Unilateral Laplace transform

Let $\mathbb{T}$ be a 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 ^{[[#potltots|[pp.793]]]} '"`UNIQ-MathJax2-QINU`"' 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 $s$. ='"`UNIQ--h-0--QINU`"'Properties of Laplace Transforms= [[Unilateral Laplace transform is a linear operator]]<br /> <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> <strong>Proposition:</strong> The Laplace transform of a [[delta derivative]]: '"`UNIQ-MathJax3-QINU`"' <div class="mw-collapsible-content"> <strong>Proof:</strong> Compute using integration by parts, '"`UNIQ-MathJax4-QINU`"' proving the claim. █ </div> </div> Assume there exist $M,\alpha > 0$ with '"`UNIQ-MathJax5-QINU`"' for all $k=0,1,2,\ldots$. Then for all $z$ where it exists, ^{[[#potltots|[pp.796]]]} '"`UNIQ-MathJax6-QINU`"' 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 ^{[[#potltots|[pp.797]]]} '"`UNIQ-MathJax7-QINU`"' <div class="mw-collapsible-content"> <strong>Proof:</strong> █ </div> </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. =='"`UNIQ--h-1--QINU`"'Convergence== We define the minimal graininess function '"`UNIQ-MathJax8-QINU`"' Let $h\geq 0$. We also define the Hilger real part of $z \in \mathbb{C}$ by '"`UNIQ-MathJax9-QINU`"' and the Hilger imaginary part of $z \in \mathbb{C}$ by '"`UNIQ-MathJax10-QINU`"' where $\mathrm{Arg}$ denotes the principal argument of $1+hz$. We let '"`UNIQ-MathJax11-QINU`"' Finally given some $\lambda \in \mathbb{R}$ we define '"`UNIQ-MathJax12-QINU`"' <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> </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> █ </div> </div> ='"`UNIQ--h-2--QINU`"'Table of Laplace transforms= <center> {| class="wikitable" |+Formula for unilateral Laplace transform |- |$\mathbb{T}=$ |Unilateral Laplace transform |- |[[Real_numbers | $\mathbb{R}$]] |$\mathscr{L}_{\mathbb{R}}\{f\}(z;s)=\displaystyle\int_s^{\infty} f(\tau) e^{-z\tau} \mathrm{d}\tau$ |- |[[Integers | $\mathbb{Z}$]] | |- |[[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 |- |$f(t;s)$ |$\mathscr{L}\{f(\cdot;s)\}(z)$ |- |$e_{\alpha}(t;s)$ |$\dfrac{1}{z-\alpha}$ |- |$h_n(t;s)$ |$\dfrac{1}{z^{n+1}}$ |- |$\sinh_{\alpha}(t;s)$ |$\dfrac{\alpha}{z^2-\alpha^2}$ |- |$\cosh_{\alpha}(t;s)$ |$\dfrac{z}{z^2-\alpha^2}$ |- |$\sin_{\alpha}(t;s)$ |$\dfrac{\alpha}{z^2+\alpha^2}$ |- |$\cos_{\alpha}(t;s)$ |$\dfrac{z}{z^2+\alpha^2}$ |- |}

See also

Bilateral Laplace transform
Unilateral convolution

References

• Martin Bohner and Gusein Sh. Guseinov: The convolution on time scales (2007): (1.1)
• Martin Bohner and Başak Karpuz: The gamma function on time scales (2013)... (next): Section 3