Difference between revisions of "Exponential functions"
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(→Examples of Exponential Functions) |
(→Examples of Exponential Functions) |
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|[[Multiples_of_integers | $h\mathbb{Z}$]] | |[[Multiples_of_integers | $h\mathbb{Z}$]] | ||
| − | | $ | + | | $ e_p(t,s) = \left\{ \begin{array}{ll} |
| − | e_p(t,s) | + | \displaystyle\prod_{k=\frac{s}{h}}^{\frac{t}{h}-1} (1+hp(hk)) &; t > s \\ |
| − | + | 1 &; t=s \\ | |
| − | &= \displaystyle\prod_{k=\frac{ | + | \displaystyle\prod_{k=\frac{t}{h}}^{\frac{s}{h}-1} \dfrac{1}{1+hp(hk)} &; t < s |
| − | \end{array}$ | + | \end{array} \right.$ |
|- | |- | ||
| [[Square_integers | $\mathbb{Z}^2$]] | | [[Square_integers | $\mathbb{Z}^2$]] | ||
| − | | $ | + | | $ e_p(t,s) = \left\{\begin{array}{ll} |
| − | e_p(t,s) | + | \displaystyle\prod_{k=\sqrt{s}}^{\sqrt{t}-1} 1 + p(k^2)(2k+1) &; t > s \\ |
| − | + | 1 &; t=s\\ | |
| − | &= \ | + | \displaystyle\prod_{k=\sqrt{t}}^{\sqrt{s}-1} \dfrac{1}{1+p(k^2)(2k+1)} &; t < s |
| − | + | \end{array} \right.$ | |
| − | \end{array}$ | ||
|- | |- | ||
|[[Harmonic_numbers | $\mathbb{H}$]] | |[[Harmonic_numbers | $\mathbb{H}$]] | ||
| − | |$ | + | |$ e_p(t,s) = e_p\left( \displaystyle\sum_{k=1}^n \dfrac{1}{k}, \displaystyle\sum_{k=1}^m \dfrac{1}{k} \right) = \left\{\begin{array}{ll} |
| − | e_p(t,s) | + | \displaystyle\prod_{k=m}^{n-1} 1 + \dfrac{1}{k+1} p \left( \displaystyle\sum_{j=1}^k \dfrac{1}{j} \right) &; t > s \\ |
| − | + | 1 &; t=s \\ | |
| − | + | \displaystyle\prod_{k=n}^{m-1} 1 + \dfrac{1}{k+1} p \left( \displaystyle\sum_{j=1}^k \dfrac{1}{j} \right) &; t < s | |
| − | &= \ | + | \end{array} \right.$ |
| − | |||
| − | \end{array}$ | ||
|} | |} | ||
Revision as of 23:52, 21 May 2014
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. The exponential function $e_p \colon \mathbb{T} \times \mathbb{T} \rightarrow \mathbb{R}$ is defined as
$$e_p(t,s) := \exp \left( \displaystyle\int_s^t \xi_{\mu(\tau)}(p(\tau))\Delta \tau \right)$$
for $s,t \in \mathbb{T}$. It turns out that $e_p$ is the unique solution to the dynamic initial value problem $$y^{\Delta} = py; y(s)=1.$$
Properties of Exponential Functions
For all $p,q \in \mathcal{R}$ and $t,s \in \mathbb{T}$,
- $e_p(t,r)e_p(r,s)=e_p(t,s)$ (semigroup property)
- $e_0(t,s)=1, e_p(t,t)=1$
- $e_p(\sigma(t),s)=(1+\mu(t)p(t))e_p(t,s)$
- $\dfrac{1}{e_p(t,s)}=e_{\ominus p}(t,s)$
- $e_p(t,s)e_q(t,s)=e_{p \oplus q}(t,s)$
- $\dfrac{e_p(t,s)}{e_q(t,s)} = e_{p \ominus q}(t,s)$
- $\left( \dfrac{1}{e_p(\cdot,s)} \right)^{\Delta} = -\dfrac{p(t)}{e_p^{\sigma}(\cdot,s)}$
Examples of Exponential Functions
| $\mathbb{T}=$ | $e_p(t,s)=$ |
| $\mathbb{R}$ | $e_p(t,s)= \left\{ \begin{array}{ll} \exp \left( \displaystyle\int_s^t p(\tau) d \tau \right) &; t>s \\ 1 &; t=s \\ \exp \left( -\displaystyle\int_t^s p(\tau) d\tau \right) &; t<s \end{array} \right.$ |
| $\mathbb{Z}$ | $e_p(t,s) = \left\{ \begin{array}{ll} \displaystyle\prod_{k=s}^{t-1} 1+p(k) &; t > s \\ 1 &; t=s \\ \displaystyle\prod_{k=t}^{s-1} \dfrac{1}{1+p(k)}&; t < s \end{array} \right.$ |
| $h\mathbb{Z}$ | $ e_p(t,s) = \left\{ \begin{array}{ll} \displaystyle\prod_{k=\frac{s}{h}}^{\frac{t}{h}-1} (1+hp(hk)) &; t > s \\ 1 &; t=s \\ \displaystyle\prod_{k=\frac{t}{h}}^{\frac{s}{h}-1} \dfrac{1}{1+hp(hk)} &; t < s \end{array} \right.$ |
| $\mathbb{Z}^2$ | $ e_p(t,s) = \left\{\begin{array}{ll} \displaystyle\prod_{k=\sqrt{s}}^{\sqrt{t}-1} 1 + p(k^2)(2k+1) &; t > s \\ 1 &; t=s\\ \displaystyle\prod_{k=\sqrt{t}}^{\sqrt{s}-1} \dfrac{1}{1+p(k^2)(2k+1)} &; t < s \end{array} \right.$ |
| $\mathbb{H}$ | $ e_p(t,s) = e_p\left( \displaystyle\sum_{k=1}^n \dfrac{1}{k}, \displaystyle\sum_{k=1}^m \dfrac{1}{k} \right) = \left\{\begin{array}{ll} \displaystyle\prod_{k=m}^{n-1} 1 + \dfrac{1}{k+1} p \left( \displaystyle\sum_{j=1}^k \dfrac{1}{j} \right) &; t > s \\ 1 &; t=s \\ \displaystyle\prod_{k=n}^{m-1} 1 + \dfrac{1}{k+1} p \left( \displaystyle\sum_{j=1}^k \dfrac{1}{j} \right) &; t < s \end{array} \right.$ |
