Difference between revisions of "Unilateral Laplace transform"

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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 <ref>Bohner, Martin ; Guseinov, Gusein Sh. ; Karpuz, Başak . Properties of the Laplace transform on time scales with arbitrary graininess. Integral Transforms Spec. Funct. 22 (2011), no. 11, pp.793.</ref>
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__NOTOC__
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$$\mathscr{L}\{f\}(z;s) = \displaystyle\int_s^{\infty} f(t) e_{\ominus z}(\sigma(t),0) \Delta t,$$
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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
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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 $s$.
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$$\mathscr{L}_{\mathbb{T}}\{f\}(z;s) = \displaystyle\int_s^{\infty} f(t) e_{\ominus z}(\sigma(t),s) \Delta t,$$
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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]].  
  
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=Properties of Laplace Transforms=
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[[Unilateral Laplace transform is a linear operator]]<br />
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[[Unilateral Laplace transform of delta derivative]]<br />
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=Table of Laplace transforms=
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<center>
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{| class="wikitable"
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|+Formula for unilateral Laplace transform
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|-
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|$\mathbb{T}=$
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|Unilateral Laplace transform
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|-
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|[[Real_numbers | $\mathbb{R}$]]
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|$\mathscr{L}_{\mathbb{R}}\{f\}(z;s)=\displaystyle\int_s^{\infty} f(\tau) e^{-z\tau} \mathrm{d}\tau$
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|-
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|[[Integers | $\mathbb{Z}$]]
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|
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|-
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|[[Multiples_of_integers | $h\mathbb{Z}$]]
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|
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|-
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| [[Square_integers | $\mathbb{Z}^2$]]
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|
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|-
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|[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q > 1$]]
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|
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|-
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|[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q < 1$]]
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|
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|-
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|[[Harmonic_numbers | $\mathbb{H}$]]
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|
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|}
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</center>
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<center>
 
{| class="wikitable"
 
{| class="wikitable"
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|+Laplace Transformations
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|+Laplace Transforms of special functions
 
|-
 
|-
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|Function $f(t;s)$
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|$f(t;s)$
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|Laplace Transformation $\mathscr{L}\{f(\cdot;s)\}(z)$
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|$\mathscr{L}\{f(\cdot;s)\}(z)$
 
|-
 
|-
 
|$e_{\alpha}(t;s)$
 
|$e_{\alpha}(t;s)$
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|-
 
|-
 
|}
 
|}
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</center>
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=See also=
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[[Bilateral Laplace transform]]<br />
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[[Unilateral convolution]]<br />
  
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==Properties of Laplace Transforms==
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=References=
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*The Laplace transform is linear, i.e. for constants $\alpha, \beta$ and Laplace-transformable functions $f,g$, <ref>Bohner, Martin ; Guseinov, Gusein Sh. ; Karpuz, Başak . Properties of the Laplace transform on time scales with arbitrary graininess. Integral Transforms Spec. Funct. 22 (2011), no. 11, pp.795.</ref>
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*{{PaperReference|The convolution on time scales|2007|Martin Bohner|author2=Gusein Sh. Guseinov|prev=|next=}}: (1.1)
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$$\mathscr{L}\{\alpha f + \beta g\} = \alpha \mathscr{L}\{f\} + \beta \mathscr{L}\{g\}.$$
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*{{PaperReference|The gamma function on time scales|2013|Martin Bohner|author2=Başak Karpuz|prev=|next=Gamma function}}: Section 3
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*Assume there exist $M,\alpha > 0$ with
 
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$$|a_k| \leq M \alpha_k$$
 
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for all $k=0,1,2,\ldots$. Then for all $z$ where it exists, <ref>Bohner, Martin ; Guseinov, Gusein Sh. ; Karpuz, Başak . Properties of the Laplace transform on time scales with arbitrary graininess. Integral Transforms Spec. Funct. 22 (2011), no. 11, pp.796.</ref>
 
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$$\mathscr{L}\left\{ \displaystyle\sum_{k=0}^{\infty} a_k h_k(\cdot,s) \right\}(z;s) = \displaystyle\sum_{k=0}^{\infty} a_k \mathscr{L}\{h_k(\cdot,s)\}(z;s) = \displaystyle\sum_{k=0}^{\infty} \dfrac{a_k}{z^{k+1}},$$
 
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where $h_k$ denotes the standard time scale [[Polynomials | polynomial]].
 
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*Let $m_z(t,s):=\displaystyle\int_s^t \dfrac{\Delta \tau}{1+\mu(\tau)z}$. Then <ref>Bohner, Martin ; Guseinov, Gusein Sh. ; Karpuz, Başak . Properties of the Laplace transform on time scales with arbitrary graininess. Integral Transforms Spec. Funct. 22 (2011), no. 11, pp.797.</ref>
 
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$$\dfrac{d}{dz} \mathscr{L}\{f\}(z;s) = -\mathscr{L}\{m_z(\sigma(\cdot),s)f\}(z;s).$$
 
  
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<references />
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[[Category:Definition]]

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

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