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