List of Laplace transforms

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The following is a list of Laplace transforms for many common functions of a single variable. [1] The Laplace transform is an integral transform that takes a function of a positive real variable t (often time) to a function of a complex variable s (complex angular frequency).

Contents

Properties

The Laplace transform of a function can be obtained using the formal definition of the Laplace transform. However, some properties of the Laplace transform can be used to obtain the Laplace transform of some functions more easily.

Linearity

For functions and and for scalar , the Laplace transform satisfies

and is, therefore, regarded as a linear operator.

Time shifting

The Laplace transform of is .

Frequency shifting

is the Laplace transform of .

Explanatory notes

The unilateral Laplace transform takes as input a function whose time domain is the non-negative reals, which is why all of the time domain functions in the table below are multiples of the Heaviside step function, u(t).

The entries of the table that involve a time delay τ are required to be causal (meaning that τ > 0). 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 region of convergence for causal systems is not the same as that of anticausal systems.

The following functions and variables are used in the table below:

Table

FunctionTime domain
Laplace s-domain
Region of convergenceReference
unit impulse all sinspection
delayed impulseRe(s) > 0time shift of
unit impulse [2]
unit step Re(s) > 0integrate unit impulse
delayed unit stepRe(s) > 0time shift of
unit step [3]
ramp Re(s) > 0integrate unit
impulse twice
nth power
(for integer n)
Re(s) > 0
(n > −1)
Integrate unit
step n times
qth power
(for complex q)
Re(s) > 0
Re(q) > −1
[4] [5]
nth rootRe(s) > 0Set q = 1/n above.
nth power with frequency shiftRe(s) > −αIntegrate unit step,
apply frequency shift
delayed nth power
with frequency shift
Re(s) > −αIntegrate unit step,
apply frequency shift,
apply time shift
exponential decay Re(s) > −αFrequency shift of
unit step
two-sided exponential decay
(only for bilateral transform)
α < Re(s) < αFrequency shift of
unit step
exponential approachRe(s) > 0Unit step minus
exponential decay
sine Re(s) > 0 [6]
cosine Re(s) > 0 [6]
hyperbolic sine Re(s) > |α| [7]
hyperbolic cosine Re(s) > |α| [7]
exponentially decaying
sine wave
Re(s) > −α [6]
exponentially decaying
cosine wave
Re(s) > −α [6]
natural logarithm Re(s) > 0 [7]
Bessel function
of the first kind,
of order n
Re(s) > 0
(n > −1)
[7]
Error function Re(s) > 0 [7]

See also

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References

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