Difference between revisions of "Exponential functions"

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$$y^{\nabla}=y;y(s)=1,$$
 
$$y^{\nabla}=y;y(s)=1,$$
 
defining the [[nabla exponential | $\nabla$-exponential]] functions.
 
defining the [[nabla exponential | $\nabla$-exponential]] functions.
 
 
 
=$\nabla$-exponential Functions=
 
Define the function $\hat{\xi}_{h} \colon \mathbb{C}_h \rightarrow \mathbb{Z}_h$ by
 
$$\hat{\xi}_h(z) = \dfrac{1}{h} \log(1-zh).$$
 
Define the $\nabla$ exponential function for $s,t \in \mathbb{T}$ by
 
$$\hat{e}_p(t,s) = \exp \left( \displaystyle\int_s^t \hat{\xi}_{\nu(\tau)}(p(\tau)) \nabla \tau \right).$$
 
 
==Properties of $\nabla$-exponential functions==
 
The function $\hat{e}_p(\cdot,s)$ is the unique solution of the initial value problem
 
$$y^{\nabla} = py; y(s)=1.$$
 
For all $p,q \in \mathcal{R}_{\nu}$ and $t,s \in \mathbb{T}$,
 
*$\hat{e}_p(t,r)\hat{e}_p(r,s)=e_p(t,s)$ (semigroup property)
 
*$\hat{e}_0(t,s)=1, \hat{e}_p(t,t)=1$
 
*$\hat{e}_p(\rho(t),s)=(1-\nu(t)p(t))\hat{e}_p(t,s)$
 
*$\dfrac{1}{\hat{e}_p(t,s)}=\hat{e}_{\ominus_{\nu} p}(s,t)$
 
*$\hat{e}_p(t,s)\hat{e}_q(t,s)=\hat{e}_{p \oplus_{\nu} q}(t,s)$
 
*$\dfrac{\hat{e}_p(t,s)}{\hat{e}_q(t,s)} = \hat{e}_{p \ominus_{\nu} q}(t,s)$
 
*$\left( \dfrac{1}{\hat{e}_p(\cdot,s)} \right)^{\nabla} = -\dfrac{p(t)}{\hat{e}_p^{\rho}(\cdot,s)}$
 
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
<strong>Theorem:</strong> (Sign of the Nabla Exponential Function) Let $p \in \mathcal{R}_{\nu}$ and $s \in \mathbb{T}$. <br />
 
i.) If $p \in \mathcal{R}_{\nu}^+$, then $\hat{e}_{p}(t,s) > 0$ for all $t \in \mathbb{T}$. <br />
 
ii.) If $1-\nu(t)p(t) < 0$ for some $t \in \mathbb{T}_{\kappa}$, then
 
$$\hat{e}(\rho(t),s)\hat{e}_{p}(t,s)<0.$$
 
iii.) If $1-\nu(t)p(t) < 0$ for all $t \in \mathbb{T}$, then $\hat{e}_p(t,s)$ changes sign at every point of $\mathbb{T}$.<br />
 
iv.) The exponential function $\hat{e}_p(\cdot,s)$ is a real-valued function that is never equal to zero.
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> proof goes here █
 
</div>
 
</div>
 

Revision as of 20:53, 20 October 2014

The classical exponential function $e^{x-s}$ is the unique solution to the initial value problem $$y'=y; y(s)=1.$$ The standard way to generalize this to time scales is called the $\Delta$-exponential function, which is the solution of $$y^{\Delta}=y;y(s)=1.$$ It generalizes the above equation in the sense that the classical derivative is replaced by the $\Delta$-derivative on some time scale. If instead of using the $\Delta$-derivative one uses the $\nabla$-derivative then the resulting exponential equation is $$y^{\nabla}=y;y(s)=1,$$ defining the $\nabla$-exponential functions.