Engineering Tables/Laplace Transform Table 2

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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.