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 $\mathrm {erf} (t)\cdot u(t)$ ${e^{s^{2}/4}\operatorname {erfc} \left(s/2\right) \over s}$ $s>0\,$ 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. $\alpha \,$ , $\beta \,$ , $\tau \,$ , 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.