t) - 3ty^{\Delta}(t)+4y(t)=0$$
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$$L(t,s)=\displaystyle\int_{s}^t \dfrac{1}{\tau + 2\mu(\tau)} \Delta \tau.$$
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whose linearly independent solutions are
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$$\left\{\begin{array}{ll}
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=Properties=
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y_1(t)&=e_{\frac{2}{t}}(t,t_0) \\
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y_2(t)&=e_{\frac{2}{t}}(t,t_0) \displaystyle\int_{t_0}^t \dfrac{\Delta \tau}{\tau + 2\mu(\tau)}.
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=See also=
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\end{array} \right.$$
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[[Euler-Cauchy dynamic equation]]<br />
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This suggests that an analogue to the logarithm could be given by
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[[Jackson logarithm]]<br />
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$$L(t,t_0)=\displaystyle\int_{t_0}^t \dfrac{\Delta \tau}{\tau + 2\mu(\tau)}.$$
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[[Mozyrska-Torres logarithm]]<br />
  
 
=References=
 
=References=
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[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.]
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*{{PaperReference|The logarithm on time scales|2005|Martin Bohner|prev=Delta exponential dynamic equation|next=Bohner logarithm}}: $(2)$
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*{{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

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