Engineering Tables/Laplace Transform Table 2

ID	Function	Time domain $x(t)={\mathcal {L}}^{-1}\left\{X(s)\right\}$	Laplace domain $X(s)={\mathcal {L}}\left\{x(t)\right\}$	Region of convergence for causal systems
1	Ideal delay	$\delta (t-\tau )\$	$e^{-\tau s}\$
1a	Unit impulse	$\delta (t)\$	$1\$	$\mathrm {all} \ s\,$
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	${t^{n} \over n!}\cdot u(t)$	${1 \over s^{n+1}}$	$s>0\,$
2a.1	qth Power	${t^{q} \over \Gamma (q+1)}\cdot u(t)$	${1 \over s^{q+1}}$	$s>0\,$
2a.2	Unit step	$u(t)\$	${1 \over s}$	$s>0\,$
2b	Delayed unit step	$u(t-\tau )\$	${e^{-\tau s} \over s}$	$s>0\,$
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)\$	${1 \over s+\alpha }$	$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)\$	${\omega \over s^{2}+\omega ^{2}}$	$s>0\$
5	Cosine	$\cos(\omega t)\cdot u(t)\$	${s \over s^{2}+\omega ^{2}}$	$s>0\$
6	Hyperbolic sine	$\sinh(\alpha t)\cdot u(t)\$	${\alpha \over s^{2}-\alpha ^{2}}$	$s>|\alpha |\$
7	Hyperbolic cosine	$\cosh(\alpha t)\cdot u(t)\$	${s \over s^{2}-\alpha ^{2}}$	$s>|\alpha |\$
8	Exponentially-decaying sine	$e^{-\alpha t}\sin(\omega t)\cdot u(t)\$	${\omega \over (s+\alpha )^{2}+\omega ^{2}}$	$s>-\alpha \$
9	Exponentially-decaying cosine	$e^{-\alpha t}\cos(\omega t)\cdot u(t)\$	${s+\alpha \over (s+\alpha )^{2}+\omega ^{2}}$	$s>-\alpha \$
10	nth Root	${\sqrt[{n}]{t}}\cdot u(t)$	$s^{-(n+1)/n}\cdot \Gamma \left(1+{\frac {1}{n}}\right)$	$s>0\,$
11	Natural logarithm	$\ln \left({t \over t_{0}}\right)\cdot u(t)$	$-{t_{0} \over s}\ [\ \ln(t_{0}s)+\gamma \ ]$	$s>0\,$
12	Bessel function of the first kind, of order n	$J_{n}(\omega t)\cdot u(t)$	${\frac {\omega ^{n}\left(s+{\sqrt {s^{2}+\omega ^{2}}}\right)^{-n}}{\sqrt {s^{2}+\omega ^{2}}}}$	$s>0\,$ $(n>-1)\,$
13	Modified Bessel function of the first kind, of order n	$I_{n}(\omega t)\cdot u(t)$	${\frac {\omega ^{n}\left(s+{\sqrt {s^{2}-\omega ^{2}}}\right)^{-n}}{\sqrt {s^{2}-\omega ^{2}}}}$	$s>|\omega |\,$
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	$\mathrm {erf} (t)\cdot u(t)$	${e^{s^{2}/4}\operatorname {erfc} \left(s/2\right) \over s}$	$s>0\,$
17	Constant	$C$	$\displaystyle {\frac {C}{s}}$
Explanatory notes: $u(t)\,$ represents the Heaviside step function. $\delta (t)\,$ represents the Dirac delta function. $\Gamma (z)\,$ 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. $\alpha \,$, $\beta \,$, $\tau \,$, $\omega \,$ and $C$ 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.