Engineering Tables/Laplace Transform Table 2

ID	Function	Time domain $x(t)={ℒ}^{-1}\{X(s)\}$	Laplace domain $X(s)=ℒ\{x(t)\}$	Region of convergence for causal systems
1	ideal delay	$\delta (t-\tau )$	$e^{-\tau s}$
1a	unit impulse	$\delta (t)$	$1$	$\mathrm{a}\mathrm{l}\mathrm{l}s$
2	delayed nth power with frequency shift	$\frac{(t-\tau {)}^n}{n!}{e}^{-\alpha (t-\tau )}\cdot u(t-\tau )$	$\frac{e^{-\tau s}}{(s+\alpha {)}^{n+1}}$	$s>0$
2a	nth power	$\frac{t^n}{n!}\cdot u(t)$	$\frac{1}{s^{n+1}}$	$s>0$
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 wave	$e^{-\alpha t}\sin (\omega t)\cdot u(t)$	$\frac{\omega }{(s+\alpha {)}^2+{\omega }^2}$	$s>-\alpha$
9	Exponentially-decaying cosine wave	$e^{-\alpha t}\cos (\omega t)\cdot u(t)$	$\frac{s+\alpha }{(s+\alpha {)}^2+{\omega }^2}$	$s>-\alpha$
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	Error function	$\mathrm{e}\mathrm{r}\mathrm{f}(t)\cdot u(t)$	$\frac{e^{s^2/4}\mathrm{erfc}(s/2)}{s}$	$s>0$
Explanatory notes: $u(t)$ represents the Heaviside step function. $\delta (t)$ represents the Dirac delta function. $\mathrm{\Gamma }(z)$ represents the Gamma function. $\gamma$ is the Euler-Mascheroni constant. $t$, a real number, typically represents time, although it can represent any independent dimension. $s$ is the complex angular frequency. $\alpha$, $\beta$, $\tau$, and $\omega$ are real numbers. $n$is an integer. 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.