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| − | Let $\mathbb{T}$ be a [[time scale]]. Define $\xi_h(z) := \dfrac{1}{h} \log(1+zh)$. Let $p \in \mathcal{R}(\mathbb{T},\mathbb{R})$ be a [[regressive_function | regressive function]]. The exponential function $e_p \colon \mathbb{T} \times \mathbb{T} \rightarrow \mathbb{R}$ is defined as
| + | 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 | $\Delta$-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 | $\Delta$-derivative]] on some time scale. If instead of using the $\Delta$-derivative one uses the [[nabla derivative | $\nabla$-derivative]] then the resulting exponential equation is |
| | + | $$y^{\nabla}=y;y(s)=1,$$ |
| | + | defining the [[nabla exponential | $\nabla$-exponential]] functions. |
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| − | $$e_p(t,s) := \exp \left( \displaystyle\int_s^t \xi_{\mu(\tau)}(p(\tau))\Delta \tau \right)$$
| + | Generally speaking, given some kind of time scale derivative operator $D$, we can define exponential functions by the $D$-dynamic equation |
| − | | + | $$Dy=y; y(s)=1.$$ |
| − | for $s,t \in \mathbb{T}$. It turns out that $e_p$ is the unique solution to the dynamic initial value problem
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| − | $$y^{\Delta} = py; y(s)=1.$$ | |
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| − | == Properties of Exponential Functions ==
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| − | For all $p,q \in \mathcal{R}$ and $t,s \in \mathbb{T}$,
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| − | *$e_p(t,r)e_p(r,s)=e_p(t,s)$ (semigroup property)
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| − | *$e_0(t,s)=1, e_p(t,t)=1$
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| − | *$e_p(\sigma(t),s)=(1+\mu(t)p(t))e_p(t,s)$
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| − | *$\dfrac{1}{e_p(t,s)}=e_{\ominus p}(t,s)$
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| − | *$e_p(t,s)e_q(t,s)=e_{p \oplus q}(t,s)$
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| − | *$\dfrac{e_p(t,s)}{e_q(t,s)} = e_{p \ominus q}(t,s)$
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| − | *$\left( \dfrac{1}{e_p(\cdot,s)} \right)^{\Delta} = -\dfrac{p(t)}{e_p^{\sigma}(\cdot,s)}$
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| − | | |
| − | == Examples of Exponential Functions ==
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| − | {| class="wikitable"
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| − | |+Time Scale Exponential Functions
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| − | |-
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| − | |$\mathbb{T}=$
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| − | |$e_p(t,s)=$
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| − | |-
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| − | |[[Real_numbers | $\mathbb{R}$]]
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| − | |$e_p(t,s)= \left\{ \begin{array}{ll}
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| − | \exp \left( \displaystyle\int_s^t p(\tau) d \tau \right) &; t>s \\
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| − | 1 &; t=s \\
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| − | \exp \left( -\displaystyle\int_t^s p(\tau) d\tau \right) &; t<s
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| − | \end{array} \right.$
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| − | |-
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| − | |[[Integers]]
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| − | |$e_p(t,s) = \displaystyle\prod_{k=s}^{t-1}$
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| − | |-
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| − | |[[Multiples_of_integers | $h\mathbb{Z}$]]
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| − | | $\begin{array}{ll}
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| − | e_p(t,s) &= \exp \left( \displaystyle\int_{s}^{t} \dfrac{1}{\mu(\tau)} \log(1 + hp(\tau)) \Delta \tau \right) \\
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| − | &= \exp \left( \displaystyle\sum_{k=\frac{s}{h}}^{\frac{t}{h}-1} \log(1+hp(hk)) \right) \\
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| − | &= \displaystyle\prod_{k=\frac{s}{h}}^{\frac{t}{h}-1} \left( 1+hp(hk) \right) \\
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| − | \end{array}$
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| − | |-
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| − | | [[Square_integers | $\mathbb{Z}^2$]]
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| − | | $\begin{array}{ll}
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| − | e_p(t,s) &= \exp \left( \displaystyle\int_s^t \dfrac{1}{\mu(\tau)} \log(1 + p(\tau) \mu(\tau)) \Delta \tau \right) \\
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| − | &= \exp \left( \displaystyle\sum_{k=\sqrt{s}}^{\sqrt{t}-1} \mu(k^2) \dfrac{1}{\mu(k^2)} \log ( 1 + p(k^2)\mu(k^2)) \right) \\
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| − | &= \exp \left( \displaystyle\sum_{k=\sqrt{s}}^{\sqrt{t}-1} \log ( 1 + p(k^2)\mu(k^2)) \right) \\
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| − | &= \displaystyle\prod_{k=\sqrt{s}}^{\sqrt{t}-1} 1 + p(k^2)(2k+1)
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| − | \end{array}$
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| − | |-
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| − | |[[Harmonic_numbers | $\mathbb{H}$]]
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| − | |$\begin{array}{ll}
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| − | e_p(t,s) &= e_p \left( \displaystyle\sum_{k=1}^n \dfrac{1}{k}, \displaystyle\sum_{k=1}^m \dfrac{1}{k} \right) \\
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| − | &= \exp \left( \displaystyle\int_{ \sum^m \frac{1}{k}}^{\sum^n \frac{1}{k}} \dfrac{1}{\mu(\tau)} \log(1 + \mu(\tau) p(\tau)) \Delta \tau \right) \\
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| − | &= \exp \left( \displaystyle\sum_{k=m}^{n-1} \log \left[1 + \mu \left( \displaystyle\sum_{j=1}^k \dfrac{1}{j} \right) p \left( \displaystyle\sum_{j=1}^k \dfrac{1}{j} \right) \right] \right) \\
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| − | &= \exp \left( \displaystyle\sum_{k=m}^{n-1} \log \left[1 + \dfrac{1}{k+1} p \left( \displaystyle\sum_{j=1}^k \dfrac{1}{j} \right) \right] \right) \\
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| − | &= \displaystyle\prod_{k=m}^{n-1} 1 + \dfrac{1}{k+1} p \left( \displaystyle\sum_{j=1}^k \dfrac{1}{j} \right) \\
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| − | \end{array}$
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| − | |}
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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
$$Dy=y; y(s)=1.$$
