# Engineering Tables/Laplace Transform Table 2

ID Function Time domain
${\displaystyle x(t)={\mathcal {L}}^{-1}\left\{X(s)\right\}}$
Laplace domain
${\displaystyle X(s)={\mathcal {L}}\left\{x(t)\right\}}$
Region of convergence
for causal systems
1 Ideal delay ${\displaystyle \delta (t-\tau )\ }$ ${\displaystyle e^{-\tau s}\ }$
1a Unit impulse ${\displaystyle \delta (t)\ }$ ${\displaystyle 1\ }$ ${\displaystyle \mathrm {all} \ s\,}$
2 Delayed nth power with frequency shift ${\displaystyle {\frac {(t-\tau )^{n}}{n!}}e^{-\alpha (t-\tau )}\cdot u(t-\tau )}$ ${\displaystyle {\frac {e^{-\tau s}}{(s+\alpha )^{n+1}}}}$ ${\displaystyle s>0\,}$
2a nth Power ${\displaystyle {t^{n} \over n!}\cdot u(t)}$ ${\displaystyle {1 \over s^{n+1}}}$ ${\displaystyle s>0\,}$
2a.1 qth Power ${\displaystyle {t^{q} \over \Gamma (q+1)}\cdot u(t)}$ ${\displaystyle {1 \over s^{q+1}}}$ ${\displaystyle s>0\,}$
2a.2 Unit step ${\displaystyle u(t)\ }$ ${\displaystyle {1 \over s}}$ ${\displaystyle s>0\,}$
2b Delayed unit step ${\displaystyle u(t-\tau )\ }$ ${\displaystyle {e^{-\tau s} \over s}}$ ${\displaystyle s>0\,}$
2c Ramp ${\displaystyle t\cdot u(t)\ }$ ${\displaystyle {\frac {1}{s^{2}}}}$ ${\displaystyle s>0\,}$
2d nth Power with frequency shift ${\displaystyle {\frac {t^{n}}{n!}}e^{-\alpha t}\cdot u(t)}$ ${\displaystyle {\frac {1}{(s+\alpha )^{n+1}}}}$ ${\displaystyle s>-\alpha \,}$
2d.1 Exponential decay ${\displaystyle e^{-\alpha t}\cdot u(t)\ }$ ${\displaystyle {1 \over s+\alpha }}$ ${\displaystyle s>-\alpha \ }$
3 Exponential approach ${\displaystyle (1-e^{-\alpha t})\cdot u(t)\ }$ ${\displaystyle {\frac {\alpha }{s(s+\alpha )}}}$ ${\displaystyle s>0\ }$
4 Sine ${\displaystyle \sin(\omega t)\cdot u(t)\ }$ ${\displaystyle {\omega \over s^{2}+\omega ^{2}}}$ ${\displaystyle s>0\ }$
5 Cosine ${\displaystyle \cos(\omega t)\cdot u(t)\ }$ ${\displaystyle {s \over s^{2}+\omega ^{2}}}$ ${\displaystyle s>0\ }$
6 Hyperbolic sine ${\displaystyle \sinh(\alpha t)\cdot u(t)\ }$ ${\displaystyle {\alpha \over s^{2}-\alpha ^{2}}}$ ${\displaystyle s>|\alpha |\ }$
7 Hyperbolic cosine ${\displaystyle \cosh(\alpha t)\cdot u(t)\ }$ ${\displaystyle {s \over s^{2}-\alpha ^{2}}}$ ${\displaystyle s>|\alpha |\ }$
8 Exponentially-decaying sine ${\displaystyle e^{-\alpha t}\sin(\omega t)\cdot u(t)\ }$ ${\displaystyle {\omega \over (s+\alpha )^{2}+\omega ^{2}}}$ ${\displaystyle s>-\alpha \ }$
9 Exponentially-decaying cosine ${\displaystyle e^{-\alpha t}\cos(\omega t)\cdot u(t)\ }$ ${\displaystyle {s+\alpha \over (s+\alpha )^{2}+\omega ^{2}}}$ ${\displaystyle s>-\alpha \ }$
10 nth Root ${\displaystyle {\sqrt[{n}]{t}}\cdot u(t)}$ ${\displaystyle s^{-(n+1)/n}\cdot \Gamma \left(1+{\frac {1}{n}}\right)}$ ${\displaystyle s>0\,}$
11 Natural logarithm ${\displaystyle \ln \left({t \over t_{0}}\right)\cdot u(t)}$ ${\displaystyle -{t_{0} \over s}\ [\ \ln(t_{0}s)+\gamma \ ]}$ ${\displaystyle s>0\,}$
12 Bessel function
of the first kind, of order n
${\displaystyle J_{n}(\omega t)\cdot u(t)}$ ${\displaystyle {\frac {\omega ^{n}\left(s+{\sqrt {s^{2}+\omega ^{2}}}\right)^{-n}}{\sqrt {s^{2}+\omega ^{2}}}}}$ ${\displaystyle s>0\,}$
${\displaystyle (n>-1)\,}$
13 Modified Bessel function
of the first kind, of order n
${\displaystyle I_{n}(\omega t)\cdot u(t)}$ ${\displaystyle {\frac {\omega ^{n}\left(s+{\sqrt {s^{2}-\omega ^{2}}}\right)^{-n}}{\sqrt {s^{2}-\omega ^{2}}}}}$ ${\displaystyle s>|\omega |\,}$
14 Bessel function
of the second kind, of order 0
${\displaystyle Y_{0}(\alpha t)\cdot u(t)}$
15 Modified Bessel function
of the second kind, of order 0
${\displaystyle K_{0}(\alpha t)\cdot u(t)}$
16 Error function ${\displaystyle \mathrm {erf} (t)\cdot u(t)}$ ${\displaystyle {e^{s^{2}/4}\operatorname {erfc} \left(s/2\right) \over s}}$ ${\displaystyle s>0\,}$
Explanatory notes:
 ${\displaystyle u(t)\,}$ represents the Heaviside step function. ${\displaystyle \delta (t)\,}$ represents the Dirac delta function. ${\displaystyle \Gamma (z)\,}$ represents the Gamma function. ${\displaystyle \gamma \,}$ is the Euler-Mascheroni constant. ${\displaystyle t\,}$, a real number, typically represents time, although it can represent any independent dimension. ${\displaystyle s\,}$ is the complex angular frequency. ${\displaystyle \alpha \,}$, ${\displaystyle \beta \,}$, ${\displaystyle \tau \,}$, and ${\displaystyle \omega \,}$ are real numbers. ${\displaystyle n\,}$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.