Table of selected Laplace transforms
 Table of selected Laplace transforms Math Laplace Transform Definition In order to prevent spam, users must register before they can edit or create articles.

## 1 Table of selected Laplace transforms

ID Function Time domain
$LaTeX: x(t) = \mathcal{L}^{-1} \left\{ X(s) \right\}$
Laplace s-domain
$LaTeX: X(s) = \mathcal{L}\left\{ x(t) \right\}$
Region of convergence
for causal systems
1 ideal delay $LaTeX: \delta(t-\tau) \$ $LaTeX: e^{-\tau s} \$
1a unit impulse $LaTeX: \delta(t) \$ $LaTeX: 1 \$ $LaTeX: \mathrm{all} \ s \,$
2 delayed nth power
with frequency shift
$LaTeX: \frac{(t-\tau)^n}{n!} e^{-\alpha (t-\tau)} \cdot u(t-\tau)$ $LaTeX: \frac{e^{-\tau s}}{(s+\alpha)^{n+1}}$ $LaTeX: s > 0 \,$
2a nth power
( for integer n )
$LaTeX: { t^n \over n! } \cdot u(t)$ $LaTeX: { 1 \over s^{n+1} }$ $LaTeX: s > 0 \,$
2a.1 qth power
( for real q )
$LaTeX: { t^q \over \Gamma(q+1) } \cdot u(t)$ $LaTeX: { 1 \over s^{q+1} }$ $LaTeX: s > 0 \,$
2a.2 unit step $LaTeX: u(t) \$ $LaTeX: { 1 \over s }$ $LaTeX: s > 0 \,$
2b delayed unit step $LaTeX: u(t-\tau) \$ $LaTeX: { e^{-\tau s} \over s }$ $LaTeX: s > 0 \,$
2c ramp $LaTeX: t \cdot u(t)\$ $LaTeX: \frac{1}{s^2}$ $LaTeX: s > 0 \,$
2d nth power with frequency shift $LaTeX: \frac{t^{n}}{n!}e^{-\alpha t} \cdot u(t)$ $LaTeX: \frac{1}{(s+\alpha)^{n+1}}$ $LaTeX: s > - \alpha \,$
2d.1 exponential decay $LaTeX: e^{-\alpha t} \cdot u(t) \$ $LaTeX: { 1 \over s+\alpha }$ $LaTeX: s > - \alpha \$
3 exponential approach $LaTeX: ( 1-e^{-\alpha t}) \cdot u(t) \$ $LaTeX: \frac{\alpha}{s(s+\alpha)}$ $LaTeX: s > 0\$
4 sine $LaTeX: \sin(\omega t) \cdot u(t) \$ $LaTeX: { \omega \over s^2 + \omega^2 }$ $LaTeX: s > 0 \$
5 cosine $LaTeX: \cos(\omega t) \cdot u(t) \$ $LaTeX: { s \over s^2 + \omega^2 }$ $LaTeX: s > 0 \$
6 hyperbolic sine $LaTeX: \sinh(\alpha t) \cdot u(t) \$ $LaTeX: { \alpha \over s^2 - \alpha^2 }$ $LaTeX: s > | \alpha | \$
7 hyperbolic cosine $LaTeX: \cosh(\alpha t) \cdot u(t) \$ $LaTeX: { s \over s^2 - \alpha^2 }$ $LaTeX: s > | \alpha | \$
8 Exponentially-decaying
sine wave
$LaTeX: e^{\alpha t} \sin(\omega t) \cdot u(t) \$ $LaTeX: { \omega \over (s-\alpha )^2 + \omega^2 }$ $LaTeX: s > \alpha \$
9 Exponentially-decaying
cosine wave
$LaTeX: e^{\alpha t} \cos(\omega t) \cdot u(t) \$ $LaTeX: { s-\alpha \over (s-\alpha )^2 + \omega^2 }$ $LaTeX: s > \alpha \$
10 nth root $LaTeX: \sqrt[n]{t} \cdot u(t)$ $LaTeX: s^{-(n+1)/n} \cdot \Gamma\left(1+\frac{1}{n}\right)$ $LaTeX: s > 0 \,$
11 natural logarithm $LaTeX: \ln \left ( { t \over t_0 } \right ) \cdot u(t)$ $LaTeX: - { t_0 \over s} \ [ \ \ln(t_0 s)+\gamma \ ]$ $LaTeX: s > 0 \,$
12 Bessel function
of the first kind,
of order n
$LaTeX: J_n( \omega t) \cdot u(t)$ $LaTeX: \frac{ \omega^n \left(s+\sqrt{s^2+ \omega^2}\right)^{-n}}{\sqrt{s^2 + \omega^2}}$ $LaTeX: s > 0 \,$
$LaTeX: (n > -1) \,$
13 Modified Bessel function
of the first kind,
of order n
$LaTeX: I_n(\omega t) \cdot u(t)$ $LaTeX: \frac{ \omega^n \left(s+\sqrt{s^2-\omega^2}\right)^{-n}}{\sqrt{s^2-\omega^2}}$ $LaTeX: s > | \omega | \,$
14 Bessel function
of the second kind,
of order 0
$LaTeX: Y_0(\alpha t) \cdot u(t)$ $LaTeX: -{2 \sinh^{-1}(s/\alpha) \over \pi \sqrt{s^2+\alpha^2}}$ $LaTeX: s > 0 \,$
15 Modified Bessel function
of the second kind,
of order 0
$LaTeX: K_0(\alpha t) \cdot u(t)$
16 Error function $LaTeX: \mathrm{erf}(t) \cdot u(t)$ $LaTeX: {e^{s^2/4} \left(1 - \operatorname{erf} \left(s/2\right)\right) \over s}$ $LaTeX: s > 0 \,$

### 1.1 Explanatory notes

• $LaTeX: u(t) \,$ represents the Heaviside step function.
• $LaTeX: \delta(t) \,$ represents the Dirac delta function.
• $LaTeX: \Gamma (z) \,$ represents the Gamma function.
• $LaTeX: \gamma \,$ is the Euler-Mascheroni constant.
• $LaTeX: t \,$, a real number, typically represents time,
although it can represent any independent dimension.
• $LaTeX: s \,$ is the complex angular frequency.
• $LaTeX: \alpha \,$, $LaTeX: \beta \,$, $LaTeX: \tau \,$, and $LaTeX: \omega \,$ are real numbers.
• $LaTeX: n \,$, 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.