Difference between revisions of "Unilateral Laplace transform"

From timescalewiki
Jump to: navigation, search
(→‎Properties of Laplace Transforms)
 
(65 intermediate revisions by the same user not shown)
Line 1: Line 1:
−
Let $\mathbb{T}$ be a [[time_scale | time scale]] and let $s \in \mathbb{T}$. If $f \in C_{rd}(\mathbb{T},\mathbb{C})$ ([[continuity | rd-continuous]]) then we define the Laplace transform of $f$ about $s$ by the formula
+
__NOTOC__
−
$$\mathscr{L}\{f\}(z;s) = \displaystyle\int_s^{\infty} f(t) e_{\ominus z}(\sigma(t),0) \Delta t,$$
+
If $\mathbb{T}$ is a [[time_scale | time scale]], $s \in \mathbb{T}$, and $f$ is [[rd-continuous]], then we define the unilateral Laplace transform of $f$ about $s$ by the formula
−
where $z$ lives in a domain $D \subset \mathbb{C}$ for which the integral converges. Let $\alpha$ be a non-negative [[regressive_function | regressive]] constant larger than $s$. We use the notation "$f(t;s)$" to denote we are thinking of $f$ as a function of $t$ with parameter $s$.
+
$$\mathscr{L}_{\mathbb{T}}\{f\}(z;s) = \displaystyle\int_s^{\infty} f(t) e_{\ominus z}(\sigma(t),s) \Delta t,$$
 +
for all $z$ for which the integral converges, where $\displaystyle\int \ldots \Delta t$ denotes the [[delta integral]], $e_{\ominus z}$ denotes a [[delta exponential]] whose subscript is the [[forward circle minus]] of the constant $z$, and $\sigma$ is the [[forward jump]].  
  
 +
=Properties of Laplace Transforms=
 +
[[Unilateral Laplace transform is a linear operator]]<br />
 +
[[Unilateral Laplace transform of delta derivative]]<br />
 +
 +
=Table of Laplace transforms=
 +
<center>
 
{| class="wikitable"
 
{| class="wikitable"
−
|+Laplace Transformations
+
|+Formula for unilateral Laplace transform
 +
|-
 +
|$\mathbb{T}=$
 +
|Unilateral Laplace transform
 +
|-
 +
|[[Real_numbers | $\mathbb{R}$]]
 +
|$\mathscr{L}_{\mathbb{R}}\{f\}(z;s)=\displaystyle\int_s^{\infty} f(\tau) e^{-z\tau} \mathrm{d}\tau$
 
|-
 
|-
−
|Function $f(t;s)$
+
|[[Integers | $\mathbb{Z}$]]
−
|Laplace Transformation $\mathscr{L}\{f(\cdot;s)\}(z)$
+
|
 +
|-
 +
|[[Multiples_of_integers | $h\mathbb{Z}$]]
 +
|
 +
|-
 +
| [[Square_integers | $\mathbb{Z}^2$]]
 +
|
 +
|-
 +
|[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q > 1$]]
 +
|
 +
|-
 +
|[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q < 1$]]
 +
|
 +
|-
 +
|[[Harmonic_numbers | $\mathbb{H}$]]
 +
|
 +
|}
 +
</center>
 +
<center>
 +
{| class="wikitable"
 +
|+Laplace Transforms of special functions
 +
|-
 +
|$f(t;s)$
 +
|$\mathscr{L}\{f(\cdot;s)\}(z)$
 
|-
 
|-
 
|$e_{\alpha}(t;s)$
 
|$e_{\alpha}(t;s)$
Line 28: Line 64:
 
|-
 
|-
 
|}
 
|}
 +
</center>
 +
 +
=See also=
 +
[[Bilateral Laplace transform]]<br />
 +
[[Unilateral convolution]]<br />
 +
 +
=References=
 +
*{{PaperReference|The convolution on time scales|2007|Martin Bohner|author2=Gusein Sh. Guseinov|prev=|next=}}: (1.1)
 +
*{{PaperReference|The gamma function on time scales|2013|Martin Bohner|author2=Başak Karpuz|prev=|next=Gamma function}}: Section 3
  
−
==Properties of Laplace Transforms==
+
[[Category:Definition]]
−
*The Laplace transform is linear, i.e. for constants $\alpha, \beta$ and Laplace-transformable functions $f,g$,
 
−
$$\mathscr{L}\{\alpha f + \beta g\} = \alpha \mathscr{L}\{f\} + \beta \mathscr{L}\{g\}.$$
 
−
*Assume there exist $M,\alpha > 0$ with
 
−
$$|a_k| \leq M \alpha_k$$
 
−
for all $k=0,1,2,\ldots$. Then for all $z$ where it exists,
 
−
$$\mathscr{L}\left\{ \displaystyle\sum_{k=0}^{\infty} a_k h_k(\cdot,s) \right\}(z;s) = \displaystyle\sum_{k=0}^{\infty} a_k \mathscr{L}\{h_k(\cdot,s)\}(z;s) = \displaystyle\sum_{k=0}^{\infty} \dfrac{a_k}{z^{k+1}},$$
 
−
where $h_k$ denotes the standard time scale [[Polynomials | polynomial]].
 
−
*Let $m_z(t,s):=\displaystyle\int_s^t \dfrac{\Delta \tau}{1+\mu(\tau)z}$. Then
 
−
$$\dfrac{d}{dz} \mathscr{L}\{f\}(z;s) = -\mathscr{L}\{m_z(\sigma(\cdot),s)f\}(z;s).$$
 

Latest revision as of 15:21, 21 January 2023

If $\mathbb{T}$ is a time scale, $s \in \mathbb{T}$, and $f$ is rd-continuous, then we define the unilateral Laplace transform of $f$ about $s$ by the formula $$\mathscr{L}_{\mathbb{T}}\{f\}(z;s) = \displaystyle\int_s^{\infty} f(t) e_{\ominus z}(\sigma(t),s) \Delta t,$$ for all $z$ for which the integral converges, where $\displaystyle\int \ldots \Delta t$ denotes the delta integral, $e_{\ominus z}$ denotes a delta exponential whose subscript is the forward circle minus of the constant $z$, and $\sigma$ is the forward jump.

Properties of Laplace Transforms

Unilateral Laplace transform is a linear operator
Unilateral Laplace transform of delta derivative

Table of Laplace transforms

Formula for unilateral Laplace transform
$\mathbb{T}=$ Unilateral Laplace transform
$\mathbb{R}$ $\mathscr{L}_{\mathbb{R}}\{f\}(z;s)=\displaystyle\int_s^{\infty} f(\tau) e^{-z\tau} \mathrm{d}\tau$
$\mathbb{Z}$
$h\mathbb{Z}$
$\mathbb{Z}^2$
$\overline{q^{\mathbb{Z}}}, q > 1$
$\overline{q^{\mathbb{Z}}}, q < 1$
$\mathbb{H}$
Laplace Transforms of special functions
$f(t;s)$ $\mathscr{L}\{f(\cdot;s)\}(z)$
$e_{\alpha}(t;s)$ $\dfrac{1}{z-\alpha}$
$h_n(t;s)$ $\dfrac{1}{z^{n+1}}$
$\sinh_{\alpha}(t;s)$ $\dfrac{\alpha}{z^2-\alpha^2}$
$\cosh_{\alpha}(t;s)$ $\dfrac{z}{z^2-\alpha^2}$
$\sin_{\alpha}(t;s)$ $\dfrac{\alpha}{z^2+\alpha^2}$
$\cos_{\alpha}(t;s)$ $\dfrac{z}{z^2+\alpha^2}$

See also

Bilateral Laplace transform
Unilateral convolution

References

Retrieved from "http://timescalewiki.org/index.php?title=Unilateral_Laplace_transform&oldid=2229"