Difference between revisions of "Gaussian bell"

From timescalewiki
Jump to: navigation, search
Line 26: Line 26:
 
|-
 
|-
 
|[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q < 1$]]
 
|[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q < 1$]]
| $\displaystyle\prod_{\log_q(t)+1}^{\infty} \dfrac{1}{\left(1+(\frac{1}{q}-1)q^k \right)^{q^k}}$
+
| $\displaystyle\prod_{k=\log_q(t)+1}^{\infty} \dfrac{1}{\left(1+(\frac{1}{q}-1)q^k \right)^{q^k}}$
 
|-
 
|-
 
|[[Harmonic_numbers | $\mathbb{H}$]]
 
|[[Harmonic_numbers | $\mathbb{H}$]]

Revision as of 00:45, 9 September 2015

Let $\mathbb{T}$ be a time scale with $0 \in \mathbb{T}$. Let $p \colon \mathbb{T} \rightarrow \mathbb{R}$ be regressive and defined by $$p(t)=\ominus(t \odot 1).$$ The Gaussian bell $\mathbf{E} \colon \mathbb{T} \rightarrow \mathbb{R}$ is defined<ref name=gaussbell /> to be the exponential function $$\mathbf{E}(t)=e_{p}(t,0).$$

Time Scale Gaussian Bells
$\mathbb{T}$ $\mathbf{E}(t)$
$\mathbb{R}$ $e^{-\frac{t^2}{2}}$
$\mathbb{Z}$ $2^{\frac{-t(t-1)}{2}}$
$h\mathbb{Z}$ $\left[(1+h)^{\frac{1}{h}} \right]^{\frac{-t(t-h)}{2}}$
$\mathbb{Z}^2$
$\overline{q^{\mathbb{Z}}}, q > 1$
$\overline{q^{\mathbb{Z}}}, q < 1$ $\displaystyle\prod_{k=\log_q(t)+1}^{\infty} \dfrac{1}{\left(1+(\frac{1}{q}-1)q^k \right)^{q^k}}$
$\mathbb{H}$ $\displaystyle\prod_{k=1}^n \left( \dfrac{k}{k+1} \right)^{H_{k-1}}$

References

<references> <ref name=gaussbell>Erbe, L.; Peterson, A.;Simon, M. Square integrability of Gaussian bells on time scales. Comput. Math. Appl. 49 (2005), no. 5-6, 871--883. </ref> </references>