# 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. |||||