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Category:Book:Engineering Tables#Laplace%20Transform%20Table%202%20

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.

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Engineering Tables/Laplace Transform Table 2