Table of selected Laplace transforms

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Table of selected Laplace transforms
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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

2 See Also