Tabelas de engenharia/Tabela de transformadas de Laplace 2

ID	Função	Dominío do tempo $x(t)={ℒ}^{-1}\{X(s)\}$	Transformada de Laplace $X(s)=ℒ\{x(t)\}$	Região de convergência para sistemas casuais
1	Atraso Ideal	$\delta (t-\tau )$	$e^{-\tau s}$
1a	Impulso Unitário	$\delta (t)$	$1$	$\mathrm{a}\mathrm{l}\mathrm{l}s$
2	Atraso na enéssima potência com deslocamento de frequência	$\frac{(t-\tau {)}^n}{n!}{e}^{-\alpha (t-\tau )}\cdot u(t-\tau )$	$\frac{e^{-\tau s}}{(s+\alpha {)}^{n+1}}$	$s>0$
2a	Elevado a enéssima potência	$\frac{t^n}{n!}\cdot u(t)$	$\frac{1}{s^{n+1}}$	$s>0$
2a.1	qth Power	$\frac{t^q}{\mathrm{\Gamma }(q+1)}\cdot u(t)$	$\frac{1}{s^{q+1}}$	$s>0$
2a.2	Unit step	$u(t)$	$\frac{1}{s}$	$s>0$
2b	Delayed unit step	$u(t-\tau )$	$\frac{e^{-\tau s}}{s}$	$s>0$
2c	Ramp	$t\cdot u(t)$	$\frac{1}{s^2}$	$s>0$
2d	nth Power with frequency shift	$\frac{t^n}{n!}{e}^{-\alpha t}\cdot u(t)$	$\frac{1}{(s+\alpha {)}^{n+1}}$	$s>-\alpha$
2d.1	Exponential decay	$e^{-\alpha t}\cdot u(t)$	$\frac{1}{s+\alpha }$	$s>-\alpha$
3	Exponential approach	$(1-e^{-\alpha t})\cdot u(t)$	$\frac{\alpha }{s(s+\alpha )}$	$s>0$
4	Sine	$\sin (\omega t)\cdot u(t)$	$\frac{\omega }{s^2+{\omega }^2}$	$s>0$
5	Cosine	$\cos (\omega t)\cdot u(t)$	$\frac{s}{s^2+{\omega }^2}$	$s>0$
6	Hyperbolic sine	$\sinh (\alpha t)\cdot u(t)$	$\frac{\alpha }{s^2-{\alpha }^2}$	$s>|\alpha |$
7	Hyperbolic cosine	$\cosh (\alpha t)\cdot u(t)$	$\frac{s}{s^2-{\alpha }^2}$	$s>|\alpha |$
8	Exponentially-decaying sine	$e^{-\alpha t}\sin (\omega t)\cdot u(t)$	$\frac{\omega }{(s+\alpha {)}^2+{\omega }^2}$	$s>-\alpha$
9	Exponentially-decaying cosine	$e^{-\alpha t}\cos (\omega t)\cdot u(t)$	$\frac{s+\alpha }{(s+\alpha {)}^2+{\omega }^2}$	$s>-\alpha$
10	nth Root	$\sqrt[n]{t}\cdot u(t)$	$s^{-(n+1)/n}\cdot \mathrm{\Gamma }(1+\frac{1}{n})$	$s>0$
11	Natural logarithm	$\ln \left(\frac{t}{t_0}\right)\cdot u(t)$	$-\frac{t_0}{s}[\ln (t_{0}s)+\gamma ]$	$s>0$
12	Bessel function of the first kind, of order n	$J_n(\omega t)\cdot u(t)$	$\frac{{\omega }^n{(s+\sqrt{s^2+{\omega }^2})}^{-n}}{\sqrt{s^2+{\omega }^2}}$	$s>0$ $(n>-1)$
13	Modified Bessel function of the first kind, of order n	$I_n(\omega t)\cdot u(t)$	$\frac{{\omega }^n{(s+\sqrt{s^2-{\omega }^2})}^{-n}}{\sqrt{s^2-{\omega }^2}}$	$s>|\omega |$
14	Bessel function of the second kind, of order 0	$Y_0(\alpha t)\cdot u(t)$
15	Modified Bessel function of the second kind, of order 0	$K_0(\alpha t)\cdot u(t)$
16	Função de erro	$\mathrm{e}\mathrm{r}\mathrm{f}(t)\cdot u(t)$	$\frac{e^{s^2/4}\mathrm{erfc}(s/2)}{s}$	$s>0$
Notas explicativas: * $u(t)$ representa a função de Heaviside. * $t$, um numero real, tipicamente representa tempo, embora possa representar qualquer dimensão independentemente. * $\delta (t)$ \represents a função Delta de Dirac. * $s$ é o complex angular frequency. * $\mathrm{\Gamma }(z)$ representa a função Gama. * $\alpha$, $\beta$, $\tau$, and $\omega$ são números reais. * $\gamma$ é a constante de Euler-Mascheroni. * $n$é uma integral. A causal system is a system where the impulse response h(t) is zero for all time t prior to t = 0. In general, the ROC for causal systems is not the same as the ROC for anticausal systems. See also causality.

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