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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 <sup>[[#potltots|[pp.793]]]</sup>
| + | __NOTOC__ |
| − | $$\mathscr{L}\{f\}(z;s) = \displaystyle\int_s^{\infty} f(t) e_{\ominus z}(\sigma(t),s) \Delta t,$$ | + | 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 $z$ is 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= | | =Table of Laplace transforms= |
| − | | + | <center> |
| | {| class="wikitable" | | {| class="wikitable" |
| − | |+Formula for Laplace transform | + | |+Formula for unilateral Laplace transform |
| | |- | | |- |
| | |$\mathbb{T}=$ | | |$\mathbb{T}=$ |
| − | |$\mathscr{L}_{\mathbb{T}}\{f\}(z;s)=$ | + | |Unilateral Laplace transform |
| | |- | | |- |
| | |[[Real_numbers | $\mathbb{R}$]] | | |[[Real_numbers | $\mathbb{R}$]] |
| − | |$\mathscr{L}_{\mathbb{R}}\{f\}(z;s)=\displaystyle\int_s^{\infty} f(\tau) e^{-z\tau} d\tau$ | + | |$\mathscr{L}_{\mathbb{R}}\{f\}(z;s)=\displaystyle\int_s^{\infty} f(\tau) e^{-z\tau} \mathrm{d}\tau$ |
| | |- | | |- |
| | |[[Integers | $\mathbb{Z}$]] | | |[[Integers | $\mathbb{Z}$]] |
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| | | | | | |
| | |} | | |} |
| − | | + | </center> |
| | + | <center> |
| | {| class="wikitable" | | {| class="wikitable" |
| | |+Laplace Transforms of special functions | | |+Laplace Transforms of special functions |
| | |- | | |- |
| − | |Function $f(t;s)$ | + | |$f(t;s)$ |
| − | |Laplace Transformation $\mathscr{L}\{f(\cdot;s)\}(z)$ | + | |$\mathscr{L}\{f(\cdot;s)\}(z)$ |
| | |- | | |- |
| | |$e_{\alpha}(t;s)$ | | |$e_{\alpha}(t;s)$ |
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| | |- | | |- |
| | |} | | |} |
| | + | </center> |
| | | | |
| − | =Properties of Laplace Transforms= | + | =See also= |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
| + | [[Bilateral Laplace transform]]<br /> |
| − | <strong>Proposition:</strong> The Laplace transform is linear, i.e. for constants $\alpha, \beta$ and Laplace-transformable functions $f,g$, <sup>[[#potltots|[pp.795]]]</sup>
| + | [[Unilateral convolution]]<br /> |
| − | $$\mathscr{L}\{\alpha f + \beta g\} = \alpha \mathscr{L}\{f\} + \beta \mathscr{L}\{g\}.$$
| |
| − | <div class="mw-collapsible-content">
| |
| − | <strong>Proof:</strong> █
| |
| − | </div>
| |
| − | </div>
| |
| | | | |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
| + | =References= |
| − | <strong>Proposition:</strong> The following formula holds:
| + | *{{PaperReference|The convolution on time scales|2007|Martin Bohner|author2=Gusein Sh. Guseinov|prev=|next=}}: (1.1) |
| − | $$e_{\ominus z}(\sigma(t),s) = \dfrac{e_{\ominus z}(t,s)}{1+\mu(t)z} = -\dfrac{(\ominus z)(t)}{z} e_{\ominus z}(t,s).$$
| + | *{{PaperReference|The gamma function on time scales|2013|Martin Bohner|author2=Başak Karpuz|prev=|next=Gamma function}}: Section 3 |
| − | <div class="mw-collapsible-content">
| |
| − | | |
| − | </div>
| |
| − | </div>
| |
| − | | |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
| |
| − | <strong>Proposition:</strong> The Laplace transform of a [[delta derivative]]:
| |
| − | $$\mathscr{L}\{f^{\Delta}\}(z;s) = -f(s) + z\mathscr{L}\{f\}(z).$$
| |
| − | <div class="mw-collapsible-content">
| |
| − | <strong>Proof:</strong> Compute using integration by parts,
| |
| − | $$\begin{array}{ll}
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| − | \mathscr{L}\{f^{\Delta}\}(z) &= \displaystyle\int_0^{\infty} f^{\Delta}(\tau) e_{\ominus z}(\sigma(\tau),s) \Delta \tau \\
| |
| − | &=
| |
| − | \end{array}$$
| |
| − | proving the claim. █
| |
| − | </div>
| |
| − | </div>
| |
| | | | |
| − | Assume there exist $M,\alpha > 0$ with
| + | [[Category:Definition]] |
| − | $$|a_k| \leq M \alpha_k$$
| |
| − | for all $k=0,1,2,\ldots$. Then for all $z$ where it exists, <sup>[[#potltots|[pp.796]]]</sup>
| |
| − | $$\mathscr{L}\left\{ \displaystyle\sum_{k=0}^{\infty} a_k h_k(\cdot,s) \right\}(z;s) = \displaystyle\sum_{k=0}^{\infty} a_k \mathscr{L}\{h_k(\cdot,s)\}(z;s) = \displaystyle\sum_{k=0}^{\infty} \dfrac{a_k}{z^{k+1}},$$
| |
| − | where $h_k$ denotes the standard time scale [[Polynomials | polynomial]].
| |
| − | | |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
| |
| − | <strong>Proposition:</strong> Let $m_z(t,s):=\displaystyle\int_s^t \dfrac{\Delta \tau}{1+\mu(\tau)z}$. Then <sup>[[#potltots|[pp.797]]]</sup>
| |
| − | $$\dfrac{d}{dz} \mathscr{L}\{f\}(z;s) = -\mathscr{L}\{m_z(\sigma(\cdot),s)f\}(z;s).$$
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| − | <div class="mw-collapsible-content">
| |
| − | <strong>Proof:</strong> █
| |
| − | </div>
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| − | </div>
| |
| − | | |
| − | It is known that $\dfrac{d}{dz} e_z(t,t_0) = m_z(t,t_0)e_z(t,t_0)$ and $\dfrac{d}{dz} e_{\ominus z}(t,t_0)=-m_z(t,t_0)e_{\ominus z}(t,t_0)$. These formulas are analogues of the formulas $\dfrac{d}{dz} e^{z(t-t_0)}=(t-t_0)e^{z(t-t_0)}$ and $\dfrac{d}{dz} e^{-z(t-t_0)}=-(t-t_0)e^{-z(t-t_0)}$ which occur in the case $\mathbb{T}=\mathbb{R}$. An important difference from the classical case is that $t-t_0$ has no dependence on the variable $z$, while $m_z(t,t_0)$ does.
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| − | | |
| − | ==Convergence==
| |
| − | We define the minimal graininess function
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| − | $$\mu_*(s)=\inf_{\tau \in [s,\infty) \cap \mathbb{T}} \mu(\tau).$$
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| − | Let $h\geq 0$. We also define the Hilger real part of $z \in \mathbb{C}$ by
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| − | $$\mathrm{Re}_h(z)=\dfrac{1}{h}(|1+hz|-1)$$
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| − | and the Hilger imaginary part of $z \in \mathbb{C}$ by
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| − | $$\mathrm{Re}_h(z)=\mathrm{Arg}(1+hz),$$
| |
| − | where $\mathrm{Arg}$ denotes the principal argument of $1+hz$. We let
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| − | $$\mathbb{C}_h = \left\{ z \in \mathbb{C} \colon z \neq -\dfrac{1}{h} \right\}.$$
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| − | Finally given some $\lambda \in \mathbb{R}$ we define
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| − | $$\mathbb{C}_h(\lambda) = \left\{ z \in \mathbb{C}_h \colon \mathrm{Re}_h(z) > \lambda \right\}.$$
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| − | | |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
| |
| − | <strong>Theorem (Absolute convergence):</strong> Let $f \in C_{\mathrm{rd}}([s,\infty) \cap \mathbb{T},\mathbb{C})$ be of [[exponential_order | exponential order $\alpha$]]. Then $\mathscr{L}\{f\}(\cdot;s)$ exists on $\mathbb{C}_{\mu_*(s)}(\alpha)$ and converges absolutely.
| |
| − | <div class="mw-collapsible-content">
| |
| − | <strong>Proof:</strong> █
| |
| − | </div>
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| − | </div>
| |
| − | | |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
| |
| − | <strong>Theorem (Uniform convergence):</strong> Let $f \in C_{\mathrm{rd}}([s,\infty)\cap\mathbb{T},\mathbb{C})$ be of exponential order $\alpha$. Then the Laplace transform $\mathscr{L}\{f\}$ converges uniformly in the half-plane $C_{\mu_*(s)}(\beta)$ for any $\beta > \alpha$.
| |
| − | <div class="mw-collapsible-content">
| |
| − | <strong>Proof:</strong> █
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| − | </div>
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| − | </div>
| |
| − | | |
| − | =Inverse Transform on isolated time scales=
| |
| − | The following information is from <sup>[[#tltotsr|pp.##]]</sup>. Define the notation
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| − | $$\mathrm{Re}_h(z) := \dfrac{1}{h} (|1+hz|-1).$$
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| − | Let $\mathbb{T}$ be a time scale with $0 < \mu_{\mathrm{min}} \leq \mu(t) \leq \mu_{\mathrm{max}} < \infty.$ Let $\mu_* := \mu_{\mathrm{min}}$ and $\mu^* := \mu_{\mathrm{max}}$. Define in $\mathbb{C}$ the Hilger circles
| |
| − | $$\mathbb{H}_{\mu(t)} = \left\{ z \in \mathbb{C} \colon 0 < \left| z + \dfrac{1}{\mu(t)} \right| < \dfrac{1}{\mu(t)} \right\},$$
| |
| − | which are disks centered at $\left(-\dfrac{1}{\mu(t)},0 \right)$ with radius $\dfrac{1}{\mu(t)}$.
| |
| − | [[File:Hilger_circles.png|thumb|250px|Here $H_{\mathrm{min}}=\mathbb{H}_{\mu^*}$ and $H_{\mathrm{max}}=\mathbb{H}_{\mu_*}$.]]
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| − | These circles define the region of convergence for the Laplace transform $\mathscr{L}_\mathbb{T}$. If $\mu_*=0$, then the region of convergence is the right half plane, but if $\mu_*>0$ the region of convergence is outside of $\mathbb{H}_{\mu_*}$.
| |
| − | | |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
| |
| − | <strong>Theorem:</strong> Suppose that $F(z)$ is analytic in the region $\mathrm{Re}_{\mu}(z)>\mathrm{Re}_{\mu}(c)$ and $F(z) \rightarrow 0$ uniformly as $|z| \rightarrow \infty$ in this region. Suppose $F(z)$ has finitely many regressive poles of finite order $\{z_1,\ldots,z_n\}$ and $\tilde{F}_{\mathbb{R}}(z)$ is the transform of the function $\tilde{f}(t)$ on $\mathbb{R}$ that corresponds to the transform $F(z)=F_{\mathbb{T}}(z)$ of $f(t)$ on $\mathbb{T}$. If
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| − | $$\displaystyle\int_{c-i\infty}^{c+i\infty} |\tilde{F}_{\mathbb{R}}(z)||dz|<\infty,$$
| |
| − | then
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| − | $$f(t)=\displaystyle\sum_{i=1}^n \mathrm{Res}_{z=z_i} e_z(t,0) F(z)$$
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| − | has transform $F(z)$ for all $z$ with $\mathrm{Re}(z)>c$.
| |
| − | <div class="mw-collapsible-content">
| |
| − | <strong>Proof:</strong> █
| |
| − | </div>
| |
| − | </div>
| |
| − | | |
| − | =References=
| |
| − | <div id="potltots"></div><bibtex>
| |
| − | @inproceedings{
| |
| − | title="Properties of the Laplace transform on time scales with arbitrary graininess",
| |
| − | author="Bohner, Martin and Guseinov, Gusein Sh. and Karpuz, Başak",
| |
| − | booktitle="Integral Transforms and Special Functions. An International Journal",
| |
| − | volume="22",
| |
| − | year="2011",
| |
| − | pages="pp.785--800",
| |
| − | doi="10.1080/10652469.2010.548335",
| |
| − | url="http://www.tandfonline.com/doi/abs/10.1080/10652469.2010.548335?journalCode=gitr20#.U8bMFLGmUhI"
| |
| − | }
| |
| − | </bibtex>
| |
| − | <div id="tltotsr"></div>
| |
| − | <bibtex>
| |
| − | @inproceedings{
| |
| − | title="The Laplace transform on time scales revisited",
| |
| − | author="Davis, John M. and Gravagne, Ian A. and Jackson, Billy J., Marks, II, Robert J. and Ramos, Alice A.",
| |
| − | booktitle="Journal of Mathematical Analysis and Applications",
| |
| − | volume="332",
| |
| − | year="2007",
| |
| − | pages="pp.1291--1307",
| |
| − | doi="10.1016/j.jmaa.2006.10.089",
| |
| − | url="http://web.ecs.baylor.edu/faculty/gravagnei/archived/laplace.pdf"
| |
| − | }
| |
| − | </bibtex>
| |
