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| Boilerpipe Text | In
mathematics
, the
Laplace transform
, named after
Pierre-Simon Laplace
(
), is an
integral transform
that converts a
function
of a
real
variable
(usually
â
â
, in the
time domain
) to a function of a
complex variable
(in the complex-valued
frequency domain
, also known as
s
-domain
or
s
-plane
). The functions are often denoted using a lowercase symbol for the time-domain function and the corresponding uppercase symbol for the frequency-domain function, e.g.
and
â
â
.
The transform is useful for converting
differentiation
and
integration
in the time domain into the algebraic operations
multiplication
and
division
in the Laplace domain (analogous to how
logarithms
are useful for simplifying multiplication and division into addition and subtraction). This gives the transform many applications in
science
and
engineering
, mostly as a tool for solving linear
differential equations
[
1
]
and
dynamical systems
by replacing
ordinary differential equations
and
integral equations
with
algebraic polynomial equations
, and by replacing
convolution
with
multiplication
.
[
2
]
[
3
]
For example, through the Laplace transform, the equation of the
simple harmonic oscillator
(
Hooke's law
)
is converted into the algebraic equation
which incorporates the
initial conditions
and
â
â
, and can be solved for the unknown function
â
â
. Once solved, the inverse Laplace transform can be used to transform it to the original domain. This is often aided by referencing tables such as that given
below
.
The Laplace transform is defined (for suitable functions
â
â
) by the
integral
where
â
â
is a
complex number
.
The Laplace transform is related to many other transforms. It is essentially the same as the
Mellin transform
and is closely related to the
Fourier transform
. Unlike for the Fourier transform, the Laplace transform of a function is often an
analytic function
, meaning that it can be expressed as a
power series
that converges locally, the coefficients of which represent the
moments
of the original function. Moreover, the techniques of
complex analysis
, especially
contour integrals
, can be used for simplifying calculations.
Pierre-Simon, marquis de Laplace
The Laplace transform is named after
mathematician
and
astronomer
Pierre-Simon, Marquis de Laplace
, who used a similar transform in his work on
probability theory
.
[
4
]
[
5
]
Laplace wrote extensively about the use of
generating functions
(1814), and the integral form of the Laplace transform evolved naturally as a result.
[
6
]
Laplace's use of generating functions was similar to what is now known as the
z-transform
, and he gave little attention to the
continuous variable
case which was discussed by
Niels Henrik Abel
.
[
7
]
From 1744,
Leonhard Euler
investigated integrals of the form
as solutions of differential equations, introducing in particular the
gamma function
.
[
8
]
Joseph-Louis Lagrange
was an admirer of Euler and, in his work on integrating
probability density functions
, investigated expressions of the form
which resembles a Laplace transform.
[
9
]
[
10
]
These types of integrals seem first to have attracted Laplace's attention in 1782, where he was following in the spirit of Euler in using the integrals themselves as solutions of equations.
[
11
]
However, in 1785, Laplace took the critical step forward when, rather than simply looking for a solution in the form of an integral, he started to apply the transforms in the sense that was later to become popular. He used an integral of the form
akin to a
Mellin transform
, to transform the whole of a
difference equation
, in order to look for solutions of the transformed equation. He then went on to apply the Laplace transform in the same way and started to derive some of its properties, beginning to appreciate its potential power.
[
12
]
Laplace also recognised that
Joseph Fourier
's method of
Fourier series
for solving the
diffusion equation
could only apply to a limited region of space, because those solutions were
periodic
. In 1809, Laplace applied his transform to find solutions that diffused indefinitely in space.
[
13
]
In 1821,
Cauchy
developed an
operational calculus
for the Laplace transform that could be used to study
linear differential equations
in much the same way the transform is now used in basic engineering. This method was popularized, and perhaps rediscovered, by
Oliver Heaviside
around the turn of the century.
[
14
]
Bernhard Riemann
used the Laplace transform in his 1859 paper
On the number of primes less than a given magnitude
, in which he also developed the inversion theorem. Riemann used the Laplace transform to develop the functional equation of the
Riemann zeta function
, and his method is still used to relate the
modular transformation law
of the
Jacobi theta function
, which is readily proved via
Poisson summation
, to the functional equation.
[
15
]
Hjalmar Mellin
was among the first to study the Laplace transform, rigorously in the
Karl Weierstrass
school of analysis, and apply it to the study of
differential equations
and
special functions
, at the turn of the 20th century.
[
16
]
At around the same time, Heaviside was busy with his operational calculus.
Thomas Joannes Stieltjes
considered a generalization of the Laplace transform connected to his
work on moments
. Other contributors in this time period included
Mathias Lerch
,
[
17
]
Oliver Heaviside
, and
Thomas Bromwich
.
[
18
]
In 1929,
Vannevar Bush
and
Norbert Wiener
published
Operational Circuit Analysis
as a text for engineering analysis of electrical circuits, applying both Fourier transforms and operational calculus, and in which they included one of the first predecessors of the modern table of Laplace transforms.
In 1934,
Raymond Paley
and
Norbert Wiener
published the important work
Fourier transforms in the complex domain
, about what is now called the Laplace transform (see below). Also during the 30s, the Laplace transform was instrumental in
Godfrey Harold Hardy
and
John Edensor Littlewood
's study of
tauberian theorems
, and this application was later expounded on by
Widder (1941)
, who developed other aspects of the theory such as a new method for inversion.
Edward Charles Titchmarsh
wrote the influential
Introduction to the theory of the Fourier integral
(1937).
The current widespread use of the transform (mainly in engineering) came about during and soon after
World War II
,
[
19
]
replacing the earlier Heaviside
operational calculus
. The advantages of the Laplace transform had been emphasized by
Gustav Doetsch
.
[
20
]
for various complex frequencies in the
s
-domain
â
â
, which can be expressed as
â
â
. The axis at
contains pure cosines. Positive
contains
damped cosines
. Negative
contains
exponentially growing
cosines.
The Laplace transform of a
function
f
(
t
)
, defined for all
real numbers
t
â„ 0
, is the function
F
(
s
)
, which is a unilateral transform defined by
[
citation needed
]
   (
Eq. 1
)
where
s
is a
complex
frequency-domain parameter
with real numbers
Ï
and
Ï
.
An alternate notation for the Laplace transform is
instead of
F
.
[
3
]
Thus
in
functional notation
. This is often written, especially in engineering settings, as
â
â
, with the understanding that the
dummy variable
does not appear in the function
â
â
.
The meaning of the integral depends on types of functions of interest. A necessary condition for existence of the integral is that
f
must be
locally integrable
on
[0, â)
. For locally integrable functions that decay at infinity or are of
exponential type
(
â
â
), the integral can be understood to be a (proper)
Lebesgue integral
. However, for many applications it is necessary to regard it as a
conditionally convergent
improper integral
at
â
. Still more generally, the integral can be understood in a
weak sense
, and this is dealt with below.
One can define the Laplace transform of a finite
Borel measure
Ό
by the Lebesgue integral
[
21
]
An important special case is where
Ό
is a
probability measure
, for example, the
Dirac delta function
. In
operational calculus
, the Laplace transform of a measure is often treated as though the measure came from a probability density function
f
. In that case, to avoid potential confusion, one often writes
where the lower limit of
0
â
is shorthand notation for
This limit emphasizes that any
point mass
located at
0
is entirely captured by the Laplace transform. Although with the Lebesgue integral, it is not necessary to take such a limit, it does appear more naturally in connection with the
LaplaceâStieltjes transform
.
Bilateral Laplace transform
[
edit
]
When one says "the Laplace transform" without qualification, the unilateral or one-sided transform is usually intended. The Laplace transform can be alternatively defined as the
bilateral Laplace transform
, or
two-sided Laplace transform
, by extending the limits of integration to be the entire real axis. If that is done, the common unilateral transform becomes a special case of the bilateral transform, where the definition of the function being transformed includes being multiplied by the
Heaviside step function
.
The bilateral Laplace transform
F
(
s
)
is defined as follows:
   (
Eq. 2
)
An alternate notation for the bilateral Laplace transform is
â
â
, instead of
F
.
Inverse Laplace transform
[
edit
]
Two integrable functions have the same Laplace transform only if they differ on a set of
Lebesgue measure
zero. This means that, on the range of the transform, there is an inverse transform. In fact, besides integrable functions, the Laplace transform is a
one-to-one mapping
from one function space into another in many other function spaces as well, although there is usually no easy characterization of the range.
Typical function spaces in which this is true include the spaces of bounded continuous functions, the space
L
â
(0, â)
, or more generally
tempered distributions
on
(0, â)
. The Laplace transform is also defined and injective for suitable spaces of tempered distributions.
In these cases, the image of the Laplace transform lives in a space of
analytic functions
in the
region of convergence
. The
inverse Laplace transform
is given by the following complex integral, which is known by various names (the
Bromwich integral
, the
FourierâMellin integral
, and
Mellin's inverse formula
):
   (
Eq. 3
)
where
Îł
is a real number so that the contour path of integration is in the region of convergence of
F
(
s
)
. In most applications, the contour can be closed, allowing the use of the
residue theorem
. An alternative formula for the inverse Laplace transform is given by
Post's inversion formula
. The limit here is interpreted in the
weak-* topology
.
In practice, it is typically more convenient to decompose a Laplace transform into known transforms of functions obtained from a table and construct the inverse by inspection.
In
pure
and
applied probability
, the Laplace transform is defined as an
expected value
. If
X
is a
random variable
with probability density function
f
, then the Laplace transform of
f
is given by the expectation
where
is the
expectation
of
random variable
â
â
.
By
convention
, this is referred to as the Laplace transform of the random variable
X
itself. Here, replacing
s
by
â
t
gives the
moment generating function
of
X
. The Laplace transform has applications throughout probability theory, including
first passage times
of
stochastic processes
such as
Markov chains
, and
renewal theory
.
Of particular use is the ability to recover the
cumulative distribution function
of a continuous random variable
X
by means of the Laplace transform as follows:
[
22
]
Algebraic construction
[
edit
]
The Laplace transform can be alternatively defined in a purely algebraic manner by applying a
field of fractions
construction to the convolution
ring
of functions on the positive half-line. The resulting
space of abstract operators
is exactly equivalent to Laplace space, but in this construction the forward and reverse transforms never need to be explicitly defined (avoiding the related difficulties with proving convergence).
[
23
]
Region of convergence
[
edit
]
If
f
is a locally integrable function (or more generally a Borel measure locally of bounded variation), then the Laplace transform
F
(
s
)
of
f
converges provided that the limit
exists.
The Laplace transform
converges absolutely
if the integral
exists as a proper Lebesgue integral. The Laplace transform is usually understood as
conditionally convergent
, meaning that it converges in the former but not in the latter sense.
The set of values for which
F
(
s
)
converges absolutely is either of the form
Re(
s
) >
a
or
Re(
s
) â„
a
, where
a
is an
extended real constant
with
ââ â€
a
†â
(a consequence of the
dominated convergence theorem
). The constant
a
is known as the abscissa of absolute convergence, and depends on the growth behavior of
f
(
t
)
.
[
24
]
Analogously, the two-sided transform converges absolutely in a strip of the form
a
< Re(
s
) <
b
, and possibly including the lines
Re(
s
) =
a
or
Re(
s
) =
b
.
[
25
]
The subset of values of
s
for which the Laplace transform converges absolutely is called the region of absolute convergence, or the domain of absolute convergence. In the two-sided case, it is sometimes called the strip of absolute convergence. The Laplace transform is analytic in the region of absolute convergence: this is a consequence of
Fubini's theorem
and
Morera's theorem
.
Similarly, the set of values for which
F
(
s
)
converges (conditionally or absolutely) is known as the region of conditional convergence, or simply the
region of convergence
(ROC). If the Laplace transform converges (conditionally) at
s
=
s
0
, then it automatically converges for all
s
with
Re(
s
) > Re(
s
0
)
. Therefore, the region of convergence is a half-plane of the form
Re(
s
) >
a
, possibly including some points of the boundary line
Re(
s
) =
a
.
In the region of convergence
Re(
s
) > Re(
s
0
)
, the Laplace transform of
f
can be expressed by
integrating by parts
as the integral
That is,
F
(
s
)
can effectively be expressed, in the region of convergence, as the absolutely convergent Laplace transform of some other function. In particular, it is analytic. In its most general form, the Laplace transform gives a one-to-one correspondence between the holomorphic functions which, for some
â
â
, are defined on
and are bounded there in absolute value by a polynomial, and the
distributions
on the real line supported on
â
â
which become
tempered distributions
after multiplied by
for some
â
â
.
[
26
]
There are several
PaleyâWiener theorems
concerning the relationship between the decay properties of
f
, and the properties of the Laplace transform within the region of convergence.
In engineering applications, a function corresponding to a
linear time-invariant (LTI) system
is
stable
if every bounded input produces a bounded output. This is equivalent to the absolute convergence of the Laplace transform of the impulse response function in the region
Re(
s
) â„ 0
. As a result, LTI systems are stable, provided that the poles of the Laplace transform of the impulse response function have negative real part.
This ROC is used in knowing about the causality and stability of a system.
Properties and theorems
[
edit
]
The Laplace transform's key property is that it converts
differentiation
and
integration
in the time domain into multiplication and division by
s
in the Laplace domain. Thus, the Laplace variable
s
is also known as an
operator variable
in the Laplace domain: either the
derivative operator
or (for
s
â1
)
the
integration operator
.
Given the functions
f
(
t
)
and
g
(
t
)
, and their respective Laplace transforms
F
(
s
)
and
G
(
s
)
,
the following table is a list of properties of unilateral Laplace transform:
[
27
]
Properties of the unilateral Laplace transform
Property
Time domain
s
domain
Comment
Linearity
Can be proved using basic rules of integration.
Frequency-domain derivative
F
âČ
is the first derivative of
F
with respect to
s
.
Frequency-domain general derivative
More general form,
n
th derivative of
F
(
s
)
.
Derivative
f
is assumed to be a
differentiable function
, and its derivative is assumed to be of exponential type. This can then be obtained by integration by parts
Second derivative
f
is assumed twice differentiable and the second derivative to be of exponential type. Follows by applying the Differentiation property to
f
âČ(
t
)
.
General derivative
f
is assumed to be
n
-times differentiable, with
n
th derivative of exponential type. Follows by
mathematical induction
.
Frequency-domain
integration
This is deduced using the nature of frequency differentiation and conditional convergence.
Time-domain integration
u
(
t
)
is the Heaviside step function and
(
u
â
f
)(
t
)
is the
convolution
of
u
(
t
)
and
f
(
t
)
.
Frequency shifting
Time shifting
a
> 0
,
u
(
t
)
is the Heaviside step function
Time scaling
a
> 0
Multiplication
The integration is done along the vertical line
Re(
Ï
) =
c
that lies entirely within the region of convergence of
F
.
[
28
]
Convolution
Circular convolution
For periodic functions with period
T
.
Complex conjugation
Periodic function
f
(
t
)
is a periodic function of period
T
so that
f
(
t
) =
f
(
t
+
T
)
, for all
t
â„ 0
. This is the result of the time shifting property and the
geometric series
.
Periodic summation
Initial value theorem
Final value theorem
â
â
, if all
poles
of
are in the left half-plane.
The final value theorem is useful because it gives the long-term behaviour without having to perform
partial fraction
decompositions (or other difficult algebra). If
F
(
s
)
has a pole in the right-hand plane or poles on the imaginary axis (e.g., if
or
â
â
), then the behaviour of this formula is undefined.
Relation to power series
[
edit
]
The Laplace transform can be viewed as a
continuous
analogue of a
power series
.
[
29
]
If
a
(
n
)
is a discrete function of a positive integer
n
, then the power series associated to
a
(
n
)
is the series
where
x
is a real variable (see
Z-transform
). Replacing summation over
n
with integration over
t
, a continuous version of the power series becomes
where the discrete function
a
(
n
)
is replaced by the continuous one
f
(
t
)
.
Changing the base of the power from
x
to
e
gives
For this to converge for, say, all bounded functions
f
, it is necessary to require that
ln
x
< 0
. Making the substitution
â
s
= ln
x
gives just the Laplace transform:
In other words, the Laplace transform is a continuous analog of a power series, in which the discrete parameter
n
is replaced by the continuous parameter
t
, and
x
is replaced by
e
â
s
.
Analogously to a power series, if
â
â
, then the power series converges to an analytic function in
â
â
, if
â
â
, the Laplace transform converges to an analytic function for
â
â
.
[
30
]
Relation to moments
[
edit
]
The quantities
are the
moments
of the function
f
. If the first
n
moments of
f
converge absolutely, then by repeated
differentiation under the integral
,
This is of special significance in probability theory, where the moments of a random variable
X
are given by the expectation values
â
â
. Then, the relation holds
Transform of a function's derivative
[
edit
]
It is often convenient to use the differentiation property of the Laplace transform to find the transform of a function's derivative. This can be derived from the basic expression for a Laplace transform as follows:
yielding
and in the bilateral case,
The general result
where
denotes the
n
th derivative of
f
, can then be established with an inductive argument.
Evaluating integrals over the positive real axis
[
edit
]
A useful property of the Laplace transform is the following:
under suitable assumptions on the behaviour of
â
â
and
â
â
in a right neighbourhood of
and on the decay rate of
â
â
and
â
â
in a left neighbourhood of
â
â
. The above formula is a variation of integration by parts, with the operators
and
being replaced by
and
â
â
. Let us prove the equivalent formulation:
By plugging in
the left-hand side turns into:
but assuming Fubini's theorem holds, by reversing the order of integration we get the wanted right-hand side.
This method can be used to compute integrals that would otherwise be difficult to compute using elementary methods of real calculus. For example,
Relationship to other transforms
[
edit
]
LaplaceâStieltjes transform
[
edit
]
The (unilateral) LaplaceâStieltjes transform of a function
g
 : â â â
is defined by the
LebesgueâStieltjes integral
The function
g
is assumed to be of
bounded variation
. If
g
is the
antiderivative
of
f
:
then the LaplaceâStieltjes transform of
g
and the Laplace transform of
f
coincide. In general, the LaplaceâStieltjes transform is the Laplace transform of the
Stieltjes measure
associated to
g
. So in practice, the only distinction between the two transforms is that the Laplace transform is thought of as operating on the density function of the measure, whereas the LaplaceâStieltjes transform is thought of as operating on its
cumulative distribution function
.
[
31
]
Let
be a complex-valued Lebesgue integrable function supported on
â
â
, and let
be its Laplace transform. Then, within the region of convergence, we have
which is the Fourier transform of the function
â
â
.
[
32
]
Indeed, the
Fourier transform
is a special case (under certain conditions) of the bilateral Laplace transform. The main difference is that the Fourier transform of a function is a complex function of a
real
variable (frequency
â
â
), the Laplace transform of a function is a complex function of a
complex
variable (damping factor
and frequency
â
â
). The Laplace transform is usually restricted to transformation of functions of
t
with
t
â„ 0
. A consequence of this restriction is that the Laplace transform of a function is a
holomorphic function
of the variable
s
. Unlike the Fourier transform, the Laplace transform of a
distribution
is generally a
well-behaved
function. Techniques of complex variables can also be used to directly study Laplace transforms. As a holomorphic function, the Laplace transform has a
power series
representation. This power series expresses a function as a linear superposition of
moments
of the function. This perspective has applications in probability theory.
Formally, the Fourier transform is equivalent to evaluating the bilateral Laplace transform with imaginary argument
s
=
iÏ
[
33
]
[
34
]
when the condition explained below is fulfilled,
This convention of the Fourier transform (
â
â
in
Fourier transform § Other conventions
) requires a factor of
â
1
/
2
Ï
â
on the inverse Fourier transform. This relationship between the Laplace and Fourier transforms is often used to determine the
frequency spectrum
of a
signal
or dynamical system.
The above relation is valid as stated
if and only if
the region of convergence (ROC) of
F
(
s
)
contains the imaginary axis,
Ï
= 0
.
For example, the function
f
(
t
) = cos(
Ï
0
t
)
has a Laplace transform
F
(
s
) =
s
/(
s
2
+
Ï
0
2
)
whose ROC is
Re(
s
) > 0
. As
s
=
iÏ
0
is a pole of
F
(
s
)
, substituting
s
=
iÏ
in
F
(
s
)
does not yield the Fourier transform of
f
(
t
)
u
(
t
)
, which contains terms proportional to the
Dirac delta functions
ÎŽ
(
Ï
±
Ï
0
)
.
However, a relation of the form
holds under much weaker conditions. For instance, this holds for the above example provided that the limit is understood as a
weak limit
of measures (see
vague topology
). General conditions relating the limit of the Laplace transform of a function on the boundary to the Fourier transform take the form of
PaleyâWiener theorems
.
The Mellin transform and its inverse are related to the two-sided Laplace transform by a change of variables.
If in the Mellin transform
we set
Ξ
=
e
â
t
we get a two-sided Laplace transform.
The unilateral or one-sided Z-transform is the Laplace transform of an ideally sampled signal with the substitution of
where
T
= 1/
f
s
is the
sampling interval
(in units of time e.g., seconds) and
f
s
is the
sampling rate
(in
samples per second
or
hertz
).
Let
be a sampling impulse train (also called a
Dirac comb
) and
be the sampled representation of the continuous-time
x
(
t
)
The Laplace transform of the sampled signal
x
q
(
t
)
is
This is the precise definition of the unilateral Z-transform of the discrete function
x
[
n
]
with the substitution of
z
â
e
sT
.
Comparing the last two equations, we find the relationship between the unilateral Z-transform and the Laplace transform of the sampled signal,
The similarity between the Z- and Laplace transforms is expanded upon in the theory of
time scale calculus
.
The integral form of the
Borel transform
is a special case of the Laplace transform for
f
an
entire function
of exponential type, meaning that
for some constants
A
and
B
. The generalized Borel transform allows a different weighting function to be used, rather than the exponential function, to transform functions not of exponential type.
Nachbin's theorem
gives necessary and sufficient conditions for the Borel transform to be well defined.
Fundamental relationships
[
edit
]
Since an ordinary Laplace transform can be written as a special case of a two-sided transform, and since the two-sided transform can be written as the sum of two one-sided transforms, the theory of the Laplace-, Fourier-, Mellin-, and Z-transforms are at bottom the same subject. However, a different point of view and different characteristic problems are associated with each of these four major integral transforms.
Table of selected Laplace transforms
[
edit
]
The following table provides Laplace transforms for many common functions of a single variable.
[
35
]
[
36
]
For definitions and explanations, see the
Explanatory Notes
at the end of the table.
Because the Laplace transform is a linear operator,
The Laplace transform of a sum is the sum of Laplace transforms of each term.
The Laplace transform of a multiple of a function is that multiple times the Laplace transformation of that function.
Using this linearity, and various
trigonometric
,
hyperbolic
, and complex number (etc.) properties and/or identities, some Laplace transforms can be obtained from others more quickly than by using the definition directly.
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
Ï
are required to be
causal
(meaning that
Ï
> 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
.
Selected Laplace transforms
Function
Time domain
Laplace
s
-domain
Region of convergence
Reference
unit impulse
all
s
inspection
delayed impulse
all
s
time shift of
unit impulse
unit step
integrate unit impulse
delayed unit step
time shift of
unit step
product of delayed function and delayed step
u-substitution,
rectangular impulse
ramp
integrate unit
impulse twice
n
th power
(for integer
n
)
(
n
> â1
)
integrate unit
step
n
times
q
th power
(for complex
q
)
[
37
]
[
38
]
n
th root
Set
q
= 1/
n
above.
n
th power with frequency shift
Integrate unit step,
apply frequency shift
delayed
n
th power
with frequency shift
integrate unit step,
apply frequency shift,
apply time shift
exponential decay
Frequency shift of
unit step
two-sided
exponential decay
(only for bilateral transform)
Frequency shift of
unit step
exponential approach
unit step minus
exponential decay
sine
[
39
]
cosine
[
39
]
hyperbolic sine
[
40
]
hyperbolic cosine
[
40
]
exponentially decaying
sine wave
[
39
]
exponentially decaying
cosine wave
[
39
]
natural logarithm
[
40
]
Bessel function
of the first kind,
of order
n
(
n
> â1
)
[
41
]
Error function
[
41
]
Explanatory notes:
u
(
t
)
represents the
Heaviside step function
.
ÎŽ
represents the
Dirac delta function
.
Î(
z
)
represents the
gamma function
.
Îł
is the
EulerâMascheroni constant
.
t
, a real number, typically represents
time
, although it can represent
any
independent dimension.
s
is the
complex
frequency domain parameter, and
Re(
s
)
is its
real part
.
α
,
ÎČ
,
Ï
, and
Ï
are
real numbers
.
n
is an
integer
.
s
-domain equivalent circuits and impedances
[
edit
]
The Laplace transform is often used in
circuit analysis
by conversions to the
s
-domain of circuit elements. Circuit elements can be transformed into
impedances
, very similar to
phasor
impedances.
Here is a summary of equivalents:
s
-domain equivalent circuits
Note that the resistor is exactly the same in the time domain and the
s
-domain. The sources are put in if there are initial conditions on the circuit elements. For example, if a capacitor has an initial voltage across it, or if the inductor has an initial current through it, the sources inserted in the
s
-domain account for that.
The equivalents for current and voltage sources are derived from the transformations in the table above.
Examples and applications
[
edit
]
The Laplace transform is used frequently in
engineering
and
physics
; the output of a
linear time-invariant system
can be calculated by convolving its unit impulse response with the input signal. Performing this calculation in Laplace space turns the convolution into a multiplication; the latter being easier to solve because of its algebraic form. For more information, see
control theory
. The Laplace transform is invertible on a large class of functions. Given a mathematical or functional description of an input or output to a
system
, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.
[
42
]
The Laplace transform can also be used to solve differential equations and is used extensively in
mechanical engineering
and
electrical engineering
. The Laplace transform reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of algebra. The original differential equation can then be solved by applying the inverse Laplace transform. English electrical engineer
Oliver Heaviside
first proposed a similar scheme, although without using the Laplace transform; and the resulting operational calculus is credited as the Heaviside calculus.
Evaluating improper integrals
[
edit
]
Let
â
â
. Then (see the table above)
From which one gets:
In the limit
â
â
, one gets
provided that the interchange of limits can be justified. This is often possible as a consequence of the
final value theorem
. Even when the interchange cannot be justified the calculation can be suggestive. For example, with
a
â 0 â
b
, proceeding formally one has
Complex impedance of a capacitor
[
edit
]
In the theory of
electrical circuits
, the current flow in a
capacitor
is proportional to the capacitance and rate of change in the electrical potential (with equations as for the
SI
unit system). Symbolically, this is expressed by the differential equation
where
C
is the capacitance of the capacitor,
i
=
i
(
t
)
is the
electric current
through the capacitor as a function of time, and
v
=
v
(
t
)
is the
voltage
across the terminals of the capacitor, also as a function of time.
Taking the Laplace transform of this equation, we obtain
where
and
Solving for
V
(
s
)
we have
The definition of the complex impedance
Z
(in
ohms
) is the ratio of the complex voltage
V
divided by the complex current
I
while holding the initial state
V
0
at zero:
Using this definition and the previous equation, we find:
which is the correct expression for the complex impedance of a capacitor. In addition, the Laplace transform has large applications in control theory.
Consider a linear time-invariant system with
transfer function
The
impulse response
is the inverse Laplace transform of this transfer function:
Partial fraction expansion
To evaluate this inverse transform, we begin by expanding
H
(
s
)
using the method of partial fraction expansion,
The unknown constants
P
and
R
are the
residues
located at the corresponding poles of the transfer function. Each residue represents the relative contribution of that
singularity
to the transfer function's overall shape.
By the
residue theorem
, the inverse Laplace transform depends only upon the poles and their residues. To find the residue
P
, we multiply both sides of the equation by
s
+
α
to get
Then by letting
s
= â
α
, the contribution from
R
vanishes and all that is left is
Similarly, the residue
R
is given by
Note that
and so the substitution of
R
and
P
into the expanded expression for
H
(
s
)
gives
Finally, using the linearity property and the known transform for exponential decay (see
Item
#
3
in the
Table of Laplace Transforms
, above), we can take the inverse Laplace transform of
H
(
s
)
to obtain
which is the impulse response of the system.
Convolution
The same result can be achieved using the
convolution property
as if the system is a series of filters with transfer functions
1/(
s
+
α
)
and
1/(
s
+
ÎČ
)
. That is, the inverse of
is
Time function
Laplace transform
Starting with the Laplace transform,
we find the inverse by first rearranging terms in the fraction:
We are now able to take the inverse Laplace transform of our terms:
This is just the
sine of the sum
of the arguments, yielding:
We can apply similar logic to find that
Statistical mechanics
[
edit
]
In
statistical mechanics
, the Laplace transform of the density of states
defines the
partition function
.
[
43
]
That is, the canonical partition function
is given by
and the inverse is given by
Spatial (not time) structure from astronomical spectrum
[
edit
]
The wide and general applicability of the Laplace transform and its inverse is illustrated by an application in astronomy which provides some information on the
spatial distribution
of matter of an
astronomical
source of
radiofrequency
thermal radiation
too distant to
resolve
as more than a point, given its
flux density
spectrum
, rather than relating the
time
domain with the spectrum (frequency domain).
Assuming certain properties of the object, e.g. spherical shape and constant temperature, calculations based on carrying out an inverse Laplace transformation on the spectrum of the object can produce the only possible
model
of the distribution of matter in it (density as a function of distance from the center) consistent with the spectrum.
[
44
]
When independent information on the structure of an object is available, the inverse Laplace transform method has been found to be in good agreement.
Birth and death processes
[
edit
]
Consider a
random walk
, with steps
occurring with probabilities
â
â
.
[
45
]
Suppose also that the time step is a
Poisson process
, with parameter
â
â
. Then the probability of the walk being at the lattice point
at time
is
This leads to a system of
integral equations
(or equivalently a system of differential equations). However, because it is a system of convolution equations, the Laplace transform converts it into a
system of linear equations
for
namely:
which may now be solved by standard methods.
The Laplace transform of the measure
on
is given by
It is intuitively clear that, for small
â
â
, the exponentially decaying integrand will become more sensitive to the concentration of the measure
on larger subsets of the domain. To make this more precise, introduce the distribution function:
Formally, we expect a limit of the following kind:
Tauberian theorems
are theorems relating the asymptotics of the Laplace transform, as
â
â
, to those of the distribution of
as
â
â
. They are thus of importance in asymptotic formulae of
probability
and
statistics
, where often the spectral side has asymptotics that are simpler to infer.
[
45
]
Two Tauberian theorems of note are the
HardyâLittlewood Tauberian theorem
and
Wiener's Tauberian theorem
. The Wiener theorem generalizes the
Ikehara Tauberian theorem
, which is the following statement:
Let
â
â
be a non-negative,
monotonic
nondecreasing function of
â
â
, defined for
â
â
. Suppose that
converges for
â
â
to the function
â
â
and that, for some non-negative number
â
â
,
has an extension as a
continuous function
for
â
â
.
Then the
limit
as
â
â
goes to infinity of
â
â
is equal toÂ
â
â
.
This statement can be applied in particular to the
logarithmic derivative
of
Riemann zeta function
, and thus provides an extremely short way to prove the
prime number theorem
.
[
46
]
Analog signal processing
Bernstein's theorem on monotone functions
Continuous-repayment mortgage
Dirichlet integral
Differential equation
Generating function
Hamburger moment problem
HardyâLittlewood Tauberian theorem
LaplaceâCarson transform
Moment-generating function
Nonlocal operator
Partial fraction decomposition
Post's inversion formula
Signal-flow graph
Transfer function
Z-transform
^
Lynn, Paul A. (1986), "The Laplace Transform and the
z
-transform",
Electronic Signals and Systems
, London: Macmillan Education UK, pp.Â
225â
272,
doi
:
10.1007/978-1-349-18461-3_6
,
ISBN
Â
978-0-333-39164-8
,
Laplace Transform and the z-transform are closely related to the Fourier Transform. Laplace Transform is somewhat more general in scope than the Fourier Transform, and is widely used by engineers for describing continuous circuits and systems, including automatic control systems.
^
"Differential Equations â Laplace Transforms"
,
Pauls Online Math Notes
, retrieved
2020-08-08
^
a
b
Weisstein, Eric W.,
"Laplace Transform"
,
Wolfram MathWorld
, retrieved
2020-08-08
^
Deakin, Michael A. B. (1981),
"The Development of the Laplace Transform, 1737-1937: I. Euler to Spitzer, 1737-1880"
,
Archive for History of Exact Sciences
,
25
(4):
343â
390,
ISSN
Â
0003-9519
^
"Des Fonctions génératrices" [On generating functions],
Théorie analytique des Probabilités
[
Analytical Probability Theory
] (in French) (2nd ed.), Paris, 1814, chap.I sect.2-20
^
Jaynes, E. T. (Edwin T.) (2003),
Probability theory : the logic of science
, Bretthorst, G. Larry, Cambridge, UK: Cambridge University Press,
ISBN
Â
0511065892
,
OCLC
Â
57254076
^
Abel, Niels H.
(1820), "Sur les fonctions génératrices et leurs déterminantes",
Ćuvres ComplĂštes
(in French), vol. II (published 1839), pp.Â
77â
88
1881 edition
^
Euler 1744
,
Euler 1753
,
Euler 1769
^
Lagrange 1773
^
Grattan-Guinness 1997
, p. 260
^
Grattan-Guinness 1997
, p. 261
^
Grattan-Guinness 1997
, pp. 261â262
^
Grattan-Guinness 1997
, pp. 262â266
^
Heaviside, Oliver
(January 2008),
"The solution of definite integrals by differential transformation"
,
Electromagnetic Theory
, vol. III, London, section 526,
ISBN
Â
9781605206189
{{
citation
}}
: CS1 maint: location missing publisher (
link
)
^
Edwards, H. M.
(1974),
Riemann's Zeta Function
, New York: Academic Press,
ISBN
Â
0-12-232750-0
,
Zbl
Â
0315.10035
^
Gardner, Murray F.; Barnes, John L. (1942),
Transients in Linear Systems studied by the Laplace Transform
, New York: Wiley
, Appendix C
^
Lerch, Mathias
(1903), "Sur un point de la théorie des fonctions génératrices d'Abel" [Proof of the inversion formula],
Acta Mathematica
(in French),
27
:
339â
351,
doi
:
10.1007/BF02421315
,
hdl
:
10338.dmlcz/501554
^
Bromwich, Thomas J.
(1916),
"Normal coordinates in dynamical systems"
,
Proceedings of the London Mathematical Society
,
15
:
401â
448,
doi
:
10.1112/plms/s2-15.1.401
^
An influential book was:
Gardner, Murray F.; Barnes, John L. (1942),
Transients in Linear Systems studied by the Laplace Transform
, New York: Wiley
^
Doetsch, Gustav (1937),
Theorie und Anwendung der Laplacesche Transformation
[
Theory and Application of the Laplace Transform
] (in German), Berlin: Springer
translation 1943
^
Feller 1971
, §XIII.1.
^
The cumulative distribution function is the integral of the probability density function.
^
MikusiĆski, Jan (14 July 2014),
Operational Calculus
, Elsevier,
ISBN
Â
9781483278933
^
Widder 1941
, Chapter II, §1
^
Widder 1941
, Chapter VI, §2
^
Beffa, Federico (2024),
"Laplace Transform of Distributions"
,
Weakly Nonlinear Systems
, Cham: Springer Nature Switzerland, pp.Â
75â
85,
doi
:
10.1007/978-3-031-40681-2_5
,
ISBN
Â
978-3-031-40680-5
, retrieved
2026-01-14
{{
citation
}}
: CS1 maint: work parameter with ISBN (
link
)
^
Korn & Korn 1967
, pp. 226â227
^
Bracewell 2000
, Table 14.1, p. 385
^
Archived at
Ghostarchive
and the
Wayback Machine
:
Mattuck, Arthur (7 November 2008),
"Where the Laplace Transform comes from"
,
YouTube
^
Widder 1941
, p. 38.
^
Feller 1971
, p. 432
^
Laurent Schwartz
(1966),
Mathematics for the physical sciences
, Addison-Wesley
, p 224.
^
Titchmarsh, E.
(1986) [1948],
Introduction to the theory of Fourier integrals
(2nd ed.),
Clarendon Press
, p. 6,
ISBN
Â
978-0-8284-0324-5
^
Takacs 1953
, p. 93
^
Riley, K. F.; Hobson, M. P.; Bence, S. J. (2010),
Mathematical methods for physics and engineering
(3rd ed.), Cambridge University Press, p. 455,
ISBN
Â
978-0-521-86153-3
^
Distefano, J. J.; Stubberud, A. R.; Williams, I. J. (1995),
Feedback systems and control
, Schaum's outlines (2nd ed.), McGraw-Hill, p. 78,
ISBN
Â
978-0-07-017052-0
^
Lipschutz, S.; Spiegel, M. R.; Liu, J. (2009),
Mathematical Handbook of Formulas and Tables
, Schaum's Outline Series (3rd ed.), McGraw-Hill, p. 183,
ISBN
Â
978-0-07-154855-7
â provides the case for real
q
.
^
http://mathworld.wolfram.com/LaplaceTransform.html
â Wolfram Mathword provides case for complex
q
^
a
b
c
d
Bracewell 1978
, p. 227.
^
a
b
c
Williams 1973
, p. 88.
^
a
b
Williams 1973
, p. 89.
^
Korn & Korn 1967
, §8.1
^
RK Pathria; Paul Beal (1996),
Statistical mechanics
(2nd ed.), Butterworth-Heinemann, p.Â
56
,
ISBN
Â
9780750624695
^
Salem, M.; Seaton, M. J. (1974), "I. Continuum spectra and brightness contours",
Monthly Notices of the Royal Astronomical Society
,
167
:
493â
510,
Bibcode
:
1974MNRAS.167..493S
,
doi
:
10.1093/mnras/167.3.493
, and
Salem, M. (1974), "II. Three-dimensional models",
Monthly Notices of the Royal Astronomical Society
,
167
:
511â
516,
Bibcode
:
1974MNRAS.167..511S
,
doi
:
10.1093/mnras/167.3.511
^
a
b
Feller,
Introduction to Probability Theory, volume II,pp=479-483
^
S. Ikehara
(1931), "An extension of Landau's theorem in the analytic theory of numbers",
Journal of Mathematics and Physics
,
10
(
1â
4):
1â
12,
doi
:
10.1002/sapm19311011
,
Zbl
Â
0001.12902
Bracewell, Ronald N. (1978),
The Fourier Transform and its Applications
(2nd ed.), McGraw-Hill Kogakusha,
ISBN
Â
978-0-07-007013-4
Bracewell, R. N. (2000),
The Fourier Transform and Its Applications
(3rd ed.), Boston: McGraw-Hill,
ISBN
Â
978-0-07-116043-8
Feller, William
(1971),
An introduction to probability theory and its applications. Vol. II.
, Second edition, New York:
John Wiley & Sons
,
MR
Â
0270403
Korn, G. A.;
Korn, T. M.
(1967),
Mathematical Handbook for Scientists and Engineers
(2nd ed.), McGraw-Hill Companies,
ISBN
Â
978-0-07-035370-1
Widder, David Vernon (1941),
The Laplace Transform
, Princeton Mathematical Series, v. 6,
Princeton University Press
,
MR
Â
0005923
Williams, J. (1973),
Laplace Transforms
, Problem Solvers, George Allen & Unwin,
ISBN
Â
978-0-04-512021-5
Takacs, J. (1953), "Fourier amplitudok meghatarozasa operatorszamitassal",
Magyar Hiradastechnika
(in Hungarian),
IV
(
7â
8):
93â
96
Euler, L.
(1744), "De constructione aequationum" [The Construction of Equations],
Opera Omnia
, 1st series (in Latin),
22
:
150â
161
Euler, L.
(1753), "Methodus aequationes differentiales" [A Method for Solving Differential Equations],
Opera Omnia
, 1st series (in Latin),
22
:
181â
213
Euler, L.
(1992) [1769], "Institutiones calculi integralis, Volume 2" [Institutions of Integral Calculus],
Opera Omnia
, 1st series (in Latin),
12
, Basel: BirkhÀuser,
ISBN
Â
978-3764314743
{{
citation
}}
: CS1 maint: work parameter with ISBN (
link
)
, Chapters 3â5
Euler, Leonhard
(1769),
Institutiones calculi integralis
[
Institutions of Integral Calculus
] (in Latin), vol. II, Paris: Petropoli, ch. 3â5, pp. 57â153
Grattan-Guinness, I
(1997), "Laplace's integral solutions to partial differential equations", in Gillispie, C. C. (ed.),
Pierre Simon Laplace 1749â1827: A Life in Exact Science
, Princeton: Princeton University Press,
ISBN
Â
978-0-691-01185-1
Lagrange, J. L.
(1773),
Mémoire sur l'utilité de la méthode
, Ćuvres de Lagrange, vol. 2, pp.Â
171â
234
Arendt, Wolfgang; Batty, Charles J.K.; Hieber, Matthias; Neubrander, Frank (2002),
Vector-Valued Laplace Transforms and Cauchy Problems
, BirkhÀuser Basel,
ISBN
Â
978-3-7643-6549-3
Davies, Brian (2002),
Integral transforms and their applications
(Third ed.), New York: Springer,
ISBN
Â
978-0-387-95314-4
Deakin, M. A. B. (1981), "The development of the Laplace transform",
Archive for History of Exact Sciences
,
25
(4):
343â
390,
doi
:
10.1007/BF01395660
,
S2CID
Â
117913073
Deakin, M. A. B. (1982), "The development of the Laplace transform",
Archive for History of Exact Sciences
,
26
(4):
351â
381,
doi
:
10.1007/BF00418754
,
S2CID
Â
123071842
Doetsch, Gustav
(1974),
Introduction to the Theory and Application of the Laplace Transformation
, Springer,
ISBN
Â
978-0-387-06407-9
Mathews, Jon; Walker, Robert L. (1970),
Mathematical methods of physics
(2nd ed.), New York: W. A. Benjamin,
ISBN
Â
0-8053-7002-1
Polyanin, A. D.; Manzhirov, A. V. (1998),
Handbook of Integral Equations
, Boca Raton: CRC Press,
ISBN
Â
978-0-8493-2876-3
Schwartz, Laurent
(1952), "Transformation de Laplace des distributions",
Comm. SĂ©m. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.]
(in French),
1952
:
196â
206,
MR
Â
0052555
Schwartz, Laurent
(2008) [1966],
Mathematics for the Physical Sciences
, Dover Books on Mathematics, New York: Dover Publications, pp.Â
215â
241,
ISBN
Â
978-0-486-46662-0
â see Chapter VI. The Laplace transform
Siebert, William McC. (1986),
Circuits, Signals, and Systems
, Cambridge, Massachusetts: MIT Press,
ISBN
Â
978-0-262-19229-3
Widder, David Vernon (1945), "What is the Laplace transform?",
The American Mathematical Monthly
,
52
(8):
419â
425,
doi
:
10.2307/2305640
,
ISSN
Â
0002-9890
,
JSTOR
Â
2305640
,
MR
Â
0013447
Weidman, J.A.C.; Fornberg, Bengt (2023), "Fully numerical Laplace transform methods",
Numerical Algorithms
,
92
:
985â
1006,
doi
:
10.1007/s11075-022-01368-x
"Laplace transform"
,
Encyclopedia of Mathematics
,
EMS Press
, 2001 [1994]
Online Computation
of the transform or inverse transform, wims.unice.fr
Tables of Integral Transforms
at EqWorld: The World of Mathematical Equations.
Weisstein, Eric W.
,
"Laplace Transform"
,
MathWorld
Good explanations of the initial and final value theorems
Archived
2009-01-08 at the
Wayback Machine
Laplace Transforms
at MathPages
Computational Knowledge Engine
allows to easily calculate Laplace Transforms and its inverse Transform.
Laplace Calculator
to calculate Laplace Transforms online easily.
Code to visualize Laplace Transforms
and many example videos. |
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## Contents
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- [(Top)](https://en.wikipedia.org/wiki/Laplace_transform)
- [1 History](https://en.wikipedia.org/wiki/Laplace_transform#History)
- [2 Formal definition](https://en.wikipedia.org/wiki/Laplace_transform#Formal_definition)
Toggle Formal definition subsection
- [2\.1 Bilateral Laplace transform](https://en.wikipedia.org/wiki/Laplace_transform#Bilateral_Laplace_transform)
- [2\.2 Inverse Laplace transform](https://en.wikipedia.org/wiki/Laplace_transform#Inverse_Laplace_transform)
- [2\.3 Probability theory](https://en.wikipedia.org/wiki/Laplace_transform#Probability_theory)
- [2\.4 Algebraic construction](https://en.wikipedia.org/wiki/Laplace_transform#Algebraic_construction)
- [3 Region of convergence](https://en.wikipedia.org/wiki/Laplace_transform#Region_of_convergence)
- [4 Properties and theorems](https://en.wikipedia.org/wiki/Laplace_transform#Properties_and_theorems)
Toggle Properties and theorems subsection
- [4\.1 Relation to power series](https://en.wikipedia.org/wiki/Laplace_transform#Relation_to_power_series)
- [4\.2 Relation to moments](https://en.wikipedia.org/wiki/Laplace_transform#Relation_to_moments)
- [4\.3 Transform of a function's derivative](https://en.wikipedia.org/wiki/Laplace_transform#Transform_of_a_function's_derivative)
- [4\.4 Evaluating integrals over the positive real axis](https://en.wikipedia.org/wiki/Laplace_transform#Evaluating_integrals_over_the_positive_real_axis)
- [5 Relationship to other transforms](https://en.wikipedia.org/wiki/Laplace_transform#Relationship_to_other_transforms)
Toggle Relationship to other transforms subsection
- [5\.1 LaplaceâStieltjes transform](https://en.wikipedia.org/wiki/Laplace_transform#Laplace%E2%80%93Stieltjes_transform)
- [5\.2 Fourier transform](https://en.wikipedia.org/wiki/Laplace_transform#Fourier_transform)
- [5\.3 Mellin transform](https://en.wikipedia.org/wiki/Laplace_transform#Mellin_transform)
- [5\.4 Z-transform](https://en.wikipedia.org/wiki/Laplace_transform#Z-transform)
- [5\.5 Borel transform](https://en.wikipedia.org/wiki/Laplace_transform#Borel_transform)
- [5\.6 Fundamental relationships](https://en.wikipedia.org/wiki/Laplace_transform#Fundamental_relationships)
- [6 Table of selected Laplace transforms](https://en.wikipedia.org/wiki/Laplace_transform#Table_of_selected_Laplace_transforms)
- [7 *s*\-domain equivalent circuits and impedances](https://en.wikipedia.org/wiki/Laplace_transform#s-domain_equivalent_circuits_and_impedances)
- [8 Examples and applications](https://en.wikipedia.org/wiki/Laplace_transform#Examples_and_applications)
Toggle Examples and applications subsection
- [8\.1 Evaluating improper integrals](https://en.wikipedia.org/wiki/Laplace_transform#Evaluating_improper_integrals)
- [8\.2 Complex impedance of a capacitor](https://en.wikipedia.org/wiki/Laplace_transform#Complex_impedance_of_a_capacitor)
- [8\.3 Impulse response](https://en.wikipedia.org/wiki/Laplace_transform#Impulse_response)
- [8\.4 Phase delay](https://en.wikipedia.org/wiki/Laplace_transform#Phase_delay)
- [8\.5 Statistical mechanics](https://en.wikipedia.org/wiki/Laplace_transform#Statistical_mechanics)
- [8\.6 Spatial (not time) structure from astronomical spectrum](https://en.wikipedia.org/wiki/Laplace_transform#Spatial_\(not_time\)_structure_from_astronomical_spectrum)
- [8\.7 Birth and death processes](https://en.wikipedia.org/wiki/Laplace_transform#Birth_and_death_processes)
- [8\.8 Tauberian theory](https://en.wikipedia.org/wiki/Laplace_transform#Tauberian_theory)
- [9 See also](https://en.wikipedia.org/wiki/Laplace_transform#See_also)
- [10 Notes](https://en.wikipedia.org/wiki/Laplace_transform#Notes)
- [11 References](https://en.wikipedia.org/wiki/Laplace_transform#References)
Toggle References subsection
- [11\.1 Modern](https://en.wikipedia.org/wiki/Laplace_transform#Modern)
- [11\.2 Historical](https://en.wikipedia.org/wiki/Laplace_transform#Historical)
- [12 Further reading](https://en.wikipedia.org/wiki/Laplace_transform#Further_reading)
- [13 External links](https://en.wikipedia.org/wiki/Laplace_transform#External_links)
Toggle the table of contents
# Laplace transform
61 languages
- [á ááá](https://am.wikipedia.org/wiki/%E1%88%8B%E1%8D%95%E1%88%8B%E1%88%B5_%E1%88%BD%E1%8C%8D%E1%8C%8D%E1%88%AD "áááá” áœááá â Amharic")
- [ۧÙŰč۱ۚÙŰ©](https://ar.wikipedia.org/wiki/%D8%AA%D8%AD%D9%88%D9%8A%D9%84_%D9%84%D8%A7%D8%A8%D9%84%D8%A7%D8%B3 "ŰȘŰÙÙÙ ÙۧۚÙۧ۳ â Arabic")
- [Asturianu](https://ast.wikipedia.org/wiki/Tresformada_de_Laplace "Tresformada de Laplace â Asturian")
- [ĐŃлгаŃŃĐșĐž](https://bg.wikipedia.org/wiki/%D0%9F%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D0%B5_%D0%BD%D0%B0_%D0%9B%D0%B0%D0%BF%D0%BB%D0%B0%D1%81 "ĐŃĐ”ĐŸĐ±ŃĐ°Đ·ĐŸĐČĐ°ĐœĐžĐ” ĐœĐ° ĐĐ°ĐżĐ»Đ°Ń â Bulgarian")
- [àŠŹàŠŸàŠàŠČàŠŸ](https://bn.wikipedia.org/wiki/%E0%A6%B2%E0%A6%BE%E0%A6%AA%E0%A7%8D%E0%A6%B2%E0%A6%BE%E0%A6%B8_%E0%A6%B0%E0%A7%82%E0%A6%AA%E0%A6%BE%E0%A6%A8%E0%A7%8D%E0%A6%A4%E0%A6%B0 "àŠČàŠŸàŠȘà§àŠČàŠŸàŠž àŠ°à§àŠȘàŠŸàŠšà§àŠ€àŠ° â Bangla")
- [Bosanski](https://bs.wikipedia.org/wiki/Laplaceova_transformacija "Laplaceova transformacija â Bosnian")
- [CatalĂ ](https://ca.wikipedia.org/wiki/Transformada_de_Laplace "Transformada de Laplace â Catalan")
- [ÄeĆĄtina](https://cs.wikipedia.org/wiki/Laplaceova_transformace "Laplaceova transformace â Czech")
- [Dansk](https://da.wikipedia.org/wiki/Laplacetransformation "Laplacetransformation â Danish")
- [Deutsch](https://de.wikipedia.org/wiki/Laplace-Transformation "Laplace-Transformation â German")
- [ÎλληΜÎčÎșÎŹ](https://el.wikipedia.org/wiki/%CE%9C%CE%B5%CF%84%CE%B1%CF%83%CF%87%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CF%83%CE%BC%CF%8C%CF%82_%CE%9B%CE%B1%CF%80%CE%BB%CE%AC%CF%82 "ÎΔÏαÏÏηΌαÏÎčÏÎŒÏÏ ÎαÏÎ»ÎŹÏ â Greek")
- [Esperanto](https://eo.wikipedia.org/wiki/Laplaca_transformo "Laplaca transformo â Esperanto")
- [Español](https://es.wikipedia.org/wiki/Transformada_de_Laplace "Transformada de Laplace â Spanish")
- [Eesti](https://et.wikipedia.org/wiki/Laplace%27i_teisendus "Laplace'i teisendus â Estonian")
- [Euskara](https://eu.wikipedia.org/wiki/Laplaceren_transformazio "Laplaceren transformazio â Basque")
- [Ùۧ۱۳Û](https://fa.wikipedia.org/wiki/%D8%AA%D8%A8%D8%AF%DB%8C%D9%84_%D9%84%D8%A7%D9%BE%D9%84%D8%A7%D8%B3 "ŰȘۚۯÛÙ ÙŰ§ÙŸÙۧ۳ â Persian")
- [Suomi](https://fi.wikipedia.org/wiki/Laplace-muunnos "Laplace-muunnos â Finnish")
- [Français](https://fr.wikipedia.org/wiki/Transformation_de_Laplace "Transformation de Laplace â French")
- [Galego](https://gl.wikipedia.org/wiki/Transformada_de_Laplace "Transformada de Laplace â Galician")
- [柹柶èȘ / Hak-kĂą-ngĂź](https://hak.wikipedia.org/wiki/L%C3%A0-ph%C3%BA-l%C3%A0-s%E1%B9%B3%CC%82_pien-von "LĂ -phĂș-lĂ -sáčłÌ pien-von â Hakka Chinese")
- [ŚąŚŚšŚŚȘ](https://he.wikipedia.org/wiki/%D7%94%D7%AA%D7%9E%D7%A8%D7%AA_%D7%9C%D7%A4%D7%9C%D7%A1 "ŚŚȘŚŚšŚȘ ŚŚ€ŚŚĄ â Hebrew")
- [à€čà€żà€šà„à€Šà„](https://hi.wikipedia.org/wiki/%E0%A4%B2%E0%A4%BE%E0%A4%AA%E0%A5%8D%E0%A4%B2%E0%A4%BE%E0%A4%B8_%E0%A4%B0%E0%A5%82%E0%A4%AA%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4%E0%A4%B0 "à€Čà€Ÿà€Șà„à€Čà€Ÿà€ž à€°à„à€Șà€Ÿà€šà„à€€à€° â Hindi")
- [Hrvatski](https://hr.wikipedia.org/wiki/Laplaceova_transformacija "Laplaceova transformacija â Croatian")
- [Magyar](https://hu.wikipedia.org/wiki/Laplace-transzform%C3%A1ci%C3%B3 "Laplace-transzformĂĄciĂł â Hungarian")
- [ŐŐĄŐ”Ő„ÖŐ„Ő¶](https://hy.wikipedia.org/wiki/%D4%BC%D5%A1%D5%BA%D5%AC%D5%A1%D5%BD%D5%AB_%D5%B1%D6%87%D5%A1%D6%83%D5%B8%D5%AD%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6 "ÔŒŐĄŐșŐŹŐĄŐœŐ« Ő±ÖŐĄÖŐžŐŐžÖŐ©Ő”ŐžÖŐ¶ â Armenian")
- [Interlingua](https://ia.wikipedia.org/wiki/Transformation_de_Laplace "Transformation de Laplace â Interlingua")
- [Bahasa Indonesia](https://id.wikipedia.org/wiki/Transformasi_Laplace "Transformasi Laplace â Indonesian")
- [Italiano](https://it.wikipedia.org/wiki/Trasformata_di_Laplace "Trasformata di Laplace â Italian")
- [æ„æŹèȘ](https://ja.wikipedia.org/wiki/%E3%83%A9%E3%83%97%E3%83%A9%E3%82%B9%E5%A4%89%E6%8F%9B "ă©ăă©ăčć€æ â Japanese")
- [Jawa](https://jv.wikipedia.org/wiki/Transformasi_Laplace "Transformasi Laplace â Javanese")
- [Taqbaylit](https://kab.wikipedia.org/wiki/Tabeddilt_n_Lapla%E1%B9%A3 "Tabeddilt n LaplaáčŁ â Kabyle")
- [ÒĐ°Đ·Đ°ÒŃĐ°](https://kk.wikipedia.org/wiki/%D0%9B%D0%B0%D0%BF%D0%BB%D0%B0%D1%81_%D1%82%D2%AF%D1%80%D0%BB%D0%B5%D0%BD%D0%B4%D1%96%D1%80%D1%83%D1%96 "ĐĐ°ĐżĐ»Đ°Ń ŃÒŻŃĐ»Đ”ĐœĐŽŃŃŃŃ â Kazakh")
- [áá¶áá¶ááááá](https://km.wikipedia.org/wiki/%E1%9E%94%E1%9F%86%E1%9E%9B%E1%9F%82%E1%9E%84%E1%9E%A1%E1%9E%B6%E1%9E%94%E1%9F%92%E1%9E%9B%E1%9E%B6%E1%9E%9F "ááááááĄá¶áááá¶á â Khmer")
- [íê”ìŽ](https://ko.wikipedia.org/wiki/%EB%9D%BC%ED%94%8C%EB%9D%BC%EC%8A%A4_%EB%B3%80%ED%99%98 "ëŒíëŒì€ ëłí â Korean")
- [LietuviĆł](https://lt.wikipedia.org/wiki/Laplaso_transformacija "Laplaso transformacija â Lithuanian")
- [àŽźàŽČàŽŻàŽŸàŽłàŽ](https://ml.wikipedia.org/wiki/%E0%B4%B2%E0%B4%BE%E0%B4%AA%E0%B5%8D%E0%B4%B2%E0%B5%87%E0%B4%B8%E0%B5%8D_%E0%B4%AA%E0%B4%B0%E0%B4%BF%E0%B4%B5%E0%B5%BC%E0%B4%A4%E0%B5%8D%E0%B4%A4%E0%B4%A8%E0%B4%82 "àŽČàŽŸàŽȘà”àŽČà”àŽžà” àŽȘàŽ°àŽżàŽ”à”ŒàŽ€à”àŽ€àŽšàŽ â Malayalam")
- [à€źà€°à€Ÿà€ à„](https://mr.wikipedia.org/wiki/%E0%A4%B2%E0%A5%85%E0%A4%AA%E0%A5%8D%E0%A4%B2%E0%A5%87%E0%A4%B8_%E0%A4%AA%E0%A4%B0%E0%A4%BF%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%A4%E0%A4%A8 "à€Čà„
à€Șà„à€Čà„à€ž à€Șà€°à€żà€”à€°à„à€€à€š â Marathi")
- [Nederlands](https://nl.wikipedia.org/wiki/Laplacetransformatie "Laplacetransformatie â Dutch")
- [Norsk nynorsk](https://nn.wikipedia.org/wiki/Laplace-transformasjon "Laplace-transformasjon â Norwegian Nynorsk")
- [Norsk bokmĂ„l](https://no.wikipedia.org/wiki/Laplacetransformasjon "Laplacetransformasjon â Norwegian BokmĂ„l")
- [Polski](https://pl.wikipedia.org/wiki/Transformacja_Laplace%E2%80%99a "Transformacja Laplaceâa â Polish")
- [PiemontĂšis](https://pms.wikipedia.org/wiki/Trasform%C3%A0_%C3%ABd_Laplace "TrasformĂ Ă«d Laplace â Piedmontese")
- [PortuguĂȘs](https://pt.wikipedia.org/wiki/Transformada_de_Laplace "Transformada de Laplace â Portuguese")
- [RomĂąnÄ](https://ro.wikipedia.org/wiki/Transformat%C4%83_Laplace "TransformatÄ Laplace â Romanian")
- [Đ ŃŃŃĐșĐžĐč](https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D0%B5_%D0%9B%D0%B0%D0%BF%D0%BB%D0%B0%D1%81%D0%B0 "ĐŃĐ”ĐŸĐ±ŃĐ°Đ·ĐŸĐČĐ°ĐœĐžĐ” ĐаплаŃĐ° â Russian")
- [Srpskohrvatski / ŃŃĐżŃĐșĐŸŃ
ŃĐČĐ°ŃŃĐșĐž](https://sh.wikipedia.org/wiki/Laplaceova_transformacija "Laplaceova transformacija â Serbo-Croatian")
- [Simple English](https://simple.wikipedia.org/wiki/Laplace_transform "Laplace transform â Simple English")
- [SlovenĆĄÄina](https://sl.wikipedia.org/wiki/Laplaceova_transformacija "Laplaceova transformacija â Slovenian")
- [Shqip](https://sq.wikipedia.org/wiki/Transformimi_i_Laplasit "Transformimi i Laplasit â Albanian")
- [ĐĄŃĐżŃĐșĐž / srpski](https://sr.wikipedia.org/wiki/%D0%9B%D0%B0%D0%BF%D0%BB%D0%B0%D1%81%D0%BE%D0%B2%D0%B0_%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D1%98%D0%B0 "ĐаплаŃĐŸĐČĐ° ŃŃĐ°ĐœŃŃĐŸŃĐŒĐ°ŃĐžŃĐ° â Serbian")
- [Sunda](https://su.wikipedia.org/wiki/Transformasi_Laplace "Transformasi Laplace â Sundanese")
- [Svenska](https://sv.wikipedia.org/wiki/Laplacetransform "Laplacetransform â Swedish")
- [àź€àźźàźżàźŽàŻ](https://ta.wikipedia.org/wiki/%E0%AE%87%E0%AE%B2%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AE%BF%E0%AE%B3%E0%AE%BE%E0%AE%9A%E0%AF%81_%E0%AE%AE%E0%AE%BE%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AF%81 "àźàźČàźȘàŻàźȘàźżàźłàźŸàźàŻ àźźàźŸàź±àŻàź±àŻ â Tamil")
- [àčàžàžą](https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%81%E0%B8%9B%E0%B8%A5%E0%B8%87%E0%B8%A5%E0%B8%B2%E0%B8%9B%E0%B8%A5%E0%B8%B1%E0%B8%AA "àžàžČàžŁàčàžàž„àžàž„àžČàžàž„àž±àžȘ â Thai")
- [TĂŒrkçe](https://tr.wikipedia.org/wiki/Laplace_d%C3%B6n%C3%BC%C5%9F%C3%BCm%C3%BC "Laplace dönĂŒĆĂŒmĂŒ â Turkish")
- [ĐŁĐșŃĐ°ŃĐœŃŃĐșĐ°](https://uk.wikipedia.org/wiki/%D0%9F%D0%B5%D1%80%D0%B5%D1%82%D0%B2%D0%BE%D1%80%D0%B5%D0%BD%D0%BD%D1%8F_%D0%9B%D0%B0%D0%BF%D0%BB%D0%B0%D1%81%D0%B0 "ĐĐ”ŃĐ”ŃĐČĐŸŃĐ”ĐœĐœŃ ĐаплаŃĐ° â Ukrainian")
- [Tiáșżng Viá»t](https://vi.wikipedia.org/wiki/Ph%C3%A9p_bi%E1%BA%BFn_%C4%91%E1%BB%95i_Laplace "PhĂ©p biáșżn Äá»i Laplace â Vietnamese")
- [ćŽèŻ](https://wuu.wikipedia.org/wiki/%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF%E5%8F%98%E6%8D%A2 "ææźææŻćæą â Wu")
- [é©ćèȘ / BĂąn-lĂąm-gĂ](https://zh-min-nan.wikipedia.org/wiki/Laplace_pi%C3%A0n-o%C4%81%E2%81%BF "Laplace piĂ n-oÄâż â Minnan")
- [çČ”èȘ](https://zh-yue.wikipedia.org/wiki/%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF%E8%AE%8A%E6%8F%9B "ææźææŻèźæ â Cantonese")
- [äžæ](https://zh.wikipedia.org/wiki/%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF%E5%8F%98%E6%8D%A2 "ææźææŻćæą â Chinese")
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From Wikipedia, the free encyclopedia
Integral transform useful in probability theory, physics, and engineering
In [mathematics](https://en.wikipedia.org/wiki/Mathematics "Mathematics"), the **Laplace transform**, named after [Pierre-Simon Laplace](https://en.wikipedia.org/wiki/Pierre-Simon_Laplace "Pierre-Simon Laplace") ([/lÉËplÉËs/](https://en.wikipedia.org/wiki/Help:IPA/English "Help:IPA/English")), is an [integral transform](https://en.wikipedia.org/wiki/Integral_transform "Integral transform") that converts a [function](https://en.wikipedia.org/wiki/Function_\(mathematics\) "Function (mathematics)") of a [real](https://en.wikipedia.org/wiki/Real_number "Real number") [variable](https://en.wikipedia.org/wiki/Variable_\(mathematics\) "Variable (mathematics)") (usually â t {\\displaystyle t}  â , in the *[time domain](https://en.wikipedia.org/wiki/Time_domain "Time domain")*) to a function of a [complex variable](https://en.wikipedia.org/wiki/Complex_number "Complex number") s {\\displaystyle s}  (in the complex-valued [frequency domain](https://en.wikipedia.org/wiki/Frequency_domain "Frequency domain"), also known as ***s*\-domain** or ***s*\-plane**). The functions are often denoted using a lowercase symbol for the time-domain function and the corresponding uppercase symbol for the frequency-domain function, e.g. x ( t ) {\\displaystyle x(t)}  and â X ( s ) {\\displaystyle X(s)}  â .
The transform is useful for converting [differentiation](https://en.wikipedia.org/wiki/Derivative "Derivative") and [integration](https://en.wikipedia.org/wiki/Integral "Integral") in the time domain into the algebraic operations [multiplication](https://en.wikipedia.org/wiki/Multiplication "Multiplication") and [division](https://en.wikipedia.org/wiki/Division_\(mathematics\) "Division (mathematics)") in the Laplace domain (analogous to how [logarithms](https://en.wikipedia.org/wiki/Logarithm "Logarithm") are useful for simplifying multiplication and division into addition and subtraction). This gives the transform many applications in [science](https://en.wikipedia.org/wiki/Science "Science") and [engineering](https://en.wikipedia.org/wiki/Engineering "Engineering"), mostly as a tool for solving linear [differential equations](https://en.wikipedia.org/wiki/Differential_equation "Differential equation")[\[1\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-Lynn_1986_pp._225%E2%80%93272-1) and [dynamical systems](https://en.wikipedia.org/wiki/Dynamical_system "Dynamical system") by replacing [ordinary differential equations](https://en.wikipedia.org/wiki/Ordinary_differential_equation "Ordinary differential equation") and [integral equations](https://en.wikipedia.org/wiki/Integral_equation "Integral equation") with [algebraic polynomial equations](https://en.wikipedia.org/wiki/Algebraic_equation "Algebraic equation"), and by replacing [convolution](https://en.wikipedia.org/wiki/Convolution "Convolution") with [multiplication](https://en.wikipedia.org/wiki/Multiplication "Multiplication").[\[2\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-2)[\[3\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-:1-3)
For example, through the Laplace transform, the equation of the [simple harmonic oscillator](https://en.wikipedia.org/wiki/Simple_harmonic_oscillator "Simple harmonic oscillator") ([Hooke's law](https://en.wikipedia.org/wiki/Hooke%27s_law "Hooke's law")) x âł ( t ) \+ k x ( t ) \= 0 {\\displaystyle x''(t)+kx(t)=0}  is converted into the algebraic equation s 2 X ( s ) â s x ( 0 ) â x âČ ( 0 ) \+ k X ( s ) \= 0 , {\\displaystyle s^{2}X(s)-sx(0)-x'(0)+kX(s)=0,}  which incorporates the [initial conditions](https://en.wikipedia.org/wiki/Initial_conditions "Initial conditions") x ( 0 ) {\\displaystyle x(0)}  and â x âČ ( 0 ) {\\displaystyle x'(0)}  â , and can be solved for the unknown function â X ( s ) {\\displaystyle X(s)}  â . Once solved, the inverse Laplace transform can be used to transform it to the original domain. This is often aided by referencing tables such as that given [below](https://en.wikipedia.org/wiki/Laplace_transform#Table_of_selected_Laplace_transforms).
The Laplace transform is defined (for suitable functions â f {\\displaystyle f}  â ) by the [integral](https://en.wikipedia.org/wiki/Integral "Integral") L { f } ( s ) \= â« 0 â f ( t ) e â s t d t , {\\displaystyle {\\mathcal {L}}\\{f\\}(s)=\\int \_{0}^{\\infty }f(t)e^{-st}\\,dt,}  where â s {\\displaystyle s}  â is a [complex number](https://en.wikipedia.org/wiki/Complex_number "Complex number").
The Laplace transform is related to many other transforms. It is essentially the same as the [Mellin transform](https://en.wikipedia.org/wiki/Mellin_transform "Mellin transform") and is closely related to the [Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform "Fourier transform"). Unlike for the Fourier transform, the Laplace transform of a function is often an [analytic function](https://en.wikipedia.org/wiki/Analytic_function "Analytic function"), meaning that it can be expressed as a [power series](https://en.wikipedia.org/wiki/Power_series "Power series") that converges locally, the coefficients of which represent the [moments](https://en.wikipedia.org/wiki/Moment_\(mathematics\) "Moment (mathematics)") of the original function. Moreover, the techniques of [complex analysis](https://en.wikipedia.org/wiki/Complex_analysis "Complex analysis"), especially [contour integrals](https://en.wikipedia.org/wiki/Contour_integral "Contour integral"), can be used for simplifying calculations.
## History
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[](https://en.wikipedia.org/wiki/File:Laplace,_Pierre-Simon,_marquis_de.jpg)
Pierre-Simon, marquis de Laplace
The Laplace transform is named after [mathematician](https://en.wikipedia.org/wiki/Mathematician "Mathematician") and [astronomer](https://en.wikipedia.org/wiki/Astronomer "Astronomer") [Pierre-Simon, Marquis de Laplace](https://en.wikipedia.org/wiki/Pierre-Simon_Laplace "Pierre-Simon Laplace"), who used a similar transform in his work on [probability theory](https://en.wikipedia.org/wiki/Probability_theory "Probability theory").[\[4\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-4)[\[5\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-5) Laplace wrote extensively about the use of [generating functions](https://en.wikipedia.org/wiki/Generating_function "Generating function") (1814), and the integral form of the Laplace transform evolved naturally as a result.[\[6\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-6)
Laplace's use of generating functions was similar to what is now known as the [z-transform](https://en.wikipedia.org/wiki/Z-transform "Z-transform"), and he gave little attention to the [continuous variable](https://en.wikipedia.org/wiki/Continuous_variable "Continuous variable") case which was discussed by [Niels Henrik Abel](https://en.wikipedia.org/wiki/Niels_Henrik_Abel "Niels Henrik Abel").[\[7\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-7)
From 1744, [Leonhard Euler](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler") investigated integrals of the form z \= â« X ( x ) e a x d x and z \= â« X ( x ) x A d x {\\displaystyle z=\\int X(x)e^{ax}\\,dx\\quad {\\text{ and }}\\quad z=\\int X(x)x^{A}\\,dx}  as solutions of differential equations, introducing in particular the [gamma function](https://en.wikipedia.org/wiki/Gamma_function "Gamma function").[\[8\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-8) [Joseph-Louis Lagrange](https://en.wikipedia.org/wiki/Joseph-Louis_Lagrange "Joseph-Louis Lagrange") was an admirer of Euler and, in his work on integrating [probability density functions](https://en.wikipedia.org/wiki/Probability_density_function "Probability density function"), investigated expressions of the form â« X ( x ) e â a x a x d x , {\\displaystyle \\int X(x)e^{-ax}a^{x}\\,dx,}  which resembles a Laplace transform.[\[9\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-9)[\[10\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-10)
These types of integrals seem first to have attracted Laplace's attention in 1782, where he was following in the spirit of Euler in using the integrals themselves as solutions of equations.[\[11\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-11) However, in 1785, Laplace took the critical step forward when, rather than simply looking for a solution in the form of an integral, he started to apply the transforms in the sense that was later to become popular. He used an integral of the form â« x s Ï ( x ) d x , {\\displaystyle \\int x^{s}\\varphi (x)\\,dx,}  akin to a [Mellin transform](https://en.wikipedia.org/wiki/Mellin_transform "Mellin transform"), to transform the whole of a [difference equation](https://en.wikipedia.org/wiki/Difference_equation "Difference equation"), in order to look for solutions of the transformed equation. He then went on to apply the Laplace transform in the same way and started to derive some of its properties, beginning to appreciate its potential power.[\[12\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-12)
Laplace also recognised that [Joseph Fourier](https://en.wikipedia.org/wiki/Joseph_Fourier "Joseph Fourier")'s method of [Fourier series](https://en.wikipedia.org/wiki/Fourier_series "Fourier series") for solving the [diffusion equation](https://en.wikipedia.org/wiki/Diffusion_equation "Diffusion equation") could only apply to a limited region of space, because those solutions were [periodic](https://en.wikipedia.org/wiki/Periodic_function "Periodic function"). In 1809, Laplace applied his transform to find solutions that diffused indefinitely in space.[\[13\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-13) In 1821, [Cauchy](https://en.wikipedia.org/wiki/Cauchy "Cauchy") developed an [operational calculus](https://en.wikipedia.org/wiki/Operational_calculus "Operational calculus") for the Laplace transform that could be used to study [linear differential equations](https://en.wikipedia.org/wiki/Linear_differential_equation "Linear differential equation") in much the same way the transform is now used in basic engineering. This method was popularized, and perhaps rediscovered, by [Oliver Heaviside](https://en.wikipedia.org/wiki/Oliver_Heaviside "Oliver Heaviside") around the turn of the century.[\[14\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-14)
[Bernhard Riemann](https://en.wikipedia.org/wiki/Bernhard_Riemann "Bernhard Riemann") used the Laplace transform in his 1859 paper *[On the number of primes less than a given magnitude](https://en.wikipedia.org/wiki/On_the_number_of_primes_less_than_a_given_magnitude "On the number of primes less than a given magnitude")*, in which he also developed the inversion theorem. Riemann used the Laplace transform to develop the functional equation of the [Riemann zeta function](https://en.wikipedia.org/wiki/Riemann_zeta_function "Riemann zeta function"), and his method is still used to relate the [modular transformation law](https://en.wikipedia.org/wiki/Modular_form "Modular form") of the [Jacobi theta function](https://en.wikipedia.org/wiki/Jacobi_theta_function "Jacobi theta function"), which is readily proved via [Poisson summation](https://en.wikipedia.org/wiki/Poisson_summation "Poisson summation"), to the functional equation.[\[15\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-15)
[Hjalmar Mellin](https://en.wikipedia.org/wiki/Hjalmar_Mellin "Hjalmar Mellin") was among the first to study the Laplace transform, rigorously in the [Karl Weierstrass](https://en.wikipedia.org/wiki/Karl_Weierstrass "Karl Weierstrass") school of analysis, and apply it to the study of [differential equations](https://en.wikipedia.org/wiki/Differential_equations "Differential equations") and [special functions](https://en.wikipedia.org/wiki/Special_functions "Special functions"), at the turn of the 20th century.[\[16\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-16) At around the same time, Heaviside was busy with his operational calculus. [Thomas Joannes Stieltjes](https://en.wikipedia.org/wiki/Thomas_Joannes_Stieltjes "Thomas Joannes Stieltjes") considered a generalization of the Laplace transform connected to his [work on moments](https://en.wikipedia.org/wiki/Stieltjes_moment_problem "Stieltjes moment problem"). Other contributors in this time period included [Mathias Lerch](https://en.wikipedia.org/wiki/Mathias_Lerch "Mathias Lerch"),[\[17\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-17) [Oliver Heaviside](https://en.wikipedia.org/wiki/Oliver_Heaviside "Oliver Heaviside"), and [Thomas Bromwich](https://en.wikipedia.org/wiki/Thomas_John_I%27Anson_Bromwich "Thomas John I'Anson Bromwich").[\[18\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-18)
In 1929, [Vannevar Bush](https://en.wikipedia.org/wiki/Vannevar_Bush "Vannevar Bush") and [Norbert Wiener](https://en.wikipedia.org/wiki/Norbert_Wiener "Norbert Wiener") published *Operational Circuit Analysis* as a text for engineering analysis of electrical circuits, applying both Fourier transforms and operational calculus, and in which they included one of the first predecessors of the modern table of Laplace transforms. In 1934, [Raymond Paley](https://en.wikipedia.org/wiki/Raymond_Paley "Raymond Paley") and [Norbert Wiener](https://en.wikipedia.org/wiki/Norbert_Wiener "Norbert Wiener") published the important work *Fourier transforms in the complex domain*, about what is now called the Laplace transform (see below). Also during the 30s, the Laplace transform was instrumental in [Godfrey Harold Hardy](https://en.wikipedia.org/wiki/G_H_Hardy "G H Hardy") and [John Edensor Littlewood](https://en.wikipedia.org/wiki/John_Edensor_Littlewood "John Edensor Littlewood")'s study of [tauberian theorems](https://en.wikipedia.org/wiki/Tauberian_theorem "Tauberian theorem"), and this application was later expounded on by [Widder (1941)](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFWidder1941), who developed other aspects of the theory such as a new method for inversion. [Edward Charles Titchmarsh](https://en.wikipedia.org/wiki/Edward_Charles_Titchmarsh "Edward Charles Titchmarsh") wrote the influential *Introduction to the theory of the Fourier integral* (1937).
The current widespread use of the transform (mainly in engineering) came about during and soon after [World War II](https://en.wikipedia.org/wiki/World_War_II "World War II"),[\[19\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-19) replacing the earlier Heaviside [operational calculus](https://en.wikipedia.org/wiki/Operational_calculus "Operational calculus"). The advantages of the Laplace transform had been emphasized by [Gustav Doetsch](https://en.wikipedia.org/wiki/Gustav_Doetsch "Gustav Doetsch").[\[20\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-20)
## Formal definition
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[](https://en.wikipedia.org/wiki/File:Complex_frequency_s-domain_negative.jpg)
â
(
e
â
s
t
)
{\\displaystyle \\Re (e^{-st})}
for various complex frequencies in the *s*\-domain
â
(
s
\=
Ï
\+
i
Ï
)
{\\displaystyle (s=\\sigma +i\\omega )}
â
, which can be expressed as
â
e
â
Ï
t
cos
âĄ
(
Ï
t
)
{\\displaystyle e^{-\\sigma t}\\cos(\\omega t)}
â
. The axis at
Ï
\=
0
{\\displaystyle \\sigma =0}
contains pure cosines. Positive
Ï
{\\displaystyle \\sigma }
contains [damped cosines](https://en.wikipedia.org/wiki/Damped_sinusoid "Damped sinusoid"). Negative
Ï
{\\displaystyle \\sigma }
contains [exponentially growing](https://en.wikipedia.org/wiki/Exponential_growth "Exponential growth") cosines.
The Laplace transform of a [function](https://en.wikipedia.org/wiki/Function_\(mathematics\) "Function (mathematics)") *f*(*t*), defined for all [real numbers](https://en.wikipedia.org/wiki/Real_number "Real number") *t* â„ 0, is the function *F*(*s*), which is a unilateral transform defined by\[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\]
F ( s ) \= â« 0 â f ( t ) e â s t d t , {\\displaystyle F(s)=\\int \_{0}^{\\infty }f(t)e^{-st}\\,dt,}  (Eq. 1)
where *s* is a [complex](https://en.wikipedia.org/wiki/Complex_number "Complex number") frequency-domain parameter s \= Ï \+ i Ï {\\displaystyle s=\\sigma +i\\omega }  with real numbers Ï and Ï.
An alternate notation for the Laplace transform is L { f } {\\displaystyle {\\mathcal {L}}\\{f\\}}  instead of *F*.[\[3\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-:1-3) Thus F ( s ) \= L { f } ( s ) {\\displaystyle F(s)={\\mathcal {L}}\\{f\\}(s)}  in [functional notation](https://en.wikipedia.org/wiki/Functional_notation "Functional notation"). This is often written, especially in engineering settings, as â F ( s ) \= L { f ( t ) } {\\displaystyle F(s)={\\mathcal {L}}\\{f(t)\\}}  â , with the understanding that the [dummy variable](https://en.wikipedia.org/wiki/Bound_variable "Bound variable") t {\\displaystyle t}  does not appear in the function â F ( s ) {\\displaystyle F(s)}  â .
The meaning of the integral depends on types of functions of interest. A necessary condition for existence of the integral is that f must be [locally integrable](https://en.wikipedia.org/wiki/Locally_integrable "Locally integrable") on \[0, â). For locally integrable functions that decay at infinity or are of [exponential type](https://en.wikipedia.org/wiki/Exponential_type "Exponential type") (â \| f ( t ) \| †A e B \| t \| {\\displaystyle \\vert f(t)\\vert \\leq Ae^{B\\vert t\\vert }}  â ), the integral can be understood to be a (proper) [Lebesgue integral](https://en.wikipedia.org/wiki/Lebesgue_integral "Lebesgue integral"). However, for many applications it is necessary to regard it as a [conditionally convergent](https://en.wikipedia.org/wiki/Conditionally_convergent "Conditionally convergent") [improper integral](https://en.wikipedia.org/wiki/Improper_integral "Improper integral") at â. Still more generally, the integral can be understood in a [weak sense](https://en.wikipedia.org/wiki/Distribution_\(mathematics\) "Distribution (mathematics)"), and this is dealt with below.
One can define the Laplace transform of a finite [Borel measure](https://en.wikipedia.org/wiki/Borel_measure "Borel measure") ÎŒ by the Lebesgue integral[\[21\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-21) L { ÎŒ } ( s ) \= â« \[ 0 , â ) e â s t d ÎŒ ( t ) . {\\displaystyle {\\mathcal {L}}\\{\\mu \\}(s)=\\int \_{\[0,\\infty )}e^{-st}\\,d\\mu (t).} 
An important special case is where ÎŒ is a [probability measure](https://en.wikipedia.org/wiki/Probability_measure "Probability measure"), for example, the [Dirac delta function](https://en.wikipedia.org/wiki/Dirac_delta_function "Dirac delta function"). In [operational calculus](https://en.wikipedia.org/wiki/Operational_calculus "Operational calculus"), the Laplace transform of a measure is often treated as though the measure came from a probability density function f. In that case, to avoid potential confusion, one often writes L { f } ( s ) \= â« 0 â â f ( t ) e â s t d t , {\\displaystyle {\\mathcal {L}}\\{f\\}(s)=\\int \_{0^{-}}^{\\infty }f(t)e^{-st}\\,dt,}  where the lower limit of 0â is shorthand notation for lim Δ â 0 \+ â« â Δ â . {\\displaystyle \\lim \_{\\varepsilon \\to 0^{+}}\\int \_{-\\varepsilon }^{\\infty }.} 
This limit emphasizes that any [point mass](https://en.wikipedia.org/wiki/Point_particle "Point particle") located at 0 is entirely captured by the Laplace transform. Although with the Lebesgue integral, it is not necessary to take such a limit, it does appear more naturally in connection with the [LaplaceâStieltjes transform](https://en.wikipedia.org/wiki/Laplace%E2%80%93Stieltjes_transform "LaplaceâStieltjes transform").
### Bilateral Laplace transform
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=3 "Edit section: Bilateral Laplace transform")\]
Main article: [Two-sided Laplace transform](https://en.wikipedia.org/wiki/Two-sided_Laplace_transform "Two-sided Laplace transform")
When one says "the Laplace transform" without qualification, the unilateral or one-sided transform is usually intended. The Laplace transform can be alternatively defined as the *bilateral Laplace transform*, or [two-sided Laplace transform](https://en.wikipedia.org/wiki/Two-sided_Laplace_transform "Two-sided Laplace transform"), by extending the limits of integration to be the entire real axis. If that is done, the common unilateral transform becomes a special case of the bilateral transform, where the definition of the function being transformed includes being multiplied by the [Heaviside step function](https://en.wikipedia.org/wiki/Heaviside_step_function "Heaviside step function").
The bilateral Laplace transform *F*(*s*) is defined as follows:
F ( s ) \= â« â â â e â s t f ( t ) d t . {\\displaystyle F(s)=\\int \_{-\\infty }^{\\infty }e^{-st}f(t)\\,dt.}  (Eq. 2)
An alternate notation for the bilateral Laplace transform is â B { f } {\\displaystyle {\\mathcal {B}}\\{f\\}}  â , instead of F.
### Inverse Laplace transform
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=4 "Edit section: Inverse Laplace transform")\]
Main article: [Inverse Laplace transform](https://en.wikipedia.org/wiki/Inverse_Laplace_transform "Inverse Laplace transform")
Two integrable functions have the same Laplace transform only if they differ on a set of [Lebesgue measure](https://en.wikipedia.org/wiki/Lebesgue_measure "Lebesgue measure") zero. This means that, on the range of the transform, there is an inverse transform. In fact, besides integrable functions, the Laplace transform is a [one-to-one mapping](https://en.wikipedia.org/wiki/One-to-one_function "One-to-one function") from one function space into another in many other function spaces as well, although there is usually no easy characterization of the range.
Typical function spaces in which this is true include the spaces of bounded continuous functions, the space [*L*â(0, â)](https://en.wikipedia.org/wiki/Lp_space "Lp space"), or more generally [tempered distributions](https://en.wikipedia.org/wiki/Tempered_distributions "Tempered distributions") on (0, â). The Laplace transform is also defined and injective for suitable spaces of tempered distributions.
In these cases, the image of the Laplace transform lives in a space of [analytic functions](https://en.wikipedia.org/wiki/Analytic_function "Analytic function") in the [region of convergence](https://en.wikipedia.org/wiki/Laplace_transform#Region_of_convergence). The [inverse Laplace transform](https://en.wikipedia.org/wiki/Inverse_Laplace_transform "Inverse Laplace transform") is given by the following complex integral, which is known by various names (the **Bromwich integral**, the **FourierâMellin integral**, and **Mellin's inverse formula**):
f ( t ) \= L â 1 { F } ( t ) \= 1 2 Ï i lim T â â â« Îł â i T Îł \+ i T e s t F ( s ) d s , {\\displaystyle f(t)={\\mathcal {L}}^{-1}\\{F\\}(t)={\\frac {1}{2\\pi i}}\\lim \_{T\\to \\infty }\\int \_{\\gamma -iT}^{\\gamma +iT}e^{st}F(s)\\,ds,}  (Eq. 3)
where Îł is a real number so that the contour path of integration is in the region of convergence of *F*(*s*). In most applications, the contour can be closed, allowing the use of the [residue theorem](https://en.wikipedia.org/wiki/Residue_theorem "Residue theorem"). An alternative formula for the inverse Laplace transform is given by [Post's inversion formula](https://en.wikipedia.org/wiki/Post%27s_inversion_formula "Post's inversion formula"). The limit here is interpreted in the [weak-\* topology](https://en.wikipedia.org/wiki/Weak_topology#Weak-*_topology "Weak topology").
In practice, it is typically more convenient to decompose a Laplace transform into known transforms of functions obtained from a table and construct the inverse by inspection.
### Probability theory
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In [pure](https://en.wikipedia.org/wiki/Probability_theory "Probability theory") and [applied probability](https://en.wikipedia.org/wiki/Applied_probability "Applied probability"), the Laplace transform is defined as an [expected value](https://en.wikipedia.org/wiki/Expected_value "Expected value"). If X is a [random variable](https://en.wikipedia.org/wiki/Random_variable "Random variable") with probability density function f, then the Laplace transform of f is given by the expectation L { f } ( s ) \= E ⥠\[ e â s X \] , {\\displaystyle {\\mathcal {L}}\\{f\\}(s)=\\operatorname {E} \\left\[e^{-sX}\\right\],} ![{\\displaystyle {\\mathcal {L}}\\{f\\}(s)=\\operatorname {E} \\left\[e^{-sX}\\right\],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3da28029aa437333b370a72f00e2745d1f8e8c73) where E ⥠\[ r \] {\\displaystyle \\operatorname {E} \[r\]} ![{\\displaystyle \\operatorname {E} \[r\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/715165949e716ef97cb5b61ba0982325cef86a1b) is the [expectation](https://en.wikipedia.org/wiki/Expected_value "Expected value") of [random variable](https://en.wikipedia.org/wiki/Random_variable "Random variable") â r {\\displaystyle r}  â .
By [convention](https://en.wikipedia.org/wiki/Abuse_of_notation "Abuse of notation"), this is referred to as the Laplace transform of the random variable X itself. Here, replacing s by â*t* gives the [moment generating function](https://en.wikipedia.org/wiki/Moment_generating_function "Moment generating function") of X. The Laplace transform has applications throughout probability theory, including [first passage times](https://en.wikipedia.org/wiki/First_passage_time "First passage time") of [stochastic processes](https://en.wikipedia.org/wiki/Stochastic_process "Stochastic process") such as [Markov chains](https://en.wikipedia.org/wiki/Markov_chain "Markov chain"), and [renewal theory](https://en.wikipedia.org/wiki/Renewal_theory "Renewal theory").
Of particular use is the ability to recover the [cumulative distribution function](https://en.wikipedia.org/wiki/Cumulative_distribution_function "Cumulative distribution function") of a continuous random variable X by means of the Laplace transform as follows:[\[22\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-22) F X ( x ) \= L â 1 { 1 s E ⥠\[ e â s X \] } ( x ) \= L â 1 { 1 s L { f } ( s ) } ( x ) . {\\displaystyle F\_{X}(x)={\\mathcal {L}}^{-1}\\left\\{{\\frac {1}{s}}\\operatorname {E} \\left\[e^{-sX}\\right\]\\right\\}(x)={\\mathcal {L}}^{-1}\\left\\{{\\frac {1}{s}}{\\mathcal {L}}\\{f\\}(s)\\right\\}(x).} ![{\\displaystyle F\_{X}(x)={\\mathcal {L}}^{-1}\\left\\{{\\frac {1}{s}}\\operatorname {E} \\left\[e^{-sX}\\right\]\\right\\}(x)={\\mathcal {L}}^{-1}\\left\\{{\\frac {1}{s}}{\\mathcal {L}}\\{f\\}(s)\\right\\}(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9728fa6f7b7c0cf6c7fd18917b64a4bbea2c7f)
### Algebraic construction
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The Laplace transform can be alternatively defined in a purely algebraic manner by applying a [field of fractions](https://en.wikipedia.org/wiki/Field_of_fractions "Field of fractions") construction to the convolution [ring](https://en.wikipedia.org/wiki/Ring_\(abstract_algebra\) "Ring (abstract algebra)") of functions on the positive half-line. The resulting [space of abstract operators](https://en.wikipedia.org/wiki/Convolution_quotient "Convolution quotient") is exactly equivalent to Laplace space, but in this construction the forward and reverse transforms never need to be explicitly defined (avoiding the related difficulties with proving convergence).[\[23\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-23)
## Region of convergence
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=7 "Edit section: Region of convergence")\]
See also: [Poleâzero plot § Continuous-time systems](https://en.wikipedia.org/wiki/Pole%E2%80%93zero_plot#Continuous-time_systems "Poleâzero plot")
If *f* is a locally integrable function (or more generally a Borel measure locally of bounded variation), then the Laplace transform *F*(*s*) of *f* converges provided that the limit lim R â â â« 0 R f ( t ) e â s t d t {\\displaystyle \\lim \_{R\\to \\infty }\\int \_{0}^{R}f(t)e^{-st}\\,dt}  exists.
The Laplace transform [converges absolutely](https://en.wikipedia.org/wiki/Absolute_convergence "Absolute convergence") if the integral â« 0 â \| f ( t ) e â s t \| d t {\\displaystyle \\int \_{0}^{\\infty }\\left\|f(t)e^{-st}\\right\|\\,dt}  exists as a proper Lebesgue integral. The Laplace transform is usually understood as [conditionally convergent](https://en.wikipedia.org/wiki/Conditional_convergence "Conditional convergence"), meaning that it converges in the former but not in the latter sense.
The set of values for which *F*(*s*) converges absolutely is either of the form Re(*s*) \> *a* or Re(*s*) â„ *a*, where *a* is an [extended real constant](https://en.wikipedia.org/wiki/Extended_real_number "Extended real number") with ââ †*a* †â (a consequence of the [dominated convergence theorem](https://en.wikipedia.org/wiki/Dominated_convergence_theorem "Dominated convergence theorem")). The constant *a* is known as the abscissa of absolute convergence, and depends on the growth behavior of *f*(*t*).[\[24\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-24) Analogously, the two-sided transform converges absolutely in a strip of the form *a* \< Re(*s*) \< *b*, and possibly including the lines Re(*s*) = *a* or Re(*s*) = *b*.[\[25\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-25) The subset of values of *s* for which the Laplace transform converges absolutely is called the region of absolute convergence, or the domain of absolute convergence. In the two-sided case, it is sometimes called the strip of absolute convergence. The Laplace transform is analytic in the region of absolute convergence: this is a consequence of [Fubini's theorem](https://en.wikipedia.org/wiki/Fubini%27s_theorem "Fubini's theorem") and [Morera's theorem](https://en.wikipedia.org/wiki/Morera%27s_theorem "Morera's theorem").
Similarly, the set of values for which *F*(*s*) converges (conditionally or absolutely) is known as the region of conditional convergence, or simply the **region of convergence** (ROC). If the Laplace transform converges (conditionally) at *s* = *s*0, then it automatically converges for all *s* with Re(*s*) \> Re(*s*0). Therefore, the region of convergence is a half-plane of the form Re(*s*) \> *a*, possibly including some points of the boundary line Re(*s*) = *a*.
In the region of convergence Re(*s*) \> Re(*s*0), the Laplace transform of *f* can be expressed by [integrating by parts](https://en.wikipedia.org/wiki/Integration_by_parts "Integration by parts") as the integral F ( s ) \= ( s â s 0 ) â« 0 â e â ( s â s 0 ) t ÎČ ( t ) d t , ÎČ ( u ) \= â« 0 u e â s 0 t f ( t ) d t . {\\displaystyle F(s)=(s-s\_{0})\\int \_{0}^{\\infty }e^{-(s-s\_{0})t}\\beta (t)\\,dt,\\quad \\beta (u)=\\int \_{0}^{u}e^{-s\_{0}t}f(t)\\,dt.} 
That is, *F*(*s*) can effectively be expressed, in the region of convergence, as the absolutely convergent Laplace transform of some other function. In particular, it is analytic. In its most general form, the Laplace transform gives a one-to-one correspondence between the holomorphic functions which, for some â Ï â R {\\displaystyle \\sigma \\in \\mathbb {R} }  â , are defined on { s â C \| R e ( s ) \> Ï } {\\displaystyle \\{s\\in \\mathbb {C} \\ \\vert \\ \\mathrm {Re} (s)\>\\sigma \\}}  and are bounded there in absolute value by a polynomial, and the [distributions](https://en.wikipedia.org/wiki/Distribution_\(mathematical_analysis\) "Distribution (mathematical analysis)") on the real line supported on â \[ 0 , â ) {\\displaystyle \[0,\\infty )}  â which become [tempered distributions](https://en.wikipedia.org/wiki/Tempered_distribution "Tempered distribution") after multiplied by e â Ï t {\\displaystyle e^{-\\sigma t}}  for some â Ï {\\displaystyle \\sigma }  â .[\[26\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-26)
There are several [PaleyâWiener theorems](https://en.wikipedia.org/wiki/Paley%E2%80%93Wiener_theorem "PaleyâWiener theorem") concerning the relationship between the decay properties of *f*, and the properties of the Laplace transform within the region of convergence.
In engineering applications, a function corresponding to a [linear time-invariant (LTI) system](https://en.wikipedia.org/wiki/Linear_time-invariant_system "Linear time-invariant system") is *stable* if every bounded input produces a bounded output. This is equivalent to the absolute convergence of the Laplace transform of the impulse response function in the region Re(*s*) â„ 0. As a result, LTI systems are stable, provided that the poles of the Laplace transform of the impulse response function have negative real part.
This ROC is used in knowing about the causality and stability of a system.
## Properties and theorems
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=8 "Edit section: Properties and theorems")\]
The Laplace transform's key property is that it converts [differentiation](https://en.wikipedia.org/wiki/Derivative "Derivative") and [integration](https://en.wikipedia.org/wiki/Integral "Integral") in the time domain into multiplication and division by *s* in the Laplace domain. Thus, the Laplace variable *s* is also known as an *operator variable* in the Laplace domain: either the *derivative operator* or (for *s*â1) the *integration operator*.
Given the functions *f*(*t*) and *g*(*t*), and their respective Laplace transforms *F*(*s*) and *G*(*s*), f ( t ) \= L â 1 { F ( s ) } , g ( t ) \= L â 1 { G ( s ) } , {\\displaystyle {\\begin{aligned}f(t)&={\\mathcal {L}}^{-1}\\{F(s)\\},\\\\g(t)&={\\mathcal {L}}^{-1}\\{G(s)\\},\\end{aligned}}} 
the following table is a list of properties of unilateral Laplace transform:[\[27\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-27)
| Property | Time domain | *s* domain | Comment |
|---|---|---|---|
| [Linearity](https://en.wikipedia.org/wiki/Linearity "Linearity") | a f ( t ) \+ b g ( t ) {\\displaystyle af(t)+bg(t)\\ }  | | |
[Initial value theorem](https://en.wikipedia.org/wiki/Initial_value_theorem "Initial value theorem")
f
(
0
\+
)
\=
lim
s
â
â
s
F
(
s
)
.
{\\displaystyle f(0^{+})=\\lim \_{s\\to \\infty }{sF(s)}.}
[Final value theorem](https://en.wikipedia.org/wiki/Final_value_theorem "Final value theorem")
â
f
(
â
)
\=
lim
s
â
0
s
F
(
s
)
{\\displaystyle f(\\infty )=\\lim \_{s\\to 0}{sF(s)}}
â
, if all [poles](https://en.wikipedia.org/wiki/Pole_\(complex_analysis\) "Pole (complex analysis)") of
s
F
(
s
)
{\\displaystyle sF(s)}
are in the left half-plane.
The final value theorem is useful because it gives the long-term behaviour without having to perform [partial fraction](https://en.wikipedia.org/wiki/Partial_fraction "Partial fraction") decompositions (or other difficult algebra). If *F*(*s*) has a pole in the right-hand plane or poles on the imaginary axis (e.g., if
f
(
t
)
\=
e
t
{\\displaystyle f(t)=e^{t}}
or
â
f
(
t
)
\=
sin
âĄ
(
t
)
{\\displaystyle f(t)=\\sin(t)}
â
), then the behaviour of this formula is undefined.
### Relation to power series
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=9 "Edit section: Relation to power series")\]
The Laplace transform can be viewed as a [continuous](https://en.wikipedia.org/wiki/Continuous_function "Continuous function") analogue of a [power series](https://en.wikipedia.org/wiki/Power_series "Power series").[\[29\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-29) If *a*(*n*) is a discrete function of a positive integer *n*, then the power series associated to *a*(*n*) is the series â n \= 0 â a ( n ) x n {\\displaystyle \\sum \_{n=0}^{\\infty }a(n)x^{n}}  where *x* is a real variable (see *[Z-transform](https://en.wikipedia.org/wiki/Z-transform "Z-transform")*). Replacing summation over *n* with integration over *t*, a continuous version of the power series becomes â« 0 â f ( t ) x t d t {\\displaystyle \\int \_{0}^{\\infty }f(t)x^{t}\\,dt}  where the discrete function *a*(*n*) is replaced by the continuous one *f*(*t*).
Changing the base of the power from *x* to *e* gives â« 0 â f ( t ) ( e ln ⥠x ) t d t {\\displaystyle \\int \_{0}^{\\infty }f(t)\\left(e^{\\ln {x}}\\right)^{t}\\,dt} 
For this to converge for, say, all bounded functions *f*, it is necessary to require that ln *x* \< 0. Making the substitution â*s* = ln *x* gives just the Laplace transform: â« 0 â f ( t ) e â s t d t {\\displaystyle \\int \_{0}^{\\infty }f(t)e^{-st}\\,dt} 
In other words, the Laplace transform is a continuous analog of a power series, in which the discrete parameter *n* is replaced by the continuous parameter *t*, and *x* is replaced by *e*â*s*.
Analogously to a power series, if â a ( n ) \= O ( Ï â n ) {\\displaystyle a(n)=O(\\rho ^{-n})}  â , then the power series converges to an analytic function in â \| x \| \< Ï {\\displaystyle \\vert x\\vert \<\\rho }  â , if â f ( t ) \= O ( e â Ï t ) {\\displaystyle f(t)=O(e^{-\\sigma t})}  â , the Laplace transform converges to an analytic function for â â ( s ) \> Ï {\\displaystyle \\Re (s)\>\\sigma }  â .[\[30\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-FOOTNOTEWidder194138-30)
### Relation to moments
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=10 "Edit section: Relation to moments")\]
Main article: [Moment-generating function](https://en.wikipedia.org/wiki/Moment-generating_function "Moment-generating function")
The quantities ÎŒ n \= â« 0 â t n f ( t ) d t {\\displaystyle \\mu \_{n}=\\int \_{0}^{\\infty }t^{n}f(t)\\,dt}  are the *moments* of the function *f*. If the first *n* moments of *f* converge absolutely, then by repeated [differentiation under the integral](https://en.wikipedia.org/wiki/Differentiation_under_the_integral "Differentiation under the integral"), ( â 1 ) n ( L f ) ( n ) ( 0 ) \= ÎŒ n . {\\displaystyle (-1)^{n}({\\mathcal {L}}f)^{(n)}(0)=\\mu \_{n}.}  This is of special significance in probability theory, where the moments of a random variable *X* are given by the expectation values â ÎŒ n \= E ⥠\[ X n \] {\\displaystyle \\mu \_{n}=\\operatorname {E} \[X^{n}\]} ![{\\displaystyle \\mu \_{n}=\\operatorname {E} \[X^{n}\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a470b23edb981c95e00f5690370662b6b102341) â . Then, the relation holds ÎŒ n \= ( â 1 ) n d n d s n E ⥠\[ e â s X \] ( 0 ) . {\\displaystyle \\mu \_{n}=(-1)^{n}{\\frac {d^{n}}{ds^{n}}}\\operatorname {E} \\left\[e^{-sX}\\right\](0).} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aebff76a8e2a99d41a4d654999b7309d1a2f5916)
### Transform of a function's derivative
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=11 "Edit section: Transform of a function's derivative")\]
It is often convenient to use the differentiation property of the Laplace transform to find the transform of a function's derivative. This can be derived from the basic expression for a Laplace transform as follows: L { f ( t ) } \= â« 0 â â e â s t f ( t ) d t \= \[ f ( t ) e â s t â s \] 0 â â â â« 0 â â e â s t â s f âČ ( t ) d t (by parts) \= \[ â f ( 0 â ) â s \] \+ 1 s L { f âČ ( t ) } , {\\displaystyle {\\begin{aligned}{\\mathcal {L}}\\left\\{f(t)\\right\\}&=\\int \_{0^{-}}^{\\infty }e^{-st}f(t)\\,dt\\\\\[6pt\]&=\\left\[{\\frac {f(t)e^{-st}}{-s}}\\right\]\_{0^{-}}^{\\infty }-\\int \_{0^{-}}^{\\infty }{\\frac {e^{-st}}{-s}}f'(t)\\,dt\\quad {\\text{(by parts)}}\\\\\[6pt\]&=\\left\[-{\\frac {f(0^{-})}{-s}}\\right\]+{\\frac {1}{s}}{\\mathcal {L}}\\left\\{f'(t)\\right\\},\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}{\\mathcal {L}}\\left\\{f(t)\\right\\}&=\\int \_{0^{-}}^{\\infty }e^{-st}f(t)\\,dt\\\\\[6pt\]&=\\left\[{\\frac {f(t)e^{-st}}{-s}}\\right\]\_{0^{-}}^{\\infty }-\\int \_{0^{-}}^{\\infty }{\\frac {e^{-st}}{-s}}f'(t)\\,dt\\quad {\\text{(by parts)}}\\\\\[6pt\]&=\\left\[-{\\frac {f(0^{-})}{-s}}\\right\]+{\\frac {1}{s}}{\\mathcal {L}}\\left\\{f'(t)\\right\\},\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/574460c0fe29a550c4b4a66b9aa82601ea9f956b) yielding L { f âČ ( t ) } \= s â
L { f ( t ) } â f ( 0 â ) , {\\displaystyle {\\mathcal {L}}\\{f'(t)\\}=s\\cdot {\\mathcal {L}}\\{f(t)\\}-f(0^{-}),}  and in the bilateral case, L { f âČ ( t ) } \= s â« â â â e â s t f ( t ) d t \= s â
L { f ( t ) } . {\\displaystyle {\\mathcal {L}}\\{f'(t)\\}=s\\int \_{-\\infty }^{\\infty }e^{-st}f(t)\\,dt=s\\cdot {\\mathcal {L}}\\{f(t)\\}.} 
The general result L { f ( n ) ( t ) } \= s n â
L { f ( t ) } â s n â 1 f ( 0 â ) â ⯠â f ( n â 1 ) ( 0 â ) , {\\displaystyle {\\mathcal {L}}\\left\\{f^{(n)}(t)\\right\\}=s^{n}\\cdot {\\mathcal {L}}\\{f(t)\\}-s^{n-1}f(0^{-})-\\cdots -f^{(n-1)}(0^{-}),}  where f ( n ) {\\displaystyle f^{(n)}}  denotes the *n*th derivative of *f*, can then be established with an inductive argument.
### Evaluating integrals over the positive real axis
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=12 "Edit section: Evaluating integrals over the positive real axis")\]
A useful property of the Laplace transform is the following: â« 0 â f ( x ) g ( x ) d x \= â« 0 â ( L f ) ( s ) â
( L â 1 g ) ( s ) d s {\\displaystyle \\int \_{0}^{\\infty }f(x)g(x)\\,dx=\\int \_{0}^{\\infty }({\\mathcal {L}}f)(s)\\cdot ({\\mathcal {L}}^{-1}g)(s)\\,ds}  under suitable assumptions on the behaviour of â f {\\displaystyle f}  â and â g {\\displaystyle g}  â in a right neighbourhood of 0 {\\displaystyle 0}  and on the decay rate of â f {\\displaystyle f}  â and â g {\\displaystyle g}  â in a left neighbourhood of â â {\\displaystyle \\infty }  â . The above formula is a variation of integration by parts, with the operators d d x {\\displaystyle {\\frac {d}{dx}}}  and â« d x {\\displaystyle \\int \\,dx}  being replaced by L {\\displaystyle {\\mathcal {L}}}  and â L â 1 {\\displaystyle {\\mathcal {L}}^{-1}}  â . Let us prove the equivalent formulation: â« 0 â ( L f ) ( x ) g ( x ) d x \= â« 0 â f ( s ) ( L g ) ( s ) d s . {\\displaystyle \\int \_{0}^{\\infty }({\\mathcal {L}}f)(x)g(x)\\,dx=\\int \_{0}^{\\infty }f(s)({\\mathcal {L}}g)(s)\\,ds.} 
By plugging in ( L f ) ( x ) \= â« 0 â f ( s ) e â s x d s {\\displaystyle ({\\mathcal {L}}f)(x)=\\int \_{0}^{\\infty }f(s)e^{-sx}\\,ds}  the left-hand side turns into: â« 0 â â« 0 â f ( s ) g ( x ) e â s x d s d x , {\\displaystyle \\int \_{0}^{\\infty }\\int \_{0}^{\\infty }f(s)g(x)e^{-sx}\\,ds\\,dx,}  but assuming Fubini's theorem holds, by reversing the order of integration we get the wanted right-hand side.
This method can be used to compute integrals that would otherwise be difficult to compute using elementary methods of real calculus. For example, â« 0 â sin ⥠x x d x \= â« 0 â L ( 1 ) ( x ) sin ⥠x d x \= â« 0 â 1 â
L ( sin ) ( x ) d x \= â« 0 â d x x 2 \+ 1 \= Ï 2 . {\\displaystyle \\int \_{0}^{\\infty }{\\frac {\\sin x}{x}}dx=\\int \_{0}^{\\infty }{\\mathcal {L}}(1)(x)\\sin xdx=\\int \_{0}^{\\infty }1\\cdot {\\mathcal {L}}(\\sin )(x)dx=\\int \_{0}^{\\infty }{\\frac {dx}{x^{2}+1}}={\\frac {\\pi }{2}}.} 
## Relationship to other transforms
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=13 "Edit section: Relationship to other transforms")\]
### LaplaceâStieltjes transform
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=14 "Edit section: LaplaceâStieltjes transform")\]
The (unilateral) LaplaceâStieltjes transform of a function *g* : â â â is defined by the [LebesgueâStieltjes integral](https://en.wikipedia.org/wiki/Lebesgue%E2%80%93Stieltjes_integral "LebesgueâStieltjes integral") { L â g } ( s ) \= â« 0 â e â s t d g ( t ) . {\\displaystyle \\{{\\mathcal {L}}^{\*}g\\}(s)=\\int \_{0}^{\\infty }e^{-st}\\,d\\,g(t)~.} 
The function *g* is assumed to be of [bounded variation](https://en.wikipedia.org/wiki/Bounded_variation "Bounded variation"). If *g* is the [antiderivative](https://en.wikipedia.org/wiki/Antiderivative "Antiderivative") of *f*: g ( x ) \= â« 0 x f ( t ) d t {\\displaystyle g(x)=\\int \_{0}^{x}f(t)\\,d\\,t} 
then the LaplaceâStieltjes transform of g and the Laplace transform of f coincide. In general, the LaplaceâStieltjes transform is the Laplace transform of the [Stieltjes measure](https://en.wikipedia.org/wiki/Stieltjes_measure "Stieltjes measure") associated to g. So in practice, the only distinction between the two transforms is that the Laplace transform is thought of as operating on the density function of the measure, whereas the LaplaceâStieltjes transform is thought of as operating on its [cumulative distribution function](https://en.wikipedia.org/wiki/Cumulative_distribution_function "Cumulative distribution function").[\[31\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-31)
### Fourier transform
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=15 "Edit section: Fourier transform")\]
Further information: [Fourier transform § Laplace transform](https://en.wikipedia.org/wiki/Fourier_transform#Laplace_transform "Fourier transform")
Let f {\\displaystyle f}  be a complex-valued Lebesgue integrable function supported on â \[ 0 , â ) {\\displaystyle \[0,\\infty )}  â , and let F ( s ) \= L f ( s ) {\\displaystyle F(s)={\\mathcal {L}}f(s)}  be its Laplace transform. Then, within the region of convergence, we have F ( Ï \+ i Ï ) \= â« 0 â f ( t ) e â Ï t e â i Ï t d t , {\\displaystyle F(\\sigma +i\\tau )=\\int \_{0}^{\\infty }f(t)e^{-\\sigma t}e^{-i\\tau t}\\,dt,}  which is the Fourier transform of the function â f ( t ) e â Ï t {\\displaystyle f(t)e^{-\\sigma t}}  â .[\[32\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-32)
Indeed, the [Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform "Fourier transform") is a special case (under certain conditions) of the bilateral Laplace transform. The main difference is that the Fourier transform of a function is a complex function of a *real* variable (frequency â Ï {\\displaystyle \\tau }  â ), the Laplace transform of a function is a complex function of a *complex* variable (damping factor Ï {\\displaystyle \\sigma }  and frequency â Ï {\\displaystyle \\tau }  â ). The Laplace transform is usually restricted to transformation of functions of *t* with *t* â„ 0. A consequence of this restriction is that the Laplace transform of a function is a [holomorphic function](https://en.wikipedia.org/wiki/Holomorphic_function "Holomorphic function") of the variable *s*. Unlike the Fourier transform, the Laplace transform of a [distribution](https://en.wikipedia.org/wiki/Distribution_\(mathematics\) "Distribution (mathematics)") is generally a [well-behaved](https://en.wikipedia.org/wiki/Well-behaved "Well-behaved") function. Techniques of complex variables can also be used to directly study Laplace transforms. As a holomorphic function, the Laplace transform has a [power series](https://en.wikipedia.org/wiki/Power_series "Power series") representation. This power series expresses a function as a linear superposition of [moments](https://en.wikipedia.org/wiki/Moment_\(mathematics\) "Moment (mathematics)") of the function. This perspective has applications in probability theory.
Formally, the Fourier transform is equivalent to evaluating the bilateral Laplace transform with imaginary argument *s* = *iÏ*[\[33\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-33)[\[34\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-34) when the condition explained below is fulfilled, f ^ ( Ï ) \= F { f ( t ) } \= L { f ( t ) } \| s \= i Ï \= F ( s ) \| s \= i Ï \= â« â â â e â i Ï t f ( t ) d t . {\\displaystyle {\\begin{aligned}{\\hat {f}}(\\omega )&={\\mathcal {F}}\\{f(t)\\}\\\\\[4pt\]&={\\mathcal {L}}\\{f(t)\\}\|\_{s=i\\omega }=F(s)\|\_{s=i\\omega }\\\\\[4pt\]&=\\int \_{-\\infty }^{\\infty }e^{-i\\omega t}f(t)\\,dt~.\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}{\\hat {f}}(\\omega )&={\\mathcal {F}}\\{f(t)\\}\\\\\[4pt\]&={\\mathcal {L}}\\{f(t)\\}\|\_{s=i\\omega }=F(s)\|\_{s=i\\omega }\\\\\[4pt\]&=\\int \_{-\\infty }^{\\infty }e^{-i\\omega t}f(t)\\,dt~.\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44fd8f6a7374f8e3ae9b00c7ca0107108784f9cd)
This convention of the Fourier transform (â f ^ 3 ( Ï ) {\\displaystyle {\\hat {f}}\_{3}(\\omega )}  â in *[Fourier transform § Other conventions](https://en.wikipedia.org/wiki/Fourier_transform#Other_conventions "Fourier transform")*) requires a factor of â 1/2*Ï*â on the inverse Fourier transform. This relationship between the Laplace and Fourier transforms is often used to determine the [frequency spectrum](https://en.wikipedia.org/wiki/Frequency_spectrum "Frequency spectrum") of a [signal](https://en.wikipedia.org/wiki/Signal_\(information_theory\) "Signal (information theory)") or dynamical system.
The above relation is valid as stated [if and only if](https://en.wikipedia.org/wiki/If_and_only_if "If and only if") the region of convergence (ROC) of *F*(*s*) contains the imaginary axis, *Ï* = 0.
For example, the function *f*(*t*) = cos(*Ï*0*t*) has a Laplace transform *F*(*s*) = *s*/(*s*2 + *Ï*02) whose ROC is Re(*s*) \> 0. As *s* = *iÏ*0 is a pole of *F*(*s*), substituting *s* = *iÏ* in *F*(*s*) does not yield the Fourier transform of *f*(*t*)*u*(*t*), which contains terms proportional to the [Dirac delta functions](https://en.wikipedia.org/wiki/Dirac_delta_functions "Dirac delta functions") *ÎŽ*(*Ï* ± *Ï*0).
However, a relation of the form lim Ï â 0 \+ F ( Ï \+ i Ï ) \= f ^ ( Ï ) {\\displaystyle \\lim \_{\\sigma \\to 0^{+}}F(\\sigma +i\\omega )={\\hat {f}}(\\omega )}  holds under much weaker conditions. For instance, this holds for the above example provided that the limit is understood as a [weak limit](https://en.wikipedia.org/wiki/Weak_limit "Weak limit") of measures (see [vague topology](https://en.wikipedia.org/wiki/Vague_topology "Vague topology")). General conditions relating the limit of the Laplace transform of a function on the boundary to the Fourier transform take the form of [PaleyâWiener theorems](https://en.wikipedia.org/wiki/Paley%E2%80%93Wiener_theorem "PaleyâWiener theorem").
### Mellin transform
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=16 "Edit section: Mellin transform")\]
Main article: [Mellin transform](https://en.wikipedia.org/wiki/Mellin_transform "Mellin transform")
The Mellin transform and its inverse are related to the two-sided Laplace transform by a change of variables.
If in the Mellin transform G ( s ) \= M { g ( Ξ ) } \= â« 0 â Ξ s g ( Ξ ) d Ξ Ξ {\\displaystyle G(s)={\\mathcal {M}}\\{g(\\theta )\\}=\\int \_{0}^{\\infty }\\theta ^{s}g(\\theta )\\,{\\frac {d\\theta }{\\theta }}}  we set *Ξ* = *e*â*t* we get a two-sided Laplace transform.
### Z-transform
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=17 "Edit section: Z-transform")\]
Further information: [Z-transform § Relationship to Laplace transform](https://en.wikipedia.org/wiki/Z-transform#Relationship_to_Laplace_transform "Z-transform")
The unilateral or one-sided Z-transform is the Laplace transform of an ideally sampled signal with the substitution of z \= d e f e s T , {\\displaystyle z{\\stackrel {\\mathrm {def} }{{}={}}}e^{sT},}  where *T* = 1/*fs* is the [sampling interval](https://en.wikipedia.org/wiki/Sampling_interval "Sampling interval") (in units of time e.g., seconds) and *fs* is the [sampling rate](https://en.wikipedia.org/wiki/Sampling_rate "Sampling rate") (in [samples per second](https://en.wikipedia.org/wiki/Samples_per_second "Samples per second") or [hertz](https://en.wikipedia.org/wiki/Hertz "Hertz")).
Let Î T ( t ) \= d e f â n \= 0 â ÎŽ ( t â n T ) {\\displaystyle \\Delta \_{T}(t)\\ {\\stackrel {\\mathrm {def} }{=}}\\ \\sum \_{n=0}^{\\infty }\\delta (t-nT)}  be a sampling impulse train (also called a [Dirac comb](https://en.wikipedia.org/wiki/Dirac_comb "Dirac comb")) and x q ( t ) \= d e f x ( t ) Î T ( t ) \= x ( t ) â n \= 0 â ÎŽ ( t â n T ) \= â n \= 0 â x ( n T ) ÎŽ ( t â n T ) \= â n \= 0 â x \[ n \] ÎŽ ( t â n T ) {\\displaystyle {\\begin{aligned}x\_{q}(t)&{\\stackrel {\\mathrm {def} }{{}={}}}x(t)\\Delta \_{T}(t)=x(t)\\sum \_{n=0}^{\\infty }\\delta (t-nT)\\\\&=\\sum \_{n=0}^{\\infty }x(nT)\\delta (t-nT)=\\sum \_{n=0}^{\\infty }x\[n\]\\delta (t-nT)\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}x\_{q}(t)&{\\stackrel {\\mathrm {def} }{{}={}}}x(t)\\Delta \_{T}(t)=x(t)\\sum \_{n=0}^{\\infty }\\delta (t-nT)\\\\&=\\sum \_{n=0}^{\\infty }x(nT)\\delta (t-nT)=\\sum \_{n=0}^{\\infty }x\[n\]\\delta (t-nT)\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c334ec8f4011a84a4a305f1ac7894fb6e1038cf0) be the sampled representation of the continuous-time *x*(*t*) x \[ n \] \= d e f x ( n T ) . {\\displaystyle x\[n\]{\\stackrel {\\mathrm {def} }{{}={}}}x(nT)~.} ![{\\displaystyle x\[n\]{\\stackrel {\\mathrm {def} }{{}={}}}x(nT)~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2b6eaf257e418b217759fe3b9b982993004af01)
The Laplace transform of the sampled signal *x**q*(*t*) is X q ( s ) \= â« 0 â â x q ( t ) e â s t d t \= â« 0 â â â n \= 0 â x \[ n \] ÎŽ ( t â n T ) e â s t d t \= â n \= 0 â x \[ n \] â« 0 â â ÎŽ ( t â n T ) e â s t d t \= â n \= 0 â x \[ n \] e â n s T . {\\displaystyle {\\begin{aligned}X\_{q}(s)&=\\int \_{0^{-}}^{\\infty }x\_{q}(t)e^{-st}\\,dt\\\\&=\\int \_{0^{-}}^{\\infty }\\sum \_{n=0}^{\\infty }x\[n\]\\delta (t-nT)e^{-st}\\,dt\\\\&=\\sum \_{n=0}^{\\infty }x\[n\]\\int \_{0^{-}}^{\\infty }\\delta (t-nT)e^{-st}\\,dt\\\\&=\\sum \_{n=0}^{\\infty }x\[n\]e^{-nsT}~.\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}X\_{q}(s)&=\\int \_{0^{-}}^{\\infty }x\_{q}(t)e^{-st}\\,dt\\\\&=\\int \_{0^{-}}^{\\infty }\\sum \_{n=0}^{\\infty }x\[n\]\\delta (t-nT)e^{-st}\\,dt\\\\&=\\sum \_{n=0}^{\\infty }x\[n\]\\int \_{0^{-}}^{\\infty }\\delta (t-nT)e^{-st}\\,dt\\\\&=\\sum \_{n=0}^{\\infty }x\[n\]e^{-nsT}~.\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee1fc990806a1d664c96eb253f19ff2def0f986)
This is the precise definition of the unilateral Z-transform of the discrete function *x*\[*n*\] X ( z ) \= â n \= 0 â x \[ n \] z â n {\\displaystyle X(z)=\\sum \_{n=0}^{\\infty }x\[n\]z^{-n}} ![{\\displaystyle X(z)=\\sum \_{n=0}^{\\infty }x\[n\]z^{-n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf7e6d1a9bf26a3017c79040c80295cb1f2eaa1) with the substitution of *z* â *e**sT*.
Comparing the last two equations, we find the relationship between the unilateral Z-transform and the Laplace transform of the sampled signal, X q ( s ) \= X ( z ) \| z \= e s T . {\\displaystyle X\_{q}(s)=X(z){\\Big \|}\_{z=e^{sT}}.} 
The similarity between the Z- and Laplace transforms is expanded upon in the theory of [time scale calculus](https://en.wikipedia.org/wiki/Time_scale_calculus "Time scale calculus").
### Borel transform
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=18 "Edit section: Borel transform")\]
The integral form of the [Borel transform](https://en.wikipedia.org/wiki/Borel_summation "Borel summation") F ( s ) \= â« 0 â f ( z ) e â s z d z {\\displaystyle F(s)=\\int \_{0}^{\\infty }f(z)e^{-sz}\\,dz}  is a special case of the Laplace transform for *f* an [entire function](https://en.wikipedia.org/wiki/Entire_function "Entire function") of exponential type, meaning that \| f ( z ) \| †A e B \| z \| {\\displaystyle \|f(z)\|\\leq Ae^{B\|z\|}}  for some constants *A* and *B*. The generalized Borel transform allows a different weighting function to be used, rather than the exponential function, to transform functions not of exponential type. [Nachbin's theorem](https://en.wikipedia.org/wiki/Nachbin%27s_theorem "Nachbin's theorem") gives necessary and sufficient conditions for the Borel transform to be well defined.
### Fundamental relationships
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=19 "Edit section: Fundamental relationships")\]
Since an ordinary Laplace transform can be written as a special case of a two-sided transform, and since the two-sided transform can be written as the sum of two one-sided transforms, the theory of the Laplace-, Fourier-, Mellin-, and Z-transforms are at bottom the same subject. However, a different point of view and different characteristic problems are associated with each of these four major integral transforms.
## Table of selected Laplace transforms
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=20 "Edit section: Table of selected Laplace transforms")\]
Main article: [List of Laplace transforms](https://en.wikipedia.org/wiki/List_of_Laplace_transforms "List of Laplace transforms")
The following table provides Laplace transforms for many common functions of a single variable.[\[35\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-35)[\[36\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-36) For definitions and explanations, see the *Explanatory Notes* at the end of the table.
Because the Laplace transform is a linear operator,
- The Laplace transform of a sum is the sum of Laplace transforms of each term.
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{\\displaystyle {\\mathcal {L}}\\{f(t)+g(t)\\}={\\mathcal {L}}\\{f(t)\\}+{\\mathcal {L}}\\{g(t)\\}}
- The Laplace transform of a multiple of a function is that multiple times the Laplace transformation of that function.
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{\\displaystyle {\\mathcal {L}}\\{af(t)\\}=a{\\mathcal {L}}\\{f(t)\\}}
Using this linearity, and various [trigonometric](https://en.wikipedia.org/wiki/List_of_trigonometric_identities "List of trigonometric identities"), [hyperbolic](https://en.wikipedia.org/wiki/Hyperbolic_function "Hyperbolic function"), and complex number (etc.) properties and/or identities, some Laplace transforms can be obtained from others more quickly than by using the definition directly.
The unilateral Laplace transform takes as input a function whose time domain is the [non-negative](https://en.wikipedia.org/wiki/Non-negative "Non-negative") reals, which is why all of the time domain functions in the table below are multiples of the [Heaviside step function](https://en.wikipedia.org/wiki/Heaviside_step_function "Heaviside step function"), *u*(*t*).
The entries of the table that involve a time delay *Ï* are required to be [causal](https://en.wikipedia.org/wiki/Causal_system "Causal system") (meaning that *Ï* \> 0). A causal system is a system where the [impulse response](https://en.wikipedia.org/wiki/Impulse_response "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](https://en.wikipedia.org/wiki/Anticausal_system "Anticausal system").
| Function | Time domain f ( t ) \= L â 1 { F ( s ) } {\\displaystyle f(t)={\\mathcal {L}}^{-1}\\{F(s)\\}}  |
|---|---|
## *s*\-domain equivalent circuits and impedances
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=21 "Edit section: s-domain equivalent circuits and impedances")\]
The Laplace transform is often used in [circuit analysis](https://en.wikipedia.org/wiki/Network_analysis_\(electrical_circuits\) "Network analysis (electrical circuits)") by conversions to the *s*\-domain of circuit elements. Circuit elements can be transformed into [impedances](https://en.wikipedia.org/wiki/Electrical_impedance "Electrical impedance"), very similar to [phasor](https://en.wikipedia.org/wiki/Phasor_\(sine_waves\) "Phasor (sine waves)") impedances.
Here is a summary of equivalents:
[](https://en.wikipedia.org/wiki/File:S-Domain_circuit_equivalents.svg "s-domain equivalent circuits")
*s*\-domain equivalent circuits
Note that the resistor is exactly the same in the time domain and the *s*\-domain. The sources are put in if there are initial conditions on the circuit elements. For example, if a capacitor has an initial voltage across it, or if the inductor has an initial current through it, the sources inserted in the *s*\-domain account for that.
The equivalents for current and voltage sources are derived from the transformations in the table above.
## Examples and applications
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=22 "Edit section: Examples and applications")\]
The Laplace transform is used frequently in [engineering](https://en.wikipedia.org/wiki/Engineering "Engineering") and [physics](https://en.wikipedia.org/wiki/Physics "Physics"); the output of a [linear time-invariant system](https://en.wikipedia.org/wiki/Linear_time-invariant_system "Linear time-invariant system") can be calculated by convolving its unit impulse response with the input signal. Performing this calculation in Laplace space turns the convolution into a multiplication; the latter being easier to solve because of its algebraic form. For more information, see [control theory](https://en.wikipedia.org/wiki/Control_theory "Control theory"). The Laplace transform is invertible on a large class of functions. Given a mathematical or functional description of an input or output to a [system](https://en.wikipedia.org/wiki/System "System"), the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.[\[42\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-42)
The Laplace transform can also be used to solve differential equations and is used extensively in [mechanical engineering](https://en.wikipedia.org/wiki/Mechanical_engineering "Mechanical engineering") and [electrical engineering](https://en.wikipedia.org/wiki/Electrical_engineering "Electrical engineering"). The Laplace transform reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of algebra. The original differential equation can then be solved by applying the inverse Laplace transform. English electrical engineer [Oliver Heaviside](https://en.wikipedia.org/wiki/Oliver_Heaviside "Oliver Heaviside") first proposed a similar scheme, although without using the Laplace transform; and the resulting operational calculus is credited as the Heaviside calculus.
### Evaluating improper integrals
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=23 "Edit section: Evaluating improper integrals")\]
Let â L { f ( t ) } \= F ( s ) {\\displaystyle {\\mathcal {L}}\\left\\{f(t)\\right\\}=F(s)}  â . Then (see the table above) â s L { f ( t ) t } \= â s â« 0 â f ( t ) t e â s t d t \= â â« 0 â f ( t ) e â s t d t \= â F ( s ) {\\displaystyle \\partial \_{s}{\\mathcal {L}}\\left\\{{\\frac {f(t)}{t}}\\right\\}=\\partial \_{s}\\int \_{0}^{\\infty }{\\frac {f(t)}{t}}e^{-st}\\,dt=-\\int \_{0}^{\\infty }f(t)e^{-st}dt=-F(s)} 
From which one gets: L { f ( t ) t } \= â« s â F ( p ) d p . {\\displaystyle {\\mathcal {L}}\\left\\{{\\frac {f(t)}{t}}\\right\\}=\\int \_{s}^{\\infty }F(p)\\,dp.} 
In the limit â s â 0 {\\displaystyle s\\rightarrow 0}  â , one gets â« 0 â f ( t ) t d t \= â« 0 â F ( p ) d p , {\\displaystyle \\int \_{0}^{\\infty }{\\frac {f(t)}{t}}\\,dt=\\int \_{0}^{\\infty }F(p)\\,dp,}  provided that the interchange of limits can be justified. This is often possible as a consequence of the [final value theorem](https://en.wikipedia.org/wiki/Final_value_theorem#Final_Value_Theorem_for_improperly_integrable_functions_\(Abel's_theorem_for_integrals\) "Final value theorem"). Even when the interchange cannot be justified the calculation can be suggestive. For example, with *a* â 0 â *b*, proceeding formally one has â« 0 â cos ⥠( a t ) â cos ⥠( b t ) t d t \= â« 0 â ( p p 2 \+ a 2 â p p 2 \+ b 2 ) d p \= \[ 1 2 ln ⥠p 2 \+ a 2 p 2 \+ b 2 \] 0 â \= 1 2 ln ⥠b 2 a 2 \= ln ⥠\| b a \| . {\\displaystyle {\\begin{aligned}\\int \_{0}^{\\infty }{\\frac {\\cos(at)-\\cos(bt)}{t}}\\,dt&=\\int \_{0}^{\\infty }\\left({\\frac {p}{p^{2}+a^{2}}}-{\\frac {p}{p^{2}+b^{2}}}\\right)\\,dp\\\\\[6pt\]&=\\left\[{\\frac {1}{2}}\\ln {\\frac {p^{2}+a^{2}}{p^{2}+b^{2}}}\\right\]\_{0}^{\\infty }={\\frac {1}{2}}\\ln {\\frac {b^{2}}{a^{2}}}=\\ln \\left\|{\\frac {b}{a}}\\right\|.\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}\\int \_{0}^{\\infty }{\\frac {\\cos(at)-\\cos(bt)}{t}}\\,dt&=\\int \_{0}^{\\infty }\\left({\\frac {p}{p^{2}+a^{2}}}-{\\frac {p}{p^{2}+b^{2}}}\\right)\\,dp\\\\\[6pt\]&=\\left\[{\\frac {1}{2}}\\ln {\\frac {p^{2}+a^{2}}{p^{2}+b^{2}}}\\right\]\_{0}^{\\infty }={\\frac {1}{2}}\\ln {\\frac {b^{2}}{a^{2}}}=\\ln \\left\|{\\frac {b}{a}}\\right\|.\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/150eb829ed85ccdec437af079d262fc08428699c)
### Complex impedance of a capacitor
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=24 "Edit section: Complex impedance of a capacitor")\]
In the theory of [electrical circuits](https://en.wikipedia.org/wiki/Electrical_circuit "Electrical circuit"), the current flow in a [capacitor](https://en.wikipedia.org/wiki/Capacitor "Capacitor") is proportional to the capacitance and rate of change in the electrical potential (with equations as for the [SI](https://en.wikipedia.org/wiki/International_System_of_Units "International System of Units") unit system). Symbolically, this is expressed by the differential equation i \= C d v d t , {\\displaystyle i=C{dv \\over dt},}  where *C* is the capacitance of the capacitor, *i* = *i*(*t*) is the [electric current](https://en.wikipedia.org/wiki/Electric_current "Electric current") through the capacitor as a function of time, and *v* = *v*(*t*) is the [voltage](https://en.wikipedia.org/wiki/Electrostatic_potential "Electrostatic potential") across the terminals of the capacitor, also as a function of time.
Taking the Laplace transform of this equation, we obtain I ( s ) \= C ( s V ( s ) â V 0 ) , {\\displaystyle I(s)=C(sV(s)-V\_{0}),}  where I ( s ) \= L { i ( t ) } , V ( s ) \= L { v ( t ) } , {\\displaystyle {\\begin{aligned}I(s)&={\\mathcal {L}}\\{i(t)\\},\\\\V(s)&={\\mathcal {L}}\\{v(t)\\},\\end{aligned}}}  and V 0 \= v ( 0 ) . {\\displaystyle V\_{0}=v(0).} 
Solving for *V*(*s*) we have V ( s ) \= I ( s ) s C \+ V 0 s . {\\displaystyle V(s)={I(s) \\over sC}+{V\_{0} \\over s}.} 
The definition of the complex impedance *Z* (in [ohms](https://en.wikipedia.org/wiki/Ohm "Ohm")) is the ratio of the complex voltage *V* divided by the complex current *I* while holding the initial state *V*0 at zero: Z ( s ) \= V ( s ) I ( s ) \| V 0 \= 0 . {\\displaystyle Z(s)=\\left.{V(s) \\over I(s)}\\right\|\_{V\_{0}=0}.} 
Using this definition and the previous equation, we find: Z ( s ) \= 1 s C , {\\displaystyle Z(s)={\\frac {1}{sC}},}  which is the correct expression for the complex impedance of a capacitor. In addition, the Laplace transform has large applications in control theory.
### Impulse response
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=25 "Edit section: Impulse response")\]
Consider a linear time-invariant system with [transfer function](https://en.wikipedia.org/wiki/Transfer_function "Transfer function") H ( s ) \= 1 ( s \+ α ) ( s \+ ÎČ ) . {\\displaystyle H(s)={\\frac {1}{(s+\\alpha )(s+\\beta )}}.} 
The [impulse response](https://en.wikipedia.org/wiki/Impulse_response "Impulse response") is the inverse Laplace transform of this transfer function: h ( t ) \= L â 1 { H ( s ) } . {\\displaystyle h(t)={\\mathcal {L}}^{-1}\\{H(s)\\}.} 
Partial fraction expansion
To evaluate this inverse transform, we begin by expanding *H*(*s*) using the method of partial fraction expansion, 1 ( s \+ α ) ( s \+ ÎČ ) \= P s \+ α \+ R s \+ ÎČ . {\\displaystyle {\\frac {1}{(s+\\alpha )(s+\\beta )}}={P \\over s+\\alpha }+{R \\over s+\\beta }.} 
The unknown constants *P* and *R* are the [residues](https://en.wikipedia.org/wiki/Residue_\(complex_analysis\) "Residue (complex analysis)") located at the corresponding poles of the transfer function. Each residue represents the relative contribution of that [singularity](https://en.wikipedia.org/wiki/Mathematical_singularity "Mathematical singularity") to the transfer function's overall shape.
By the [residue theorem](https://en.wikipedia.org/wiki/Residue_theorem "Residue theorem"), the inverse Laplace transform depends only upon the poles and their residues. To find the residue *P*, we multiply both sides of the equation by *s* + *α* to get 1 s \+ ÎČ \= P \+ R ( s \+ α ) s \+ ÎČ . {\\displaystyle {\\frac {1}{s+\\beta }}=P+{R(s+\\alpha ) \\over s+\\beta }.} 
Then by letting *s* = â*α*, the contribution from *R* vanishes and all that is left is P \= 1 s \+ ÎČ \| s \= â α \= 1 ÎČ â α . {\\displaystyle P=\\left.{1 \\over s+\\beta }\\right\|\_{s=-\\alpha }={1 \\over \\beta -\\alpha }.} 
Similarly, the residue *R* is given by R \= 1 s \+ α \| s \= â ÎČ \= 1 α â ÎČ . {\\displaystyle R=\\left.{1 \\over s+\\alpha }\\right\|\_{s=-\\beta }={1 \\over \\alpha -\\beta }.} 
Note that R \= â 1 ÎČ â α \= â P {\\displaystyle R={-1 \\over \\beta -\\alpha }=-P}  and so the substitution of *R* and *P* into the expanded expression for *H*(*s*) gives H ( s ) \= ( 1 ÎČ â α ) â
( 1 s \+ α â 1 s \+ ÎČ ) . {\\displaystyle H(s)=\\left({\\frac {1}{\\beta -\\alpha }}\\right)\\cdot \\left({1 \\over s+\\alpha }-{1 \\over s+\\beta }\\right).} 
Finally, using the linearity property and the known transform for exponential decay (see *Item* \#*3* in the *Table of Laplace Transforms*, above), we can take the inverse Laplace transform of *H*(*s*) to obtain h ( t ) \= L â 1 { H ( s ) } \= 1 ÎČ â α ( e â α t â e â ÎČ t ) , {\\displaystyle h(t)={\\mathcal {L}}^{-1}\\{H(s)\\}={\\frac {1}{\\beta -\\alpha }}\\left(e^{-\\alpha t}-e^{-\\beta t}\\right),}  which is the impulse response of the system.
Convolution
The same result can be achieved using the [convolution property](https://en.wikipedia.org/wiki/Convolution_theorem "Convolution theorem") as if the system is a series of filters with transfer functions 1/(*s* + *α*) and 1/(*s* + *ÎČ*). That is, the inverse of H ( s ) \= 1 ( s \+ α ) ( s \+ ÎČ ) \= 1 s \+ α â
1 s \+ ÎČ {\\displaystyle H(s)={\\frac {1}{(s+\\alpha )(s+\\beta )}}={\\frac {1}{s+\\alpha }}\\cdot {\\frac {1}{s+\\beta }}}  is L â 1 { 1 s \+ α } â L â 1 { 1 s \+ ÎČ } \= e â α t â e â ÎČ t \= â« 0 t e â α x e â ÎČ ( t â x ) d x \= e â α t â e â ÎČ t ÎČ â α . {\\displaystyle {\\mathcal {L}}^{-1}\\!\\left\\{{\\frac {1}{s+\\alpha }}\\right\\}\*{\\mathcal {L}}^{-1}\\!\\left\\{{\\frac {1}{s+\\beta }}\\right\\}=e^{-\\alpha t}\*e^{-\\beta t}=\\int \_{0}^{t}e^{-\\alpha x}e^{-\\beta (t-x)}\\,dx={\\frac {e^{-\\alpha t}-e^{-\\beta t}}{\\beta -\\alpha }}.} 
### Phase delay
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=26 "Edit section: Phase delay")\]
| Time function | Laplace transform |
|---|---|
| sin ⥠( Ï t \+ Ï ) {\\displaystyle \\sin {(\\omega t+\\varphi )}}  | |
Starting with the Laplace transform, X ( s ) \= s sin ⥠( Ï ) \+ Ï cos ⥠( Ï ) s 2 \+ Ï 2 {\\displaystyle X(s)={\\frac {s\\sin(\\varphi )+\\omega \\cos(\\varphi )}{s^{2}+\\omega ^{2}}}}  we find the inverse by first rearranging terms in the fraction: X ( s ) \= s sin ⥠( Ï ) s 2 \+ Ï 2 \+ Ï cos ⥠( Ï ) s 2 \+ Ï 2 \= sin ⥠( Ï ) ( s s 2 \+ Ï 2 ) \+ cos ⥠( Ï ) ( Ï s 2 \+ Ï 2 ) . {\\displaystyle {\\begin{aligned}X(s)&={\\frac {s\\sin(\\varphi )}{s^{2}+\\omega ^{2}}}+{\\frac {\\omega \\cos(\\varphi )}{s^{2}+\\omega ^{2}}}\\\\&=\\sin(\\varphi )\\left({\\frac {s}{s^{2}+\\omega ^{2}}}\\right)+\\cos(\\varphi )\\left({\\frac {\\omega }{s^{2}+\\omega ^{2}}}\\right).\\end{aligned}}} 
We are now able to take the inverse Laplace transform of our terms: x ( t ) \= sin ⥠( Ï ) L â 1 { s s 2 \+ Ï 2 } \+ cos ⥠( Ï ) L â 1 { Ï s 2 \+ Ï 2 } \= sin ⥠( Ï ) cos ⥠( Ï t ) \+ cos ⥠( Ï ) sin ⥠( Ï t ) . {\\displaystyle {\\begin{aligned}x(t)&=\\sin(\\varphi ){\\mathcal {L}}^{-1}\\left\\{{\\frac {s}{s^{2}+\\omega ^{2}}}\\right\\}+\\cos(\\varphi ){\\mathcal {L}}^{-1}\\left\\{{\\frac {\\omega }{s^{2}+\\omega ^{2}}}\\right\\}\\\\&=\\sin(\\varphi )\\cos(\\omega t)+\\cos(\\varphi )\\sin(\\omega t).\\end{aligned}}} 
This is just the [sine of the sum](https://en.wikipedia.org/wiki/Trigonometric_identity#Angle_sum_and_difference_identities "Trigonometric identity") of the arguments, yielding: x ( t ) \= sin ⥠( Ï t \+ Ï ) . {\\displaystyle x(t)=\\sin(\\omega t+\\varphi ).} 
We can apply similar logic to find that L â 1 { s cos âĄ Ï â Ï sin âĄ Ï s 2 \+ Ï 2 } \= cos ⥠( Ï t \+ Ï ) . {\\displaystyle {\\mathcal {L}}^{-1}\\left\\{{\\frac {s\\cos \\varphi -\\omega \\sin \\varphi }{s^{2}+\\omega ^{2}}}\\right\\}=\\cos {(\\omega t+\\varphi )}.} 
### Statistical mechanics
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=27 "Edit section: Statistical mechanics")\]
In [statistical mechanics](https://en.wikipedia.org/wiki/Statistical_mechanics "Statistical mechanics"), the Laplace transform of the density of states g ( E ) {\\displaystyle g(E)}  defines the [partition function](https://en.wikipedia.org/wiki/Partition_function_\(statistical_mechanics\) "Partition function (statistical mechanics)").[\[43\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-43) That is, the canonical partition function Z ( ÎČ ) {\\displaystyle Z(\\beta )}  is given by Z ( ÎČ ) \= â« 0 â e â ÎČ E g ( E ) d E {\\displaystyle Z(\\beta )=\\int \_{0}^{\\infty }e^{-\\beta E}g(E)\\,dE}  and the inverse is given by g ( E ) \= 1 2 Ï i â« ÎČ 0 â i â ÎČ 0 \+ i â e ÎČ E Z ( ÎČ ) d ÎČ {\\displaystyle g(E)={\\frac {1}{2\\pi i}}\\int \_{\\beta \_{0}-i\\infty }^{\\beta \_{0}+i\\infty }e^{\\beta E}Z(\\beta )\\,d\\beta } 
### Spatial (not time) structure from astronomical spectrum
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=28 "Edit section: Spatial (not time) structure from astronomical spectrum")\]
The wide and general applicability of the Laplace transform and its inverse is illustrated by an application in astronomy which provides some information on the *spatial distribution* of matter of an [astronomical](https://en.wikipedia.org/wiki/Astronomy "Astronomy") source of [radiofrequency](https://en.wikipedia.org/wiki/Radiofrequency "Radiofrequency") [thermal radiation](https://en.wikipedia.org/wiki/Thermal_radiation "Thermal radiation") too distant to [resolve](https://en.wikipedia.org/wiki/Angular_resolution "Angular resolution") as more than a point, given its [flux density](https://en.wikipedia.org/wiki/Flux_density "Flux density") [spectrum](https://en.wikipedia.org/wiki/Spectrum "Spectrum"), rather than relating the *time* domain with the spectrum (frequency domain).
Assuming certain properties of the object, e.g. spherical shape and constant temperature, calculations based on carrying out an inverse Laplace transformation on the spectrum of the object can produce the only possible [model](https://en.wikipedia.org/wiki/Mathematical_model "Mathematical model") of the distribution of matter in it (density as a function of distance from the center) consistent with the spectrum.[\[44\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-44) When independent information on the structure of an object is available, the inverse Laplace transform method has been found to be in good agreement.
### Birth and death processes
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=29 "Edit section: Birth and death processes")\]
Consider a [random walk](https://en.wikipedia.org/wiki/Random_walk "Random walk"), with steps { \+ 1 , â 1 } {\\displaystyle \\{+1,-1\\}}  occurring with probabilities â p , q \= 1 â p {\\displaystyle p,q=1-p}  â .[\[45\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-Feller-45) Suppose also that the time step is a [Poisson process](https://en.wikipedia.org/wiki/Poisson_process "Poisson process"), with parameter â λ {\\displaystyle \\lambda }  â . Then the probability of the walk being at the lattice point n {\\displaystyle n}  at time t {\\displaystyle t}  is P n ( t ) \= â« 0 t λ e â λ ( t â s ) ( p P n â 1 ( s ) \+ q P n \+ 1 ( s ) ) d s ( \+ e â λ t when n \= 0 ) . {\\displaystyle P\_{n}(t)=\\int \_{0}^{t}\\lambda e^{-\\lambda (t-s)}(pP\_{n-1}(s)+qP\_{n+1}(s))\\,ds\\quad (+e^{-\\lambda t}\\quad {\\text{when}}\\ n=0).}  This leads to a system of [integral equations](https://en.wikipedia.org/wiki/Integral_equation "Integral equation") (or equivalently a system of differential equations). However, because it is a system of convolution equations, the Laplace transform converts it into a [system of linear equations](https://en.wikipedia.org/wiki/System_of_linear_equations "System of linear equations") for Ï n ( s ) \= L ( P n ) ( s ) , {\\displaystyle \\pi \_{n}(s)={\\mathcal {L}}(P\_{n})(s),}  namely: Ï n ( s ) \= λ λ \+ s ( p Ï n â 1 ( s ) \+ q Ï n \+ 1 ( s ) ) ( \+ 1 λ \+ s when n \= 0 ) {\\displaystyle \\pi \_{n}(s)={\\frac {\\lambda }{\\lambda +s}}(p\\pi \_{n-1}(s)+q\\pi \_{n+1}(s))\\quad (+{\\frac {1}{\\lambda +s}}\\quad {\\text{when}}\\ n=0)}  which may now be solved by standard methods.
### Tauberian theory
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=30 "Edit section: Tauberian theory")\]
The Laplace transform of the measure ÎŒ {\\displaystyle \\mu }  on \[ 0 , â ) {\\displaystyle \[0,\\infty )}  is given by L ÎŒ ( s ) \= â« 0 â e â s t d ÎŒ ( t ) . {\\displaystyle {\\mathcal {L}}\\mu (s)=\\int \_{0}^{\\infty }e^{-st}d\\mu (t).}  It is intuitively clear that, for small â s \> 0 {\\displaystyle s\>0}  â , the exponentially decaying integrand will become more sensitive to the concentration of the measure ÎŒ {\\displaystyle \\mu }  on larger subsets of the domain. To make this more precise, introduce the distribution function: M ( t ) \= ÎŒ ( \[ 0 , t ) ) . {\\displaystyle M(t)=\\mu (\[0,t)).}  Formally, we expect a limit of the following kind: lim s â 0 \+ L ÎŒ ( s ) \= lim t â â M ( t ) . {\\displaystyle \\lim \_{s\\to 0^{+}}{\\mathcal {L}}\\mu (s)=\\lim \_{t\\to \\infty }M(t).}  [Tauberian theorems](https://en.wikipedia.org/wiki/Tauberian_theorem "Tauberian theorem") are theorems relating the asymptotics of the Laplace transform, as â s â 0 \+ {\\displaystyle s\\to 0^{+}}  â , to those of the distribution of ÎŒ {\\displaystyle \\mu }  as â t â â {\\displaystyle t\\to \\infty }  â . They are thus of importance in asymptotic formulae of [probability](https://en.wikipedia.org/wiki/Probability "Probability") and [statistics](https://en.wikipedia.org/wiki/Statistics "Statistics"), where often the spectral side has asymptotics that are simpler to infer.[\[45\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-Feller-45)
Two Tauberian theorems of note are the [HardyâLittlewood Tauberian theorem](https://en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood_Tauberian_theorem "HardyâLittlewood Tauberian theorem") and [Wiener's Tauberian theorem](https://en.wikipedia.org/wiki/Wiener%27s_Tauberian_theorem "Wiener's Tauberian theorem"). The Wiener theorem generalizes the [Ikehara Tauberian theorem](https://en.wikipedia.org/wiki/Ikehara_Tauberian_theorem "Ikehara Tauberian theorem"), which is the following statement:
Let â A ( x ) {\\displaystyle A(x)}  â be a non-negative, [monotonic](https://en.wikipedia.org/wiki/Monotonic_function "Monotonic function") nondecreasing function of â x {\\displaystyle x}  â , defined for â 0 †x \< â {\\displaystyle 0\\leq x\<\\infty }  â . Suppose that f ( s ) \= â« 0 â A ( x ) e â x s d x {\\displaystyle f(s)=\\int \_{0}^{\\infty }A(x)e^{-xs}\\,dx}  converges for â â ( s ) \> 1 {\\displaystyle \\Re (s)\>1}  â to the function â f ( s ) {\\displaystyle f(s)}  â and that, for some non-negative number â c {\\displaystyle c}  â , f ( s ) â c s â 1 {\\displaystyle f(s)-{\\frac {c}{s-1}}}  has an extension as a [continuous function](https://en.wikipedia.org/wiki/Continuous_function "Continuous function") for â â ( s ) â„ 1 {\\displaystyle \\Re (s)\\geq 1}  â . Then the [limit](https://en.wikipedia.org/wiki/Limit_of_a_function "Limit of a function") as â x {\\displaystyle x}  â goes to infinity of â e â x A ( x ) {\\displaystyle e^{-x}A(x)}  â is equal to â c {\\displaystyle c}  â .
This statement can be applied in particular to the [logarithmic derivative](https://en.wikipedia.org/wiki/Logarithmic_derivative "Logarithmic derivative") of [Riemann zeta function](https://en.wikipedia.org/wiki/Riemann_zeta_function "Riemann zeta function"), and thus provides an extremely short way to prove the [prime number theorem](https://en.wikipedia.org/wiki/Prime_number_theorem "Prime number theorem").[\[46\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-46)
## See also
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=31 "Edit section: See also")\]
- [](https://en.wikipedia.org/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg)[Mathematics portal](https://en.wikipedia.org/wiki/Portal:Mathematics "Portal:Mathematics")
- [Analog signal processing](https://en.wikipedia.org/wiki/Analog_signal_processing "Analog signal processing")
- [Bernstein's theorem on monotone functions](https://en.wikipedia.org/wiki/Bernstein%27s_theorem_on_monotone_functions "Bernstein's theorem on monotone functions")
- [Continuous-repayment mortgage](https://en.wikipedia.org/wiki/Continuous-repayment_mortgage#Mortgage_difference_and_differential_equation "Continuous-repayment mortgage")
- [Dirichlet integral](https://en.wikipedia.org/wiki/Dirichlet_integral "Dirichlet integral")
- [Differential equation](https://en.wikipedia.org/wiki/Differential_equation "Differential equation")
- [Generating function](https://en.wikipedia.org/wiki/Generating_function "Generating function")
- [Hamburger moment problem](https://en.wikipedia.org/wiki/Hamburger_moment_problem "Hamburger moment problem")
- [HardyâLittlewood Tauberian theorem](https://en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood_Tauberian_theorem "HardyâLittlewood Tauberian theorem")
- [LaplaceâCarson transform](https://en.wikipedia.org/wiki/Laplace%E2%80%93Carson_transform "LaplaceâCarson transform")
- [Moment-generating function](https://en.wikipedia.org/wiki/Moment-generating_function "Moment-generating function")
- [Nonlocal operator](https://en.wikipedia.org/wiki/Nonlocal_operator "Nonlocal operator")
- [Partial fraction decomposition](https://en.wikipedia.org/wiki/Partial_fraction_decomposition "Partial fraction decomposition")
- [Post's inversion formula](https://en.wikipedia.org/wiki/Post%27s_inversion_formula "Post's inversion formula")
- [Signal-flow graph](https://en.wikipedia.org/wiki/Signal-flow_graph "Signal-flow graph")
- [Transfer function](https://en.wikipedia.org/wiki/Transfer_function "Transfer function")
- [Z-transform](https://en.wikipedia.org/wiki/Z-transform "Z-transform")
## Notes
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=32 "Edit section: Notes")\]
1. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-Lynn_1986_pp._225%E2%80%93272_1-0)**
Lynn, Paul A. (1986), "The Laplace Transform and the *z*\-transform", *Electronic Signals and Systems*, London: Macmillan Education UK, pp. 225â272, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-1-349-18461-3\_6](https://doi.org/10.1007%2F978-1-349-18461-3_6), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-333-39164-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-333-39164-8 "Special:BookSources/978-0-333-39164-8")
, "Laplace Transform and the z-transform are closely related to the Fourier Transform. Laplace Transform is somewhat more general in scope than the Fourier Transform, and is widely used by engineers for describing continuous circuits and systems, including automatic control systems."
2. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-2)**
["Differential Equations â Laplace Transforms"](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx), *Pauls Online Math Notes*, retrieved 2020-08-08
3. ^ [***a***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-:1_3-0) [***b***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-:1_3-1)
Weisstein, Eric W., ["Laplace Transform"](https://mathworld.wolfram.com/LaplaceTransform.html), *Wolfram MathWorld*, retrieved 2020-08-08
4. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-4)**
Deakin, Michael A. B. (1981), ["The Development of the Laplace Transform, 1737-1937: I. Euler to Spitzer, 1737-1880"](https://www.jstor.org/stable/41133637), *Archive for History of Exact Sciences*, **25** (4): 343â390, [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0003-9519](https://search.worldcat.org/issn/0003-9519)
5. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-5)**
"Des Fonctions génératrices" \[On generating functions\], [*Théorie analytique des Probabilités*](https://archive.org/details/thorieanalytiqu01laplgoog) \[*Analytical Probability Theory*\] (in French) (2nd ed.), Paris, 1814, chap.I sect.2-20
6. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-6)**
Jaynes, E. T. (Edwin T.) (2003), *Probability theory : the logic of science*, Bretthorst, G. Larry, Cambridge, UK: Cambridge University Press, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0511065892](https://en.wikipedia.org/wiki/Special:BookSources/0511065892 "Special:BookSources/0511065892")
, [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [57254076](https://search.worldcat.org/oclc/57254076)
7. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-7)**
[Abel, Niels H.](https://en.wikipedia.org/wiki/Niels_Henrik_Abel "Niels Henrik Abel") (1820), "Sur les fonctions gĂ©nĂ©ratrices et leurs dĂ©terminantes", *Ćuvres ComplĂštes* (in French), vol. II (published 1839), pp. 77â88
[1881 edition](https://books.google.com/books?id=6FtDAQAAMAAJ&pg=RA2-PA67)
8. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-8)** [Euler 1744](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFEuler1744), [Euler 1753](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFEuler1753), [Euler 1769](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFEuler1769)
9. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-9)** [Lagrange 1773](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFLagrange1773)
10. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-10)** [Grattan-Guinness 1997](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFGrattan-Guinness1997), p. 260
11. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-11)** [Grattan-Guinness 1997](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFGrattan-Guinness1997), p. 261
12. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-12)** [Grattan-Guinness 1997](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFGrattan-Guinness1997), pp. 261â262
13. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-13)** [Grattan-Guinness 1997](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFGrattan-Guinness1997), pp. 262â266
14. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-14)**
[Heaviside, Oliver](https://en.wikipedia.org/wiki/Oliver_Heaviside "Oliver Heaviside") (January 2008), ["The solution of definite integrals by differential transformation"](https://books.google.com/books?id=y9auR0L6ZRcC&pg=PA234), *Electromagnetic Theory*, vol. III, London, section 526, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[9781605206189](https://en.wikipedia.org/wiki/Special:BookSources/9781605206189 "Special:BookSources/9781605206189")
`{{citation}}`: CS1 maint: location missing publisher ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_location_missing_publisher "Category:CS1 maint: location missing publisher"))
15. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-15)**
[Edwards, H. M.](https://en.wikipedia.org/wiki/Harold_Edwards_\(mathematician\) "Harold Edwards (mathematician)") (1974), *Riemann's Zeta Function*, New York: Academic Press, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-12-232750-0](https://en.wikipedia.org/wiki/Special:BookSources/0-12-232750-0 "Special:BookSources/0-12-232750-0")
, [Zbl](https://en.wikipedia.org/wiki/Zbl_\(identifier\) "Zbl (identifier)") [0315\.10035](https://zbmath.org/?format=complete&q=an:0315.10035)
16. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-16)**
Gardner, Murray F.; Barnes, John L. (1942), *Transients in Linear Systems studied by the Laplace Transform*, New York: Wiley
, Appendix C
17. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-17)**
[Lerch, Mathias](https://en.wikipedia.org/wiki/Mathias_Lerch "Mathias Lerch") (1903), "Sur un point de la thĂ©orie des fonctions gĂ©nĂ©ratrices d'Abel" \[Proof of the inversion formula\], *[Acta Mathematica](https://en.wikipedia.org/wiki/Acta_Mathematica "Acta Mathematica")* (in French), **27**: 339â351, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/BF02421315](https://doi.org/10.1007%2FBF02421315), [hdl](https://en.wikipedia.org/wiki/Hdl_\(identifier\) "Hdl (identifier)"):[10338\.dmlcz/501554](https://hdl.handle.net/10338.dmlcz%2F501554)
18. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-18)**
[Bromwich, Thomas J.](https://en.wikipedia.org/wiki/Thomas_John_I%27Anson_Bromwich "Thomas John I'Anson Bromwich") (1916), ["Normal coordinates in dynamical systems"](https://zenodo.org/record/2319588), *[Proceedings of the London Mathematical Society](https://en.wikipedia.org/wiki/Proceedings_of_the_London_Mathematical_Society "Proceedings of the London Mathematical Society")*, **15**: 401â448, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1112/plms/s2-15.1.401](https://doi.org/10.1112%2Fplms%2Fs2-15.1.401)
19. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-19)**
An influential book was:
Gardner, Murray F.; Barnes, John L. (1942), *Transients in Linear Systems studied by the Laplace Transform*, New York: Wiley
20. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-20)**
Doetsch, Gustav (1937), *Theorie und Anwendung der Laplacesche Transformation* \[*Theory and Application of the Laplace Transform*\] (in German), Berlin: Springer
translation 1943
21. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-21)** [Feller 1971](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFFeller1971), §XIII.1.
22. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-22)** The cumulative distribution function is the integral of the probability density function.
23. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-23)**
MikusiĆski, Jan (14 July 2014), [*Operational Calculus*](https://books.google.com/books?id=e8LSBQAAQBAJ), Elsevier, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[9781483278933](https://en.wikipedia.org/wiki/Special:BookSources/9781483278933 "Special:BookSources/9781483278933")
24. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-24)** [Widder 1941](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFWidder1941), Chapter II, §1
25. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-25)** [Widder 1941](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFWidder1941), Chapter VI, §2
26. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-26)**
Beffa, Federico (2024), ["Laplace Transform of Distributions"](https://link.springer.com/10.1007/978-3-031-40681-2_5), *Weakly Nonlinear Systems*, Cham: Springer Nature Switzerland, pp. 75â85, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-3-031-40681-2\_5](https://doi.org/10.1007%2F978-3-031-40681-2_5), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-3-031-40680-5](https://en.wikipedia.org/wiki/Special:BookSources/978-3-031-40680-5 "Special:BookSources/978-3-031-40680-5")
, retrieved 2026-01-14
`{{citation}}`: CS1 maint: work parameter with ISBN ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_work_parameter_with_ISBN "Category:CS1 maint: work parameter with ISBN"))
27. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-27)** [Korn & Korn 1967](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFKornKorn1967), pp. 226â227
28. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-28)** [Bracewell 2000](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFBracewell2000), Table 14.1, p. 385
29. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-29)**
Archived at [Ghostarchive](https://ghostarchive.org/varchive/youtube/20211211/zvbdoSeGAgI) and the [Wayback Machine](https://web.archive.org/web/20141220033002/https://www.youtube.com/watch?v=zvbdoSeGAgI&gl=US&hl=en):
Mattuck, Arthur (7 November 2008), ["Where the Laplace Transform comes from"](https://www.youtube.com/watch?v=zvbdoSeGAgI), *[YouTube](https://en.wikipedia.org/wiki/YouTube "YouTube")*
30. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-FOOTNOTEWidder194138_30-0)** [Widder 1941](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFWidder1941), p. 38.
31. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-31)** [Feller 1971](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFFeller1971), p. 432
32. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-32)**
[Laurent Schwartz](https://en.wikipedia.org/wiki/Laurent_Schwartz "Laurent Schwartz") (1966), *Mathematics for the physical sciences*, Addison-Wesley
, p 224.
33. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-33)**
[Titchmarsh, E.](https://en.wikipedia.org/wiki/Edward_Charles_Titchmarsh "Edward Charles Titchmarsh") (1986) \[1948\], *Introduction to the theory of Fourier integrals* (2nd ed.), [Clarendon Press](https://en.wikipedia.org/wiki/Clarendon_Press "Clarendon Press"), p. 6, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-8284-0324-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8284-0324-5 "Special:BookSources/978-0-8284-0324-5")
34. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-34)** [Takacs 1953](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFTakacs1953), p. 93
35. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-35)**
Riley, K. F.; Hobson, M. P.; Bence, S. J. (2010), *Mathematical methods for physics and engineering* (3rd ed.), Cambridge University Press, p. 455, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-521-86153-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-86153-3 "Special:BookSources/978-0-521-86153-3")
36. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-36)**
Distefano, J. J.; Stubberud, A. R.; Williams, I. J. (1995), *Feedback systems and control*, Schaum's outlines (2nd ed.), McGraw-Hill, p. 78, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-07-017052-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-017052-0 "Special:BookSources/978-0-07-017052-0")
37. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-37)**
Lipschutz, S.; Spiegel, M. R.; Liu, J. (2009), *Mathematical Handbook of Formulas and Tables*, Schaum's Outline Series (3rd ed.), McGraw-Hill, p. 183, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-07-154855-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-154855-7 "Special:BookSources/978-0-07-154855-7")
â provides the case for real *q*.
38. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-38)** <http://mathworld.wolfram.com/LaplaceTransform.html> â Wolfram Mathword provides case for complex *q*
39. ^ [***a***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-FOOTNOTEBracewell1978227_39-0) [***b***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-FOOTNOTEBracewell1978227_39-1) [***c***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-FOOTNOTEBracewell1978227_39-2) [***d***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-FOOTNOTEBracewell1978227_39-3) [Bracewell 1978](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFBracewell1978), p. 227.
40. ^ [***a***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-FOOTNOTEWilliams197388_40-0) [***b***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-FOOTNOTEWilliams197388_40-1) [***c***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-FOOTNOTEWilliams197388_40-2) [Williams 1973](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFWilliams1973), p. 88.
41. ^ [***a***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-FOOTNOTEWilliams197389_41-0) [***b***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-FOOTNOTEWilliams197389_41-1) [Williams 1973](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFWilliams1973), p. 89.
42. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-42)** [Korn & Korn 1967](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFKornKorn1967), §8.1
43. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-43)**
RK Pathria; Paul Beal (1996), [*Statistical mechanics*](https://archive.org/details/statisticalmecha00path_911) (2nd ed.), Butterworth-Heinemann, p. [56](https://archive.org/details/statisticalmecha00path_911/page/n66), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[9780750624695](https://en.wikipedia.org/wiki/Special:BookSources/9780750624695 "Special:BookSources/9780750624695")
44. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-44)**
Salem, M.; Seaton, M. J. (1974), "I. Continuum spectra and brightness contours", *[Monthly Notices of the Royal Astronomical Society](https://en.wikipedia.org/wiki/Monthly_Notices_of_the_Royal_Astronomical_Society "Monthly Notices of the Royal Astronomical Society")*, **167**: 493â510, [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1974MNRAS.167..493S](https://ui.adsabs.harvard.edu/abs/1974MNRAS.167..493S), [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1093/mnras/167.3.493](https://doi.org/10.1093%2Fmnras%2F167.3.493)
, and
Salem, M. (1974), "II. Three-dimensional models", *Monthly Notices of the Royal Astronomical Society*, **167**: 511â516, [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1974MNRAS.167..511S](https://ui.adsabs.harvard.edu/abs/1974MNRAS.167..511S), [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1093/mnras/167.3.511](https://doi.org/10.1093%2Fmnras%2F167.3.511)
45. ^ [***a***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-Feller_45-0) [***b***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-Feller_45-1)
Feller, *Introduction to Probability Theory, volume II,pp=479-483*
46. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-46)**
[S. Ikehara](https://en.wikipedia.org/wiki/Shikao_Ikehara "Shikao Ikehara") (1931), "An extension of Landau's theorem in the analytic theory of numbers", *Journal of Mathematics and Physics*, **10** (1â4\): 1â12, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1002/sapm19311011](https://doi.org/10.1002%2Fsapm19311011), [Zbl](https://en.wikipedia.org/wiki/Zbl_\(identifier\) "Zbl (identifier)") [0001\.12902](https://zbmath.org/?format=complete&q=an:0001.12902)
## References
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=33 "Edit section: References")\]
### Modern
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=34 "Edit section: Modern")\]
- Bracewell, Ronald N. (1978), *The Fourier Transform and its Applications* (2nd ed.), McGraw-Hill Kogakusha, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-07-007013-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-007013-4 "Special:BookSources/978-0-07-007013-4")
- Bracewell, R. N. (2000), *The Fourier Transform and Its Applications* (3rd ed.), Boston: McGraw-Hill, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-07-116043-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-116043-8 "Special:BookSources/978-0-07-116043-8")
- [Feller, William](https://en.wikipedia.org/wiki/William_Feller "William Feller") (1971), *An introduction to probability theory and its applications. Vol. II.*, Second edition, New York: [John Wiley & Sons](https://en.wikipedia.org/wiki/John_Wiley_%26_Sons "John Wiley & Sons"), [MR](https://en.wikipedia.org/wiki/MR_\(identifier\) "MR (identifier)") [0270403](https://mathscinet.ams.org/mathscinet-getitem?mr=0270403)
- Korn, G. A.; [Korn, T. M.](https://en.wikipedia.org/wiki/Theresa_M._Korn "Theresa M. Korn") (1967), *Mathematical Handbook for Scientists and Engineers* (2nd ed.), McGraw-Hill Companies, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-07-035370-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-035370-1 "Special:BookSources/978-0-07-035370-1")
- Widder, David Vernon (1941), *The Laplace Transform*, Princeton Mathematical Series, v. 6, [Princeton University Press](https://en.wikipedia.org/wiki/Princeton_University_Press "Princeton University Press"), [MR](https://en.wikipedia.org/wiki/MR_\(identifier\) "MR (identifier)") [0005923](https://mathscinet.ams.org/mathscinet-getitem?mr=0005923)
- Williams, J. (1973), *Laplace Transforms*, Problem Solvers, George Allen & Unwin, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-04-512021-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-04-512021-5 "Special:BookSources/978-0-04-512021-5")
- Takacs, J. (1953), "Fourier amplitudok meghatarozasa operatorszamitassal", *Magyar Hiradastechnika* (in Hungarian), **IV** (7â8\): 93â96
### Historical
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=35 "Edit section: Historical")\]
- [Euler, L.](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler") (1744), "De constructione aequationum" \[The Construction of Equations\], *Opera Omnia*, 1st series (in Latin), **22**: 150â161
- [Euler, L.](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler") (1753), "Methodus aequationes differentiales" \[A Method for Solving Differential Equations\], *Opera Omnia*, 1st series (in Latin), **22**: 181â213
- [Euler, L.](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler") (1992) \[1769\], "Institutiones calculi integralis, Volume 2" \[Institutions of Integral Calculus\], *Opera Omnia*, 1st series (in Latin), **12**, Basel: BirkhÀuser, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-3764314743](https://en.wikipedia.org/wiki/Special:BookSources/978-3764314743 "Special:BookSources/978-3764314743")
`{{citation}}`: CS1 maint: work parameter with ISBN ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_work_parameter_with_ISBN "Category:CS1 maint: work parameter with ISBN")), Chapters 3â5
- [Euler, Leonhard](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler") (1769), [*Institutiones calculi integralis*](https://books.google.com/books?id=BFqWNwpfqo8C) \[*Institutions of Integral Calculus*\] (in Latin), vol. II, Paris: Petropoli, ch. 3â5, pp. 57â153
- [Grattan-Guinness, I](https://en.wikipedia.org/wiki/Ivor_Grattan-Guinness "Ivor Grattan-Guinness") (1997), "Laplace's integral solutions to partial differential equations", in Gillispie, C. C. (ed.), *Pierre Simon Laplace 1749â1827: A Life in Exact Science*, Princeton: Princeton University Press, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-691-01185-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-01185-1 "Special:BookSources/978-0-691-01185-1")
- [Lagrange, J. L.](https://en.wikipedia.org/wiki/Joseph_Louis_Lagrange "Joseph Louis Lagrange") (1773), *MĂ©moire sur l'utilitĂ© de la mĂ©thode*, Ćuvres de Lagrange, vol. 2, pp. 171â234
## Further reading
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=36 "Edit section: Further reading")\]
- Arendt, Wolfgang; Batty, Charles J.K.; Hieber, Matthias; Neubrander, Frank (2002), *Vector-Valued Laplace Transforms and Cauchy Problems*, BirkhÀuser Basel, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-3-7643-6549-3](https://en.wikipedia.org/wiki/Special:BookSources/978-3-7643-6549-3 "Special:BookSources/978-3-7643-6549-3")
- Davies, Brian (2002), *Integral transforms and their applications* (Third ed.), New York: Springer, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-387-95314-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-95314-4 "Special:BookSources/978-0-387-95314-4")
- Deakin, M. A. B. (1981), "The development of the Laplace transform", *Archive for History of Exact Sciences*, **25** (4): 343â390, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/BF01395660](https://doi.org/10.1007%2FBF01395660), [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [117913073](https://api.semanticscholar.org/CorpusID:117913073)
- Deakin, M. A. B. (1982), "The development of the Laplace transform", *Archive for History of Exact Sciences*, **26** (4): 351â381, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/BF00418754](https://doi.org/10.1007%2FBF00418754), [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [123071842](https://api.semanticscholar.org/CorpusID:123071842)
- [Doetsch, Gustav](https://en.wikipedia.org/wiki/Gustav_Doetsch "Gustav Doetsch") (1974), *Introduction to the Theory and Application of the Laplace Transformation*, Springer, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-387-06407-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-06407-9 "Special:BookSources/978-0-387-06407-9")
- Mathews, Jon; Walker, Robert L. (1970), *Mathematical methods of physics* (2nd ed.), New York: W. A. Benjamin, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-8053-7002-1](https://en.wikipedia.org/wiki/Special:BookSources/0-8053-7002-1 "Special:BookSources/0-8053-7002-1")
- Polyanin, A. D.; Manzhirov, A. V. (1998), *Handbook of Integral Equations*, Boca Raton: CRC Press, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-8493-2876-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8493-2876-3 "Special:BookSources/978-0-8493-2876-3")
- [Schwartz, Laurent](https://en.wikipedia.org/wiki/Laurent_Schwartz "Laurent Schwartz") (1952), "Transformation de Laplace des distributions", *Comm. SĂ©m. Math. Univ. Lund \[Medd. Lunds Univ. Mat. Sem.\]* (in French), **1952**: 196â206, [MR](https://en.wikipedia.org/wiki/MR_\(identifier\) "MR (identifier)") [0052555](https://mathscinet.ams.org/mathscinet-getitem?mr=0052555)
- [Schwartz, Laurent](https://en.wikipedia.org/wiki/Laurent_Schwartz "Laurent Schwartz") (2008) \[1966\], [*Mathematics for the Physical Sciences*](https://books.google.com/books?id=-_AuDQAAQBAJ&pg=PA215), Dover Books on Mathematics, New York: Dover Publications, pp. 215â241, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-486-46662-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-46662-0 "Special:BookSources/978-0-486-46662-0")
â see Chapter VI. The Laplace transform
- Siebert, William McC. (1986), *Circuits, Signals, and Systems*, Cambridge, Massachusetts: MIT Press, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-262-19229-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-262-19229-3 "Special:BookSources/978-0-262-19229-3")
- Widder, David Vernon (1945), "What is the Laplace transform?", *[The American Mathematical Monthly](https://en.wikipedia.org/wiki/American_Mathematical_Monthly "American Mathematical Monthly")*, **52** (8): 419â425, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.2307/2305640](https://doi.org/10.2307%2F2305640), [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0002-9890](https://search.worldcat.org/issn/0002-9890), [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [2305640](https://www.jstor.org/stable/2305640), [MR](https://en.wikipedia.org/wiki/MR_\(identifier\) "MR (identifier)") [0013447](https://mathscinet.ams.org/mathscinet-getitem?mr=0013447)
- Weidman, J.A.C.; Fornberg, Bengt (2023), "Fully numerical Laplace transform methods", *Numerical Algorithms*, **92**: 985â1006, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/s11075-022-01368-x](https://doi.org/10.1007%2Fs11075-022-01368-x)
## External links
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=37 "Edit section: External links")\]
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Wikimedia Commons has media related to [Laplace transformation](https://commons.wikimedia.org/wiki/Category:Laplace_transformation "commons:Category:Laplace transformation").
- ["Laplace transform"](https://www.encyclopediaofmath.org/index.php?title=Laplace_transform), *[Encyclopedia of Mathematics](https://en.wikipedia.org/wiki/Encyclopedia_of_Mathematics "Encyclopedia of Mathematics")*, [EMS Press](https://en.wikipedia.org/wiki/European_Mathematical_Society "European Mathematical Society"), 2001 \[1994\]
- [Online Computation](http://wims.unice.fr/wims/wims.cgi?lang=en&+module=tool%2Fanalysis%2Ffourierlaplace) of the transform or inverse transform, wims.unice.fr
- [Tables of Integral Transforms](http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm) at EqWorld: The World of Mathematical Equations.
- [Weisstein, Eric W.](https://en.wikipedia.org/wiki/Eric_W._Weisstein "Eric W. Weisstein"), ["Laplace Transform"](https://mathworld.wolfram.com/LaplaceTransform.html), *[MathWorld](https://en.wikipedia.org/wiki/MathWorld "MathWorld")*
- [Good explanations of the initial and final value theorems](http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/) [Archived](https://web.archive.org/web/20090108132440/http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/) 2009-01-08 at the [Wayback Machine](https://en.wikipedia.org/wiki/Wayback_Machine "Wayback Machine")
- [Laplace Transforms](http://www.mathpages.com/home/kmath508/kmath508.htm) at MathPages
- [Computational Knowledge Engine](http://www.wolframalpha.com/input/?i=laplace+transform+example) allows to easily calculate Laplace Transforms and its inverse Transform.
- [Laplace Calculator](http://www.laplacetransformcalculator.com/easy-laplace-transform-calculator/) to calculate Laplace Transforms online easily.
- [Code to visualize Laplace Transforms](https://johnflux.com/2019/02/12/laplace-transform-visualized/) and many example videos.
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Laplace transform
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| Readable Markdown | In [mathematics](https://en.wikipedia.org/wiki/Mathematics "Mathematics"), the **Laplace transform**, named after [Pierre-Simon Laplace](https://en.wikipedia.org/wiki/Pierre-Simon_Laplace "Pierre-Simon Laplace") (), is an [integral transform](https://en.wikipedia.org/wiki/Integral_transform "Integral transform") that converts a [function](https://en.wikipedia.org/wiki/Function_\(mathematics\) "Function (mathematics)") of a [real](https://en.wikipedia.org/wiki/Real_number "Real number") [variable](https://en.wikipedia.org/wiki/Variable_\(mathematics\) "Variable (mathematics)") (usually â â , in the *[time domain](https://en.wikipedia.org/wiki/Time_domain "Time domain")*) to a function of a [complex variable](https://en.wikipedia.org/wiki/Complex_number "Complex number")  (in the complex-valued [frequency domain](https://en.wikipedia.org/wiki/Frequency_domain "Frequency domain"), also known as ***s*\-domain** or ***s*\-plane**). The functions are often denoted using a lowercase symbol for the time-domain function and the corresponding uppercase symbol for the frequency-domain function, e.g.  and â â .
The transform is useful for converting [differentiation](https://en.wikipedia.org/wiki/Derivative "Derivative") and [integration](https://en.wikipedia.org/wiki/Integral "Integral") in the time domain into the algebraic operations [multiplication](https://en.wikipedia.org/wiki/Multiplication "Multiplication") and [division](https://en.wikipedia.org/wiki/Division_\(mathematics\) "Division (mathematics)") in the Laplace domain (analogous to how [logarithms](https://en.wikipedia.org/wiki/Logarithm "Logarithm") are useful for simplifying multiplication and division into addition and subtraction). This gives the transform many applications in [science](https://en.wikipedia.org/wiki/Science "Science") and [engineering](https://en.wikipedia.org/wiki/Engineering "Engineering"), mostly as a tool for solving linear [differential equations](https://en.wikipedia.org/wiki/Differential_equation "Differential equation")[\[1\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-Lynn_1986_pp._225%E2%80%93272-1) and [dynamical systems](https://en.wikipedia.org/wiki/Dynamical_system "Dynamical system") by replacing [ordinary differential equations](https://en.wikipedia.org/wiki/Ordinary_differential_equation "Ordinary differential equation") and [integral equations](https://en.wikipedia.org/wiki/Integral_equation "Integral equation") with [algebraic polynomial equations](https://en.wikipedia.org/wiki/Algebraic_equation "Algebraic equation"), and by replacing [convolution](https://en.wikipedia.org/wiki/Convolution "Convolution") with [multiplication](https://en.wikipedia.org/wiki/Multiplication "Multiplication").[\[2\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-2)[\[3\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-:1-3)
For example, through the Laplace transform, the equation of the [simple harmonic oscillator](https://en.wikipedia.org/wiki/Simple_harmonic_oscillator "Simple harmonic oscillator") ([Hooke's law](https://en.wikipedia.org/wiki/Hooke%27s_law "Hooke's law"))  is converted into the algebraic equation  which incorporates the [initial conditions](https://en.wikipedia.org/wiki/Initial_conditions "Initial conditions")  and â â , and can be solved for the unknown function â â . Once solved, the inverse Laplace transform can be used to transform it to the original domain. This is often aided by referencing tables such as that given [below](https://en.wikipedia.org/wiki/Laplace_transform#Table_of_selected_Laplace_transforms).
The Laplace transform is defined (for suitable functions â â ) by the [integral](https://en.wikipedia.org/wiki/Integral "Integral")  where â â is a [complex number](https://en.wikipedia.org/wiki/Complex_number "Complex number").
The Laplace transform is related to many other transforms. It is essentially the same as the [Mellin transform](https://en.wikipedia.org/wiki/Mellin_transform "Mellin transform") and is closely related to the [Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform "Fourier transform"). Unlike for the Fourier transform, the Laplace transform of a function is often an [analytic function](https://en.wikipedia.org/wiki/Analytic_function "Analytic function"), meaning that it can be expressed as a [power series](https://en.wikipedia.org/wiki/Power_series "Power series") that converges locally, the coefficients of which represent the [moments](https://en.wikipedia.org/wiki/Moment_\(mathematics\) "Moment (mathematics)") of the original function. Moreover, the techniques of [complex analysis](https://en.wikipedia.org/wiki/Complex_analysis "Complex analysis"), especially [contour integrals](https://en.wikipedia.org/wiki/Contour_integral "Contour integral"), can be used for simplifying calculations.
[](https://en.wikipedia.org/wiki/File:Laplace,_Pierre-Simon,_marquis_de.jpg)
Pierre-Simon, marquis de Laplace
The Laplace transform is named after [mathematician](https://en.wikipedia.org/wiki/Mathematician "Mathematician") and [astronomer](https://en.wikipedia.org/wiki/Astronomer "Astronomer") [Pierre-Simon, Marquis de Laplace](https://en.wikipedia.org/wiki/Pierre-Simon_Laplace "Pierre-Simon Laplace"), who used a similar transform in his work on [probability theory](https://en.wikipedia.org/wiki/Probability_theory "Probability theory").[\[4\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-4)[\[5\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-5) Laplace wrote extensively about the use of [generating functions](https://en.wikipedia.org/wiki/Generating_function "Generating function") (1814), and the integral form of the Laplace transform evolved naturally as a result.[\[6\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-6)
Laplace's use of generating functions was similar to what is now known as the [z-transform](https://en.wikipedia.org/wiki/Z-transform "Z-transform"), and he gave little attention to the [continuous variable](https://en.wikipedia.org/wiki/Continuous_variable "Continuous variable") case which was discussed by [Niels Henrik Abel](https://en.wikipedia.org/wiki/Niels_Henrik_Abel "Niels Henrik Abel").[\[7\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-7)
From 1744, [Leonhard Euler](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler") investigated integrals of the form  as solutions of differential equations, introducing in particular the [gamma function](https://en.wikipedia.org/wiki/Gamma_function "Gamma function").[\[8\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-8) [Joseph-Louis Lagrange](https://en.wikipedia.org/wiki/Joseph-Louis_Lagrange "Joseph-Louis Lagrange") was an admirer of Euler and, in his work on integrating [probability density functions](https://en.wikipedia.org/wiki/Probability_density_function "Probability density function"), investigated expressions of the form  which resembles a Laplace transform.[\[9\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-9)[\[10\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-10)
These types of integrals seem first to have attracted Laplace's attention in 1782, where he was following in the spirit of Euler in using the integrals themselves as solutions of equations.[\[11\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-11) However, in 1785, Laplace took the critical step forward when, rather than simply looking for a solution in the form of an integral, he started to apply the transforms in the sense that was later to become popular. He used an integral of the form  akin to a [Mellin transform](https://en.wikipedia.org/wiki/Mellin_transform "Mellin transform"), to transform the whole of a [difference equation](https://en.wikipedia.org/wiki/Difference_equation "Difference equation"), in order to look for solutions of the transformed equation. He then went on to apply the Laplace transform in the same way and started to derive some of its properties, beginning to appreciate its potential power.[\[12\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-12)
Laplace also recognised that [Joseph Fourier](https://en.wikipedia.org/wiki/Joseph_Fourier "Joseph Fourier")'s method of [Fourier series](https://en.wikipedia.org/wiki/Fourier_series "Fourier series") for solving the [diffusion equation](https://en.wikipedia.org/wiki/Diffusion_equation "Diffusion equation") could only apply to a limited region of space, because those solutions were [periodic](https://en.wikipedia.org/wiki/Periodic_function "Periodic function"). In 1809, Laplace applied his transform to find solutions that diffused indefinitely in space.[\[13\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-13) In 1821, [Cauchy](https://en.wikipedia.org/wiki/Cauchy "Cauchy") developed an [operational calculus](https://en.wikipedia.org/wiki/Operational_calculus "Operational calculus") for the Laplace transform that could be used to study [linear differential equations](https://en.wikipedia.org/wiki/Linear_differential_equation "Linear differential equation") in much the same way the transform is now used in basic engineering. This method was popularized, and perhaps rediscovered, by [Oliver Heaviside](https://en.wikipedia.org/wiki/Oliver_Heaviside "Oliver Heaviside") around the turn of the century.[\[14\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-14)
[Bernhard Riemann](https://en.wikipedia.org/wiki/Bernhard_Riemann "Bernhard Riemann") used the Laplace transform in his 1859 paper *[On the number of primes less than a given magnitude](https://en.wikipedia.org/wiki/On_the_number_of_primes_less_than_a_given_magnitude "On the number of primes less than a given magnitude")*, in which he also developed the inversion theorem. Riemann used the Laplace transform to develop the functional equation of the [Riemann zeta function](https://en.wikipedia.org/wiki/Riemann_zeta_function "Riemann zeta function"), and his method is still used to relate the [modular transformation law](https://en.wikipedia.org/wiki/Modular_form "Modular form") of the [Jacobi theta function](https://en.wikipedia.org/wiki/Jacobi_theta_function "Jacobi theta function"), which is readily proved via [Poisson summation](https://en.wikipedia.org/wiki/Poisson_summation "Poisson summation"), to the functional equation.[\[15\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-15)
[Hjalmar Mellin](https://en.wikipedia.org/wiki/Hjalmar_Mellin "Hjalmar Mellin") was among the first to study the Laplace transform, rigorously in the [Karl Weierstrass](https://en.wikipedia.org/wiki/Karl_Weierstrass "Karl Weierstrass") school of analysis, and apply it to the study of [differential equations](https://en.wikipedia.org/wiki/Differential_equations "Differential equations") and [special functions](https://en.wikipedia.org/wiki/Special_functions "Special functions"), at the turn of the 20th century.[\[16\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-16) At around the same time, Heaviside was busy with his operational calculus. [Thomas Joannes Stieltjes](https://en.wikipedia.org/wiki/Thomas_Joannes_Stieltjes "Thomas Joannes Stieltjes") considered a generalization of the Laplace transform connected to his [work on moments](https://en.wikipedia.org/wiki/Stieltjes_moment_problem "Stieltjes moment problem"). Other contributors in this time period included [Mathias Lerch](https://en.wikipedia.org/wiki/Mathias_Lerch "Mathias Lerch"),[\[17\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-17) [Oliver Heaviside](https://en.wikipedia.org/wiki/Oliver_Heaviside "Oliver Heaviside"), and [Thomas Bromwich](https://en.wikipedia.org/wiki/Thomas_John_I%27Anson_Bromwich "Thomas John I'Anson Bromwich").[\[18\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-18)
In 1929, [Vannevar Bush](https://en.wikipedia.org/wiki/Vannevar_Bush "Vannevar Bush") and [Norbert Wiener](https://en.wikipedia.org/wiki/Norbert_Wiener "Norbert Wiener") published *Operational Circuit Analysis* as a text for engineering analysis of electrical circuits, applying both Fourier transforms and operational calculus, and in which they included one of the first predecessors of the modern table of Laplace transforms. In 1934, [Raymond Paley](https://en.wikipedia.org/wiki/Raymond_Paley "Raymond Paley") and [Norbert Wiener](https://en.wikipedia.org/wiki/Norbert_Wiener "Norbert Wiener") published the important work *Fourier transforms in the complex domain*, about what is now called the Laplace transform (see below). Also during the 30s, the Laplace transform was instrumental in [Godfrey Harold Hardy](https://en.wikipedia.org/wiki/G_H_Hardy "G H Hardy") and [John Edensor Littlewood](https://en.wikipedia.org/wiki/John_Edensor_Littlewood "John Edensor Littlewood")'s study of [tauberian theorems](https://en.wikipedia.org/wiki/Tauberian_theorem "Tauberian theorem"), and this application was later expounded on by [Widder (1941)](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFWidder1941), who developed other aspects of the theory such as a new method for inversion. [Edward Charles Titchmarsh](https://en.wikipedia.org/wiki/Edward_Charles_Titchmarsh "Edward Charles Titchmarsh") wrote the influential *Introduction to the theory of the Fourier integral* (1937).
The current widespread use of the transform (mainly in engineering) came about during and soon after [World War II](https://en.wikipedia.org/wiki/World_War_II "World War II"),[\[19\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-19) replacing the earlier Heaviside [operational calculus](https://en.wikipedia.org/wiki/Operational_calculus "Operational calculus"). The advantages of the Laplace transform had been emphasized by [Gustav Doetsch](https://en.wikipedia.org/wiki/Gustav_Doetsch "Gustav Doetsch").[\[20\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-20)
[](https://en.wikipedia.org/wiki/File:Complex_frequency_s-domain_negative.jpg)
 for various complex frequencies in the *s*\-domain â â , which can be expressed as â â . The axis at  contains pure cosines. Positive  contains [damped cosines](https://en.wikipedia.org/wiki/Damped_sinusoid "Damped sinusoid"). Negative  contains [exponentially growing](https://en.wikipedia.org/wiki/Exponential_growth "Exponential growth") cosines.
The Laplace transform of a [function](https://en.wikipedia.org/wiki/Function_\(mathematics\) "Function (mathematics)") *f*(*t*), defined for all [real numbers](https://en.wikipedia.org/wiki/Real_number "Real number") *t* â„ 0, is the function *F*(*s*), which is a unilateral transform defined by\[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\]
 (Eq. 1)
where *s* is a [complex](https://en.wikipedia.org/wiki/Complex_number "Complex number") frequency-domain parameter  with real numbers Ï and Ï.
An alternate notation for the Laplace transform is  instead of *F*.[\[3\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-:1-3) Thus  in [functional notation](https://en.wikipedia.org/wiki/Functional_notation "Functional notation"). This is often written, especially in engineering settings, as â â , with the understanding that the [dummy variable](https://en.wikipedia.org/wiki/Bound_variable "Bound variable")  does not appear in the function â â .
The meaning of the integral depends on types of functions of interest. A necessary condition for existence of the integral is that f must be [locally integrable](https://en.wikipedia.org/wiki/Locally_integrable "Locally integrable") on \[0, â). For locally integrable functions that decay at infinity or are of [exponential type](https://en.wikipedia.org/wiki/Exponential_type "Exponential type") (â â ), the integral can be understood to be a (proper) [Lebesgue integral](https://en.wikipedia.org/wiki/Lebesgue_integral "Lebesgue integral"). However, for many applications it is necessary to regard it as a [conditionally convergent](https://en.wikipedia.org/wiki/Conditionally_convergent "Conditionally convergent") [improper integral](https://en.wikipedia.org/wiki/Improper_integral "Improper integral") at â. Still more generally, the integral can be understood in a [weak sense](https://en.wikipedia.org/wiki/Distribution_\(mathematics\) "Distribution (mathematics)"), and this is dealt with below.
One can define the Laplace transform of a finite [Borel measure](https://en.wikipedia.org/wiki/Borel_measure "Borel measure") Ό by the Lebesgue integral[\[21\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-21) 
An important special case is where ÎŒ is a [probability measure](https://en.wikipedia.org/wiki/Probability_measure "Probability measure"), for example, the [Dirac delta function](https://en.wikipedia.org/wiki/Dirac_delta_function "Dirac delta function"). In [operational calculus](https://en.wikipedia.org/wiki/Operational_calculus "Operational calculus"), the Laplace transform of a measure is often treated as though the measure came from a probability density function f. In that case, to avoid potential confusion, one often writes  where the lower limit of 0â is shorthand notation for 
This limit emphasizes that any [point mass](https://en.wikipedia.org/wiki/Point_particle "Point particle") located at 0 is entirely captured by the Laplace transform. Although with the Lebesgue integral, it is not necessary to take such a limit, it does appear more naturally in connection with the [LaplaceâStieltjes transform](https://en.wikipedia.org/wiki/Laplace%E2%80%93Stieltjes_transform "LaplaceâStieltjes transform").
### Bilateral Laplace transform
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=3 "Edit section: Bilateral Laplace transform")\]
When one says "the Laplace transform" without qualification, the unilateral or one-sided transform is usually intended. The Laplace transform can be alternatively defined as the *bilateral Laplace transform*, or [two-sided Laplace transform](https://en.wikipedia.org/wiki/Two-sided_Laplace_transform "Two-sided Laplace transform"), by extending the limits of integration to be the entire real axis. If that is done, the common unilateral transform becomes a special case of the bilateral transform, where the definition of the function being transformed includes being multiplied by the [Heaviside step function](https://en.wikipedia.org/wiki/Heaviside_step_function "Heaviside step function").
The bilateral Laplace transform *F*(*s*) is defined as follows:
 (Eq. 2)
An alternate notation for the bilateral Laplace transform is â â , instead of F.
### Inverse Laplace transform
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=4 "Edit section: Inverse Laplace transform")\]
Two integrable functions have the same Laplace transform only if they differ on a set of [Lebesgue measure](https://en.wikipedia.org/wiki/Lebesgue_measure "Lebesgue measure") zero. This means that, on the range of the transform, there is an inverse transform. In fact, besides integrable functions, the Laplace transform is a [one-to-one mapping](https://en.wikipedia.org/wiki/One-to-one_function "One-to-one function") from one function space into another in many other function spaces as well, although there is usually no easy characterization of the range.
Typical function spaces in which this is true include the spaces of bounded continuous functions, the space [*L*â(0, â)](https://en.wikipedia.org/wiki/Lp_space "Lp space"), or more generally [tempered distributions](https://en.wikipedia.org/wiki/Tempered_distributions "Tempered distributions") on (0, â). The Laplace transform is also defined and injective for suitable spaces of tempered distributions.
In these cases, the image of the Laplace transform lives in a space of [analytic functions](https://en.wikipedia.org/wiki/Analytic_function "Analytic function") in the [region of convergence](https://en.wikipedia.org/wiki/Laplace_transform#Region_of_convergence). The [inverse Laplace transform](https://en.wikipedia.org/wiki/Inverse_Laplace_transform "Inverse Laplace transform") is given by the following complex integral, which is known by various names (the **Bromwich integral**, the **FourierâMellin integral**, and **Mellin's inverse formula**):
 (Eq. 3)
where Îł is a real number so that the contour path of integration is in the region of convergence of *F*(*s*). In most applications, the contour can be closed, allowing the use of the [residue theorem](https://en.wikipedia.org/wiki/Residue_theorem "Residue theorem"). An alternative formula for the inverse Laplace transform is given by [Post's inversion formula](https://en.wikipedia.org/wiki/Post%27s_inversion_formula "Post's inversion formula"). The limit here is interpreted in the [weak-\* topology](https://en.wikipedia.org/wiki/Weak_topology#Weak-*_topology "Weak topology").
In practice, it is typically more convenient to decompose a Laplace transform into known transforms of functions obtained from a table and construct the inverse by inspection.
In [pure](https://en.wikipedia.org/wiki/Probability_theory "Probability theory") and [applied probability](https://en.wikipedia.org/wiki/Applied_probability "Applied probability"), the Laplace transform is defined as an [expected value](https://en.wikipedia.org/wiki/Expected_value "Expected value"). If X is a [random variable](https://en.wikipedia.org/wiki/Random_variable "Random variable") with probability density function f, then the Laplace transform of f is given by the expectation ![{\\displaystyle {\\mathcal {L}}\\{f\\}(s)=\\operatorname {E} \\left\[e^{-sX}\\right\],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3da28029aa437333b370a72f00e2745d1f8e8c73) where ![{\\displaystyle \\operatorname {E} \[r\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/715165949e716ef97cb5b61ba0982325cef86a1b) is the [expectation](https://en.wikipedia.org/wiki/Expected_value "Expected value") of [random variable](https://en.wikipedia.org/wiki/Random_variable "Random variable") â â .
By [convention](https://en.wikipedia.org/wiki/Abuse_of_notation "Abuse of notation"), this is referred to as the Laplace transform of the random variable X itself. Here, replacing s by â*t* gives the [moment generating function](https://en.wikipedia.org/wiki/Moment_generating_function "Moment generating function") of X. The Laplace transform has applications throughout probability theory, including [first passage times](https://en.wikipedia.org/wiki/First_passage_time "First passage time") of [stochastic processes](https://en.wikipedia.org/wiki/Stochastic_process "Stochastic process") such as [Markov chains](https://en.wikipedia.org/wiki/Markov_chain "Markov chain"), and [renewal theory](https://en.wikipedia.org/wiki/Renewal_theory "Renewal theory").
Of particular use is the ability to recover the [cumulative distribution function](https://en.wikipedia.org/wiki/Cumulative_distribution_function "Cumulative distribution function") of a continuous random variable X by means of the Laplace transform as follows:[\[22\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-22) ![{\\displaystyle F\_{X}(x)={\\mathcal {L}}^{-1}\\left\\{{\\frac {1}{s}}\\operatorname {E} \\left\[e^{-sX}\\right\]\\right\\}(x)={\\mathcal {L}}^{-1}\\left\\{{\\frac {1}{s}}{\\mathcal {L}}\\{f\\}(s)\\right\\}(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9728fa6f7b7c0cf6c7fd18917b64a4bbea2c7f)
### Algebraic construction
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=6 "Edit section: Algebraic construction")\]
The Laplace transform can be alternatively defined in a purely algebraic manner by applying a [field of fractions](https://en.wikipedia.org/wiki/Field_of_fractions "Field of fractions") construction to the convolution [ring](https://en.wikipedia.org/wiki/Ring_\(abstract_algebra\) "Ring (abstract algebra)") of functions on the positive half-line. The resulting [space of abstract operators](https://en.wikipedia.org/wiki/Convolution_quotient "Convolution quotient") is exactly equivalent to Laplace space, but in this construction the forward and reverse transforms never need to be explicitly defined (avoiding the related difficulties with proving convergence).[\[23\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-23)
## Region of convergence
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=7 "Edit section: Region of convergence")\]
If *f* is a locally integrable function (or more generally a Borel measure locally of bounded variation), then the Laplace transform *F*(*s*) of *f* converges provided that the limit  exists.
The Laplace transform [converges absolutely](https://en.wikipedia.org/wiki/Absolute_convergence "Absolute convergence") if the integral  exists as a proper Lebesgue integral. The Laplace transform is usually understood as [conditionally convergent](https://en.wikipedia.org/wiki/Conditional_convergence "Conditional convergence"), meaning that it converges in the former but not in the latter sense.
The set of values for which *F*(*s*) converges absolutely is either of the form Re(*s*) \> *a* or Re(*s*) â„ *a*, where *a* is an [extended real constant](https://en.wikipedia.org/wiki/Extended_real_number "Extended real number") with ââ †*a* †â (a consequence of the [dominated convergence theorem](https://en.wikipedia.org/wiki/Dominated_convergence_theorem "Dominated convergence theorem")). The constant *a* is known as the abscissa of absolute convergence, and depends on the growth behavior of *f*(*t*).[\[24\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-24) Analogously, the two-sided transform converges absolutely in a strip of the form *a* \< Re(*s*) \< *b*, and possibly including the lines Re(*s*) = *a* or Re(*s*) = *b*.[\[25\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-25) The subset of values of *s* for which the Laplace transform converges absolutely is called the region of absolute convergence, or the domain of absolute convergence. In the two-sided case, it is sometimes called the strip of absolute convergence. The Laplace transform is analytic in the region of absolute convergence: this is a consequence of [Fubini's theorem](https://en.wikipedia.org/wiki/Fubini%27s_theorem "Fubini's theorem") and [Morera's theorem](https://en.wikipedia.org/wiki/Morera%27s_theorem "Morera's theorem").
Similarly, the set of values for which *F*(*s*) converges (conditionally or absolutely) is known as the region of conditional convergence, or simply the **region of convergence** (ROC). If the Laplace transform converges (conditionally) at *s* = *s*0, then it automatically converges for all *s* with Re(*s*) \> Re(*s*0). Therefore, the region of convergence is a half-plane of the form Re(*s*) \> *a*, possibly including some points of the boundary line Re(*s*) = *a*.
In the region of convergence Re(*s*) \> Re(*s*0), the Laplace transform of *f* can be expressed by [integrating by parts](https://en.wikipedia.org/wiki/Integration_by_parts "Integration by parts") as the integral 
That is, *F*(*s*) can effectively be expressed, in the region of convergence, as the absolutely convergent Laplace transform of some other function. In particular, it is analytic. In its most general form, the Laplace transform gives a one-to-one correspondence between the holomorphic functions which, for some â â , are defined on  and are bounded there in absolute value by a polynomial, and the [distributions](https://en.wikipedia.org/wiki/Distribution_\(mathematical_analysis\) "Distribution (mathematical analysis)") on the real line supported on â â which become [tempered distributions](https://en.wikipedia.org/wiki/Tempered_distribution "Tempered distribution") after multiplied by  for some â â .[\[26\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-26)
There are several [PaleyâWiener theorems](https://en.wikipedia.org/wiki/Paley%E2%80%93Wiener_theorem "PaleyâWiener theorem") concerning the relationship between the decay properties of *f*, and the properties of the Laplace transform within the region of convergence.
In engineering applications, a function corresponding to a [linear time-invariant (LTI) system](https://en.wikipedia.org/wiki/Linear_time-invariant_system "Linear time-invariant system") is *stable* if every bounded input produces a bounded output. This is equivalent to the absolute convergence of the Laplace transform of the impulse response function in the region Re(*s*) â„ 0. As a result, LTI systems are stable, provided that the poles of the Laplace transform of the impulse response function have negative real part.
This ROC is used in knowing about the causality and stability of a system.
## Properties and theorems
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=8 "Edit section: Properties and theorems")\]
The Laplace transform's key property is that it converts [differentiation](https://en.wikipedia.org/wiki/Derivative "Derivative") and [integration](https://en.wikipedia.org/wiki/Integral "Integral") in the time domain into multiplication and division by *s* in the Laplace domain. Thus, the Laplace variable *s* is also known as an *operator variable* in the Laplace domain: either the *derivative operator* or (for *s*â1) the *integration operator*.
Given the functions *f*(*t*) and *g*(*t*), and their respective Laplace transforms *F*(*s*) and *G*(*s*), 
the following table is a list of properties of unilateral Laplace transform:[\[27\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-27)
| Property | Time domain | *s* domain | Comment |
|---|---|---|---|
| [Linearity](https://en.wikipedia.org/wiki/Linearity "Linearity") |  | | |
[Initial value theorem](https://en.wikipedia.org/wiki/Initial_value_theorem "Initial value theorem")
[Final value theorem](https://en.wikipedia.org/wiki/Final_value_theorem "Final value theorem")
â â , if all [poles](https://en.wikipedia.org/wiki/Pole_\(complex_analysis\) "Pole (complex analysis)") of  are in the left half-plane.
The final value theorem is useful because it gives the long-term behaviour without having to perform [partial fraction](https://en.wikipedia.org/wiki/Partial_fraction "Partial fraction") decompositions (or other difficult algebra). If *F*(*s*) has a pole in the right-hand plane or poles on the imaginary axis (e.g., if  or â â ), then the behaviour of this formula is undefined.
### Relation to power series
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=9 "Edit section: Relation to power series")\]
The Laplace transform can be viewed as a [continuous](https://en.wikipedia.org/wiki/Continuous_function "Continuous function") analogue of a [power series](https://en.wikipedia.org/wiki/Power_series "Power series").[\[29\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-29) If *a*(*n*) is a discrete function of a positive integer *n*, then the power series associated to *a*(*n*) is the series  where *x* is a real variable (see *[Z-transform](https://en.wikipedia.org/wiki/Z-transform "Z-transform")*). Replacing summation over *n* with integration over *t*, a continuous version of the power series becomes  where the discrete function *a*(*n*) is replaced by the continuous one *f*(*t*).
Changing the base of the power from *x* to *e* gives 
For this to converge for, say, all bounded functions *f*, it is necessary to require that ln *x* \< 0. Making the substitution â*s* = ln *x* gives just the Laplace transform: 
In other words, the Laplace transform is a continuous analog of a power series, in which the discrete parameter *n* is replaced by the continuous parameter *t*, and *x* is replaced by *e*â*s*.
Analogously to a power series, if â â , then the power series converges to an analytic function in â â , if â â , the Laplace transform converges to an analytic function for â â .[\[30\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-FOOTNOTEWidder194138-30)
### Relation to moments
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=10 "Edit section: Relation to moments")\]
The quantities  are the *moments* of the function *f*. If the first *n* moments of *f* converge absolutely, then by repeated [differentiation under the integral](https://en.wikipedia.org/wiki/Differentiation_under_the_integral "Differentiation under the integral"),  This is of special significance in probability theory, where the moments of a random variable *X* are given by the expectation values â ![{\\displaystyle \\mu \_{n}=\\operatorname {E} \[X^{n}\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a470b23edb981c95e00f5690370662b6b102341)â . Then, the relation holds .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aebff76a8e2a99d41a4d654999b7309d1a2f5916)
### Transform of a function's derivative
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=11 "Edit section: Transform of a function's derivative")\]
It is often convenient to use the differentiation property of the Laplace transform to find the transform of a function's derivative. This can be derived from the basic expression for a Laplace transform as follows: ![{\\displaystyle {\\begin{aligned}{\\mathcal {L}}\\left\\{f(t)\\right\\}&=\\int \_{0^{-}}^{\\infty }e^{-st}f(t)\\,dt\\\\\[6pt\]&=\\left\[{\\frac {f(t)e^{-st}}{-s}}\\right\]\_{0^{-}}^{\\infty }-\\int \_{0^{-}}^{\\infty }{\\frac {e^{-st}}{-s}}f'(t)\\,dt\\quad {\\text{(by parts)}}\\\\\[6pt\]&=\\left\[-{\\frac {f(0^{-})}{-s}}\\right\]+{\\frac {1}{s}}{\\mathcal {L}}\\left\\{f'(t)\\right\\},\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/574460c0fe29a550c4b4a66b9aa82601ea9f956b) yielding  and in the bilateral case, 
The general result  where  denotes the *n*th derivative of *f*, can then be established with an inductive argument.
### Evaluating integrals over the positive real axis
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=12 "Edit section: Evaluating integrals over the positive real axis")\]
A useful property of the Laplace transform is the following:  under suitable assumptions on the behaviour of â â and â â in a right neighbourhood of  and on the decay rate of â â and â â in a left neighbourhood of â â . The above formula is a variation of integration by parts, with the operators  and  being replaced by  and â â . Let us prove the equivalent formulation: 
By plugging in  the left-hand side turns into:  but assuming Fubini's theorem holds, by reversing the order of integration we get the wanted right-hand side.
This method can be used to compute integrals that would otherwise be difficult to compute using elementary methods of real calculus. For example, 
## Relationship to other transforms
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=13 "Edit section: Relationship to other transforms")\]
### LaplaceâStieltjes transform
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=14 "Edit section: LaplaceâStieltjes transform")\]
The (unilateral) LaplaceâStieltjes transform of a function *g* : â â â is defined by the [LebesgueâStieltjes integral](https://en.wikipedia.org/wiki/Lebesgue%E2%80%93Stieltjes_integral "LebesgueâStieltjes integral") 
The function *g* is assumed to be of [bounded variation](https://en.wikipedia.org/wiki/Bounded_variation "Bounded variation"). If *g* is the [antiderivative](https://en.wikipedia.org/wiki/Antiderivative "Antiderivative") of *f*: 
then the LaplaceâStieltjes transform of g and the Laplace transform of f coincide. In general, the LaplaceâStieltjes transform is the Laplace transform of the [Stieltjes measure](https://en.wikipedia.org/wiki/Stieltjes_measure "Stieltjes measure") associated to g. So in practice, the only distinction between the two transforms is that the Laplace transform is thought of as operating on the density function of the measure, whereas the LaplaceâStieltjes transform is thought of as operating on its [cumulative distribution function](https://en.wikipedia.org/wiki/Cumulative_distribution_function "Cumulative distribution function").[\[31\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-31)
Let  be a complex-valued Lebesgue integrable function supported on â â , and let  be its Laplace transform. Then, within the region of convergence, we have  which is the Fourier transform of the function â â .[\[32\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-32)
Indeed, the [Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform "Fourier transform") is a special case (under certain conditions) of the bilateral Laplace transform. The main difference is that the Fourier transform of a function is a complex function of a *real* variable (frequency â â ), the Laplace transform of a function is a complex function of a *complex* variable (damping factor  and frequency â â ). The Laplace transform is usually restricted to transformation of functions of *t* with *t* â„ 0. A consequence of this restriction is that the Laplace transform of a function is a [holomorphic function](https://en.wikipedia.org/wiki/Holomorphic_function "Holomorphic function") of the variable *s*. Unlike the Fourier transform, the Laplace transform of a [distribution](https://en.wikipedia.org/wiki/Distribution_\(mathematics\) "Distribution (mathematics)") is generally a [well-behaved](https://en.wikipedia.org/wiki/Well-behaved "Well-behaved") function. Techniques of complex variables can also be used to directly study Laplace transforms. As a holomorphic function, the Laplace transform has a [power series](https://en.wikipedia.org/wiki/Power_series "Power series") representation. This power series expresses a function as a linear superposition of [moments](https://en.wikipedia.org/wiki/Moment_\(mathematics\) "Moment (mathematics)") of the function. This perspective has applications in probability theory.
Formally, the Fourier transform is equivalent to evaluating the bilateral Laplace transform with imaginary argument *s* = *iÏ*[\[33\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-33)[\[34\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-34) when the condition explained below is fulfilled, ![{\\displaystyle {\\begin{aligned}{\\hat {f}}(\\omega )&={\\mathcal {F}}\\{f(t)\\}\\\\\[4pt\]&={\\mathcal {L}}\\{f(t)\\}\|\_{s=i\\omega }=F(s)\|\_{s=i\\omega }\\\\\[4pt\]&=\\int \_{-\\infty }^{\\infty }e^{-i\\omega t}f(t)\\,dt~.\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44fd8f6a7374f8e3ae9b00c7ca0107108784f9cd)
This convention of the Fourier transform (â â in *[Fourier transform § Other conventions](https://en.wikipedia.org/wiki/Fourier_transform#Other_conventions "Fourier transform")*) requires a factor of â 1/2*Ï*â on the inverse Fourier transform. This relationship between the Laplace and Fourier transforms is often used to determine the [frequency spectrum](https://en.wikipedia.org/wiki/Frequency_spectrum "Frequency spectrum") of a [signal](https://en.wikipedia.org/wiki/Signal_\(information_theory\) "Signal (information theory)") or dynamical system.
The above relation is valid as stated [if and only if](https://en.wikipedia.org/wiki/If_and_only_if "If and only if") the region of convergence (ROC) of *F*(*s*) contains the imaginary axis, *Ï* = 0.
For example, the function *f*(*t*) = cos(*Ï*0*t*) has a Laplace transform *F*(*s*) = *s*/(*s*2 + *Ï*02) whose ROC is Re(*s*) \> 0. As *s* = *iÏ*0 is a pole of *F*(*s*), substituting *s* = *iÏ* in *F*(*s*) does not yield the Fourier transform of *f*(*t*)*u*(*t*), which contains terms proportional to the [Dirac delta functions](https://en.wikipedia.org/wiki/Dirac_delta_functions "Dirac delta functions") *ÎŽ*(*Ï* ± *Ï*0).
However, a relation of the form  holds under much weaker conditions. For instance, this holds for the above example provided that the limit is understood as a [weak limit](https://en.wikipedia.org/wiki/Weak_limit "Weak limit") of measures (see [vague topology](https://en.wikipedia.org/wiki/Vague_topology "Vague topology")). General conditions relating the limit of the Laplace transform of a function on the boundary to the Fourier transform take the form of [PaleyâWiener theorems](https://en.wikipedia.org/wiki/Paley%E2%80%93Wiener_theorem "PaleyâWiener theorem").
The Mellin transform and its inverse are related to the two-sided Laplace transform by a change of variables.
If in the Mellin transform  we set *Ξ* = *e*â*t* we get a two-sided Laplace transform.
The unilateral or one-sided Z-transform is the Laplace transform of an ideally sampled signal with the substitution of  where *T* = 1/*fs* is the [sampling interval](https://en.wikipedia.org/wiki/Sampling_interval "Sampling interval") (in units of time e.g., seconds) and *fs* is the [sampling rate](https://en.wikipedia.org/wiki/Sampling_rate "Sampling rate") (in [samples per second](https://en.wikipedia.org/wiki/Samples_per_second "Samples per second") or [hertz](https://en.wikipedia.org/wiki/Hertz "Hertz")).
Let  be a sampling impulse train (also called a [Dirac comb](https://en.wikipedia.org/wiki/Dirac_comb "Dirac comb")) and ![{\\displaystyle {\\begin{aligned}x\_{q}(t)&{\\stackrel {\\mathrm {def} }{{}={}}}x(t)\\Delta \_{T}(t)=x(t)\\sum \_{n=0}^{\\infty }\\delta (t-nT)\\\\&=\\sum \_{n=0}^{\\infty }x(nT)\\delta (t-nT)=\\sum \_{n=0}^{\\infty }x\[n\]\\delta (t-nT)\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c334ec8f4011a84a4a305f1ac7894fb6e1038cf0) be the sampled representation of the continuous-time *x*(*t*) ![{\\displaystyle x\[n\]{\\stackrel {\\mathrm {def} }{{}={}}}x(nT)~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2b6eaf257e418b217759fe3b9b982993004af01)
The Laplace transform of the sampled signal *x**q*(*t*) is ![{\\displaystyle {\\begin{aligned}X\_{q}(s)&=\\int \_{0^{-}}^{\\infty }x\_{q}(t)e^{-st}\\,dt\\\\&=\\int \_{0^{-}}^{\\infty }\\sum \_{n=0}^{\\infty }x\[n\]\\delta (t-nT)e^{-st}\\,dt\\\\&=\\sum \_{n=0}^{\\infty }x\[n\]\\int \_{0^{-}}^{\\infty }\\delta (t-nT)e^{-st}\\,dt\\\\&=\\sum \_{n=0}^{\\infty }x\[n\]e^{-nsT}~.\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee1fc990806a1d664c96eb253f19ff2def0f986)
This is the precise definition of the unilateral Z-transform of the discrete function *x*\[*n*\] ![{\\displaystyle X(z)=\\sum \_{n=0}^{\\infty }x\[n\]z^{-n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf7e6d1a9bf26a3017c79040c80295cb1f2eaa1) with the substitution of *z* â *e**sT*.
Comparing the last two equations, we find the relationship between the unilateral Z-transform and the Laplace transform of the sampled signal, 
The similarity between the Z- and Laplace transforms is expanded upon in the theory of [time scale calculus](https://en.wikipedia.org/wiki/Time_scale_calculus "Time scale calculus").
The integral form of the [Borel transform](https://en.wikipedia.org/wiki/Borel_summation "Borel summation")  is a special case of the Laplace transform for *f* an [entire function](https://en.wikipedia.org/wiki/Entire_function "Entire function") of exponential type, meaning that  for some constants *A* and *B*. The generalized Borel transform allows a different weighting function to be used, rather than the exponential function, to transform functions not of exponential type. [Nachbin's theorem](https://en.wikipedia.org/wiki/Nachbin%27s_theorem "Nachbin's theorem") gives necessary and sufficient conditions for the Borel transform to be well defined.
### Fundamental relationships
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=19 "Edit section: Fundamental relationships")\]
Since an ordinary Laplace transform can be written as a special case of a two-sided transform, and since the two-sided transform can be written as the sum of two one-sided transforms, the theory of the Laplace-, Fourier-, Mellin-, and Z-transforms are at bottom the same subject. However, a different point of view and different characteristic problems are associated with each of these four major integral transforms.
## Table of selected Laplace transforms
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=20 "Edit section: Table of selected Laplace transforms")\]
The following table provides Laplace transforms for many common functions of a single variable.[\[35\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-35)[\[36\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-36) For definitions and explanations, see the *Explanatory Notes* at the end of the table.
Because the Laplace transform is a linear operator,
- The Laplace transform of a sum is the sum of Laplace transforms of each term.
- The Laplace transform of a multiple of a function is that multiple times the Laplace transformation of that function.
Using this linearity, and various [trigonometric](https://en.wikipedia.org/wiki/List_of_trigonometric_identities "List of trigonometric identities"), [hyperbolic](https://en.wikipedia.org/wiki/Hyperbolic_function "Hyperbolic function"), and complex number (etc.) properties and/or identities, some Laplace transforms can be obtained from others more quickly than by using the definition directly.
The unilateral Laplace transform takes as input a function whose time domain is the [non-negative](https://en.wikipedia.org/wiki/Non-negative "Non-negative") reals, which is why all of the time domain functions in the table below are multiples of the [Heaviside step function](https://en.wikipedia.org/wiki/Heaviside_step_function "Heaviside step function"), *u*(*t*).
The entries of the table that involve a time delay *Ï* are required to be [causal](https://en.wikipedia.org/wiki/Causal_system "Causal system") (meaning that *Ï* \> 0). A causal system is a system where the [impulse response](https://en.wikipedia.org/wiki/Impulse_response "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](https://en.wikipedia.org/wiki/Anticausal_system "Anticausal system").
| Function | Time domain  |
|---|---|
## *s*\-domain equivalent circuits and impedances
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=21 "Edit section: s-domain equivalent circuits and impedances")\]
The Laplace transform is often used in [circuit analysis](https://en.wikipedia.org/wiki/Network_analysis_\(electrical_circuits\) "Network analysis (electrical circuits)") by conversions to the *s*\-domain of circuit elements. Circuit elements can be transformed into [impedances](https://en.wikipedia.org/wiki/Electrical_impedance "Electrical impedance"), very similar to [phasor](https://en.wikipedia.org/wiki/Phasor_\(sine_waves\) "Phasor (sine waves)") impedances.
Here is a summary of equivalents:
[](https://en.wikipedia.org/wiki/File:S-Domain_circuit_equivalents.svg "s-domain equivalent circuits")
*s*\-domain equivalent circuits
Note that the resistor is exactly the same in the time domain and the *s*\-domain. The sources are put in if there are initial conditions on the circuit elements. For example, if a capacitor has an initial voltage across it, or if the inductor has an initial current through it, the sources inserted in the *s*\-domain account for that.
The equivalents for current and voltage sources are derived from the transformations in the table above.
## Examples and applications
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=22 "Edit section: Examples and applications")\]
The Laplace transform is used frequently in [engineering](https://en.wikipedia.org/wiki/Engineering "Engineering") and [physics](https://en.wikipedia.org/wiki/Physics "Physics"); the output of a [linear time-invariant system](https://en.wikipedia.org/wiki/Linear_time-invariant_system "Linear time-invariant system") can be calculated by convolving its unit impulse response with the input signal. Performing this calculation in Laplace space turns the convolution into a multiplication; the latter being easier to solve because of its algebraic form. For more information, see [control theory](https://en.wikipedia.org/wiki/Control_theory "Control theory"). The Laplace transform is invertible on a large class of functions. Given a mathematical or functional description of an input or output to a [system](https://en.wikipedia.org/wiki/System "System"), the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.[\[42\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-42)
The Laplace transform can also be used to solve differential equations and is used extensively in [mechanical engineering](https://en.wikipedia.org/wiki/Mechanical_engineering "Mechanical engineering") and [electrical engineering](https://en.wikipedia.org/wiki/Electrical_engineering "Electrical engineering"). The Laplace transform reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of algebra. The original differential equation can then be solved by applying the inverse Laplace transform. English electrical engineer [Oliver Heaviside](https://en.wikipedia.org/wiki/Oliver_Heaviside "Oliver Heaviside") first proposed a similar scheme, although without using the Laplace transform; and the resulting operational calculus is credited as the Heaviside calculus.
### Evaluating improper integrals
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=23 "Edit section: Evaluating improper integrals")\]
Let â â . Then (see the table above) 
From which one gets: 
In the limit â â , one gets  provided that the interchange of limits can be justified. This is often possible as a consequence of the [final value theorem](https://en.wikipedia.org/wiki/Final_value_theorem#Final_Value_Theorem_for_improperly_integrable_functions_\(Abel's_theorem_for_integrals\) "Final value theorem"). Even when the interchange cannot be justified the calculation can be suggestive. For example, with *a* â 0 â *b*, proceeding formally one has ![{\\displaystyle {\\begin{aligned}\\int \_{0}^{\\infty }{\\frac {\\cos(at)-\\cos(bt)}{t}}\\,dt&=\\int \_{0}^{\\infty }\\left({\\frac {p}{p^{2}+a^{2}}}-{\\frac {p}{p^{2}+b^{2}}}\\right)\\,dp\\\\\[6pt\]&=\\left\[{\\frac {1}{2}}\\ln {\\frac {p^{2}+a^{2}}{p^{2}+b^{2}}}\\right\]\_{0}^{\\infty }={\\frac {1}{2}}\\ln {\\frac {b^{2}}{a^{2}}}=\\ln \\left\|{\\frac {b}{a}}\\right\|.\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/150eb829ed85ccdec437af079d262fc08428699c)
### Complex impedance of a capacitor
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=24 "Edit section: Complex impedance of a capacitor")\]
In the theory of [electrical circuits](https://en.wikipedia.org/wiki/Electrical_circuit "Electrical circuit"), the current flow in a [capacitor](https://en.wikipedia.org/wiki/Capacitor "Capacitor") is proportional to the capacitance and rate of change in the electrical potential (with equations as for the [SI](https://en.wikipedia.org/wiki/International_System_of_Units "International System of Units") unit system). Symbolically, this is expressed by the differential equation  where *C* is the capacitance of the capacitor, *i* = *i*(*t*) is the [electric current](https://en.wikipedia.org/wiki/Electric_current "Electric current") through the capacitor as a function of time, and *v* = *v*(*t*) is the [voltage](https://en.wikipedia.org/wiki/Electrostatic_potential "Electrostatic potential") across the terminals of the capacitor, also as a function of time.
Taking the Laplace transform of this equation, we obtain  where  and 
Solving for *V*(*s*) we have 
The definition of the complex impedance *Z* (in [ohms](https://en.wikipedia.org/wiki/Ohm "Ohm")) is the ratio of the complex voltage *V* divided by the complex current *I* while holding the initial state *V*0 at zero: 
Using this definition and the previous equation, we find:  which is the correct expression for the complex impedance of a capacitor. In addition, the Laplace transform has large applications in control theory.
Consider a linear time-invariant system with [transfer function](https://en.wikipedia.org/wiki/Transfer_function "Transfer function") 
The [impulse response](https://en.wikipedia.org/wiki/Impulse_response "Impulse response") is the inverse Laplace transform of this transfer function: 
Partial fraction expansion
To evaluate this inverse transform, we begin by expanding *H*(*s*) using the method of partial fraction expansion, 
The unknown constants *P* and *R* are the [residues](https://en.wikipedia.org/wiki/Residue_\(complex_analysis\) "Residue (complex analysis)") located at the corresponding poles of the transfer function. Each residue represents the relative contribution of that [singularity](https://en.wikipedia.org/wiki/Mathematical_singularity "Mathematical singularity") to the transfer function's overall shape.
By the [residue theorem](https://en.wikipedia.org/wiki/Residue_theorem "Residue theorem"), the inverse Laplace transform depends only upon the poles and their residues. To find the residue *P*, we multiply both sides of the equation by *s* + *α* to get 
Then by letting *s* = â*α*, the contribution from *R* vanishes and all that is left is 
Similarly, the residue *R* is given by 
Note that  and so the substitution of *R* and *P* into the expanded expression for *H*(*s*) gives 
Finally, using the linearity property and the known transform for exponential decay (see *Item* \#*3* in the *Table of Laplace Transforms*, above), we can take the inverse Laplace transform of *H*(*s*) to obtain  which is the impulse response of the system.
Convolution
The same result can be achieved using the [convolution property](https://en.wikipedia.org/wiki/Convolution_theorem "Convolution theorem") as if the system is a series of filters with transfer functions 1/(*s* + *α*) and 1/(*s* + *ÎČ*). That is, the inverse of  is 
| Time function | Laplace transform |
|---|---|
|  | |
Starting with the Laplace transform,  we find the inverse by first rearranging terms in the fraction: 
We are now able to take the inverse Laplace transform of our terms: 
This is just the [sine of the sum](https://en.wikipedia.org/wiki/Trigonometric_identity#Angle_sum_and_difference_identities "Trigonometric identity") of the arguments, yielding: 
We can apply similar logic to find that 
### Statistical mechanics
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=27 "Edit section: Statistical mechanics")\]
In [statistical mechanics](https://en.wikipedia.org/wiki/Statistical_mechanics "Statistical mechanics"), the Laplace transform of the density of states  defines the [partition function](https://en.wikipedia.org/wiki/Partition_function_\(statistical_mechanics\) "Partition function (statistical mechanics)").[\[43\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-43) That is, the canonical partition function  is given by  and the inverse is given by 
### Spatial (not time) structure from astronomical spectrum
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=28 "Edit section: Spatial (not time) structure from astronomical spectrum")\]
The wide and general applicability of the Laplace transform and its inverse is illustrated by an application in astronomy which provides some information on the *spatial distribution* of matter of an [astronomical](https://en.wikipedia.org/wiki/Astronomy "Astronomy") source of [radiofrequency](https://en.wikipedia.org/wiki/Radiofrequency "Radiofrequency") [thermal radiation](https://en.wikipedia.org/wiki/Thermal_radiation "Thermal radiation") too distant to [resolve](https://en.wikipedia.org/wiki/Angular_resolution "Angular resolution") as more than a point, given its [flux density](https://en.wikipedia.org/wiki/Flux_density "Flux density") [spectrum](https://en.wikipedia.org/wiki/Spectrum "Spectrum"), rather than relating the *time* domain with the spectrum (frequency domain).
Assuming certain properties of the object, e.g. spherical shape and constant temperature, calculations based on carrying out an inverse Laplace transformation on the spectrum of the object can produce the only possible [model](https://en.wikipedia.org/wiki/Mathematical_model "Mathematical model") of the distribution of matter in it (density as a function of distance from the center) consistent with the spectrum.[\[44\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-44) When independent information on the structure of an object is available, the inverse Laplace transform method has been found to be in good agreement.
### Birth and death processes
\[[edit](https://en.wikipedia.org/w/index.php?title=Laplace_transform&action=edit§ion=29 "Edit section: Birth and death processes")\]
Consider a [random walk](https://en.wikipedia.org/wiki/Random_walk "Random walk"), with steps  occurring with probabilities â â .[\[45\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-Feller-45) Suppose also that the time step is a [Poisson process](https://en.wikipedia.org/wiki/Poisson_process "Poisson process"), with parameter â â . Then the probability of the walk being at the lattice point  at time  is  This leads to a system of [integral equations](https://en.wikipedia.org/wiki/Integral_equation "Integral equation") (or equivalently a system of differential equations). However, because it is a system of convolution equations, the Laplace transform converts it into a [system of linear equations](https://en.wikipedia.org/wiki/System_of_linear_equations "System of linear equations") for  namely:  which may now be solved by standard methods.
The Laplace transform of the measure  on  is given by  It is intuitively clear that, for small â â , the exponentially decaying integrand will become more sensitive to the concentration of the measure  on larger subsets of the domain. To make this more precise, introduce the distribution function:  Formally, we expect a limit of the following kind:  [Tauberian theorems](https://en.wikipedia.org/wiki/Tauberian_theorem "Tauberian theorem") are theorems relating the asymptotics of the Laplace transform, as â â , to those of the distribution of  as â â . They are thus of importance in asymptotic formulae of [probability](https://en.wikipedia.org/wiki/Probability "Probability") and [statistics](https://en.wikipedia.org/wiki/Statistics "Statistics"), where often the spectral side has asymptotics that are simpler to infer.[\[45\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-Feller-45)
Two Tauberian theorems of note are the [HardyâLittlewood Tauberian theorem](https://en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood_Tauberian_theorem "HardyâLittlewood Tauberian theorem") and [Wiener's Tauberian theorem](https://en.wikipedia.org/wiki/Wiener%27s_Tauberian_theorem "Wiener's Tauberian theorem"). The Wiener theorem generalizes the [Ikehara Tauberian theorem](https://en.wikipedia.org/wiki/Ikehara_Tauberian_theorem "Ikehara Tauberian theorem"), which is the following statement:
Let â â be a non-negative, [monotonic](https://en.wikipedia.org/wiki/Monotonic_function "Monotonic function") nondecreasing function of â â , defined for â â . Suppose that  converges for â â to the function â â and that, for some non-negative number â â ,  has an extension as a [continuous function](https://en.wikipedia.org/wiki/Continuous_function "Continuous function") for â â . Then the [limit](https://en.wikipedia.org/wiki/Limit_of_a_function "Limit of a function") as â â goes to infinity of â â is equal to â â .
This statement can be applied in particular to the [logarithmic derivative](https://en.wikipedia.org/wiki/Logarithmic_derivative "Logarithmic derivative") of [Riemann zeta function](https://en.wikipedia.org/wiki/Riemann_zeta_function "Riemann zeta function"), and thus provides an extremely short way to prove the [prime number theorem](https://en.wikipedia.org/wiki/Prime_number_theorem "Prime number theorem").[\[46\]](https://en.wikipedia.org/wiki/Laplace_transform#cite_note-46)
- [Analog signal processing](https://en.wikipedia.org/wiki/Analog_signal_processing "Analog signal processing")
- [Bernstein's theorem on monotone functions](https://en.wikipedia.org/wiki/Bernstein%27s_theorem_on_monotone_functions "Bernstein's theorem on monotone functions")
- [Continuous-repayment mortgage](https://en.wikipedia.org/wiki/Continuous-repayment_mortgage#Mortgage_difference_and_differential_equation "Continuous-repayment mortgage")
- [Dirichlet integral](https://en.wikipedia.org/wiki/Dirichlet_integral "Dirichlet integral")
- [Differential equation](https://en.wikipedia.org/wiki/Differential_equation "Differential equation")
- [Generating function](https://en.wikipedia.org/wiki/Generating_function "Generating function")
- [Hamburger moment problem](https://en.wikipedia.org/wiki/Hamburger_moment_problem "Hamburger moment problem")
- [HardyâLittlewood Tauberian theorem](https://en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood_Tauberian_theorem "HardyâLittlewood Tauberian theorem")
- [LaplaceâCarson transform](https://en.wikipedia.org/wiki/Laplace%E2%80%93Carson_transform "LaplaceâCarson transform")
- [Moment-generating function](https://en.wikipedia.org/wiki/Moment-generating_function "Moment-generating function")
- [Nonlocal operator](https://en.wikipedia.org/wiki/Nonlocal_operator "Nonlocal operator")
- [Partial fraction decomposition](https://en.wikipedia.org/wiki/Partial_fraction_decomposition "Partial fraction decomposition")
- [Post's inversion formula](https://en.wikipedia.org/wiki/Post%27s_inversion_formula "Post's inversion formula")
- [Signal-flow graph](https://en.wikipedia.org/wiki/Signal-flow_graph "Signal-flow graph")
- [Transfer function](https://en.wikipedia.org/wiki/Transfer_function "Transfer function")
- [Z-transform](https://en.wikipedia.org/wiki/Z-transform "Z-transform")
1. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-Lynn_1986_pp._225%E2%80%93272_1-0)**
Lynn, Paul A. (1986), "The Laplace Transform and the *z*\-transform", *Electronic Signals and Systems*, London: Macmillan Education UK, pp. 225â272, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-1-349-18461-3\_6](https://doi.org/10.1007%2F978-1-349-18461-3_6), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-333-39164-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-333-39164-8 "Special:BookSources/978-0-333-39164-8")
, "Laplace Transform and the z-transform are closely related to the Fourier Transform. Laplace Transform is somewhat more general in scope than the Fourier Transform, and is widely used by engineers for describing continuous circuits and systems, including automatic control systems."
2. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-2)**
["Differential Equations â Laplace Transforms"](https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx), *Pauls Online Math Notes*, retrieved 2020-08-08
3. ^ [***a***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-:1_3-0) [***b***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-:1_3-1)
Weisstein, Eric W., ["Laplace Transform"](https://mathworld.wolfram.com/LaplaceTransform.html), *Wolfram MathWorld*, retrieved 2020-08-08
4. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-4)**
Deakin, Michael A. B. (1981), ["The Development of the Laplace Transform, 1737-1937: I. Euler to Spitzer, 1737-1880"](https://www.jstor.org/stable/41133637), *Archive for History of Exact Sciences*, **25** (4): 343â390, [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0003-9519](https://search.worldcat.org/issn/0003-9519)
5. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-5)**
"Des Fonctions génératrices" \[On generating functions\], [*Théorie analytique des Probabilités*](https://archive.org/details/thorieanalytiqu01laplgoog) \[*Analytical Probability Theory*\] (in French) (2nd ed.), Paris, 1814, chap.I sect.2-20
6. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-6)**
Jaynes, E. T. (Edwin T.) (2003), *Probability theory : the logic of science*, Bretthorst, G. Larry, Cambridge, UK: Cambridge University Press, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0511065892](https://en.wikipedia.org/wiki/Special:BookSources/0511065892 "Special:BookSources/0511065892")
, [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [57254076](https://search.worldcat.org/oclc/57254076)
7. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-7)**
[Abel, Niels H.](https://en.wikipedia.org/wiki/Niels_Henrik_Abel "Niels Henrik Abel") (1820), "Sur les fonctions gĂ©nĂ©ratrices et leurs dĂ©terminantes", *Ćuvres ComplĂštes* (in French), vol. II (published 1839), pp. 77â88
[1881 edition](https://books.google.com/books?id=6FtDAQAAMAAJ&pg=RA2-PA67)
8. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-8)** [Euler 1744](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFEuler1744), [Euler 1753](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFEuler1753), [Euler 1769](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFEuler1769)
9. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-9)** [Lagrange 1773](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFLagrange1773)
10. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-10)** [Grattan-Guinness 1997](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFGrattan-Guinness1997), p. 260
11. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-11)** [Grattan-Guinness 1997](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFGrattan-Guinness1997), p. 261
12. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-12)** [Grattan-Guinness 1997](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFGrattan-Guinness1997), pp. 261â262
13. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-13)** [Grattan-Guinness 1997](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFGrattan-Guinness1997), pp. 262â266
14. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-14)**
[Heaviside, Oliver](https://en.wikipedia.org/wiki/Oliver_Heaviside "Oliver Heaviside") (January 2008), ["The solution of definite integrals by differential transformation"](https://books.google.com/books?id=y9auR0L6ZRcC&pg=PA234), *Electromagnetic Theory*, vol. III, London, section 526, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[9781605206189](https://en.wikipedia.org/wiki/Special:BookSources/9781605206189 "Special:BookSources/9781605206189")
`{{citation}}`: CS1 maint: location missing publisher ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_location_missing_publisher "Category:CS1 maint: location missing publisher"))
15. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-15)**
[Edwards, H. M.](https://en.wikipedia.org/wiki/Harold_Edwards_\(mathematician\) "Harold Edwards (mathematician)") (1974), *Riemann's Zeta Function*, New York: Academic Press, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-12-232750-0](https://en.wikipedia.org/wiki/Special:BookSources/0-12-232750-0 "Special:BookSources/0-12-232750-0")
, [Zbl](https://en.wikipedia.org/wiki/Zbl_\(identifier\) "Zbl (identifier)") [0315\.10035](https://zbmath.org/?format=complete&q=an:0315.10035)
16. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-16)**
Gardner, Murray F.; Barnes, John L. (1942), *Transients in Linear Systems studied by the Laplace Transform*, New York: Wiley
, Appendix C
17. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-17)**
[Lerch, Mathias](https://en.wikipedia.org/wiki/Mathias_Lerch "Mathias Lerch") (1903), "Sur un point de la thĂ©orie des fonctions gĂ©nĂ©ratrices d'Abel" \[Proof of the inversion formula\], *[Acta Mathematica](https://en.wikipedia.org/wiki/Acta_Mathematica "Acta Mathematica")* (in French), **27**: 339â351, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/BF02421315](https://doi.org/10.1007%2FBF02421315), [hdl](https://en.wikipedia.org/wiki/Hdl_\(identifier\) "Hdl (identifier)"):[10338\.dmlcz/501554](https://hdl.handle.net/10338.dmlcz%2F501554)
18. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-18)**
[Bromwich, Thomas J.](https://en.wikipedia.org/wiki/Thomas_John_I%27Anson_Bromwich "Thomas John I'Anson Bromwich") (1916), ["Normal coordinates in dynamical systems"](https://zenodo.org/record/2319588), *[Proceedings of the London Mathematical Society](https://en.wikipedia.org/wiki/Proceedings_of_the_London_Mathematical_Society "Proceedings of the London Mathematical Society")*, **15**: 401â448, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1112/plms/s2-15.1.401](https://doi.org/10.1112%2Fplms%2Fs2-15.1.401)
19. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-19)**
An influential book was:
Gardner, Murray F.; Barnes, John L. (1942), *Transients in Linear Systems studied by the Laplace Transform*, New York: Wiley
20. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-20)**
Doetsch, Gustav (1937), *Theorie und Anwendung der Laplacesche Transformation* \[*Theory and Application of the Laplace Transform*\] (in German), Berlin: Springer
translation 1943
21. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-21)** [Feller 1971](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFFeller1971), §XIII.1.
22. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-22)** The cumulative distribution function is the integral of the probability density function.
23. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-23)**
MikusiĆski, Jan (14 July 2014), [*Operational Calculus*](https://books.google.com/books?id=e8LSBQAAQBAJ), Elsevier, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[9781483278933](https://en.wikipedia.org/wiki/Special:BookSources/9781483278933 "Special:BookSources/9781483278933")
24. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-24)** [Widder 1941](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFWidder1941), Chapter II, §1
25. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-25)** [Widder 1941](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFWidder1941), Chapter VI, §2
26. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-26)**
Beffa, Federico (2024), ["Laplace Transform of Distributions"](https://link.springer.com/10.1007/978-3-031-40681-2_5), *Weakly Nonlinear Systems*, Cham: Springer Nature Switzerland, pp. 75â85, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-3-031-40681-2\_5](https://doi.org/10.1007%2F978-3-031-40681-2_5), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-3-031-40680-5](https://en.wikipedia.org/wiki/Special:BookSources/978-3-031-40680-5 "Special:BookSources/978-3-031-40680-5")
, retrieved 2026-01-14
`{{citation}}`: CS1 maint: work parameter with ISBN ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_work_parameter_with_ISBN "Category:CS1 maint: work parameter with ISBN"))
27. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-27)** [Korn & Korn 1967](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFKornKorn1967), pp. 226â227
28. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-28)** [Bracewell 2000](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFBracewell2000), Table 14.1, p. 385
29. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-29)**
Archived at [Ghostarchive](https://ghostarchive.org/varchive/youtube/20211211/zvbdoSeGAgI) and the [Wayback Machine](https://web.archive.org/web/20141220033002/https://www.youtube.com/watch?v=zvbdoSeGAgI&gl=US&hl=en):
Mattuck, Arthur (7 November 2008), ["Where the Laplace Transform comes from"](https://www.youtube.com/watch?v=zvbdoSeGAgI), *[YouTube](https://en.wikipedia.org/wiki/YouTube "YouTube")*
30. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-FOOTNOTEWidder194138_30-0)** [Widder 1941](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFWidder1941), p. 38.
31. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-31)** [Feller 1971](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFFeller1971), p. 432
32. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-32)**
[Laurent Schwartz](https://en.wikipedia.org/wiki/Laurent_Schwartz "Laurent Schwartz") (1966), *Mathematics for the physical sciences*, Addison-Wesley
, p 224.
33. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-33)**
[Titchmarsh, E.](https://en.wikipedia.org/wiki/Edward_Charles_Titchmarsh "Edward Charles Titchmarsh") (1986) \[1948\], *Introduction to the theory of Fourier integrals* (2nd ed.), [Clarendon Press](https://en.wikipedia.org/wiki/Clarendon_Press "Clarendon Press"), p. 6, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-8284-0324-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8284-0324-5 "Special:BookSources/978-0-8284-0324-5")
34. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-34)** [Takacs 1953](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFTakacs1953), p. 93
35. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-35)**
Riley, K. F.; Hobson, M. P.; Bence, S. J. (2010), *Mathematical methods for physics and engineering* (3rd ed.), Cambridge University Press, p. 455, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-521-86153-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-86153-3 "Special:BookSources/978-0-521-86153-3")
36. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-36)**
Distefano, J. J.; Stubberud, A. R.; Williams, I. J. (1995), *Feedback systems and control*, Schaum's outlines (2nd ed.), McGraw-Hill, p. 78, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-07-017052-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-017052-0 "Special:BookSources/978-0-07-017052-0")
37. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-37)**
Lipschutz, S.; Spiegel, M. R.; Liu, J. (2009), *Mathematical Handbook of Formulas and Tables*, Schaum's Outline Series (3rd ed.), McGraw-Hill, p. 183, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-07-154855-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-154855-7 "Special:BookSources/978-0-07-154855-7")
â provides the case for real *q*.
38. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-38)** <http://mathworld.wolfram.com/LaplaceTransform.html> â Wolfram Mathword provides case for complex *q*
39. ^ [***a***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-FOOTNOTEBracewell1978227_39-0) [***b***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-FOOTNOTEBracewell1978227_39-1) [***c***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-FOOTNOTEBracewell1978227_39-2) [***d***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-FOOTNOTEBracewell1978227_39-3) [Bracewell 1978](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFBracewell1978), p. 227.
40. ^ [***a***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-FOOTNOTEWilliams197388_40-0) [***b***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-FOOTNOTEWilliams197388_40-1) [***c***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-FOOTNOTEWilliams197388_40-2) [Williams 1973](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFWilliams1973), p. 88.
41. ^ [***a***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-FOOTNOTEWilliams197389_41-0) [***b***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-FOOTNOTEWilliams197389_41-1) [Williams 1973](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFWilliams1973), p. 89.
42. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-42)** [Korn & Korn 1967](https://en.wikipedia.org/wiki/Laplace_transform#CITEREFKornKorn1967), §8.1
43. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-43)**
RK Pathria; Paul Beal (1996), [*Statistical mechanics*](https://archive.org/details/statisticalmecha00path_911) (2nd ed.), Butterworth-Heinemann, p. [56](https://archive.org/details/statisticalmecha00path_911/page/n66), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[9780750624695](https://en.wikipedia.org/wiki/Special:BookSources/9780750624695 "Special:BookSources/9780750624695")
44. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-44)**
Salem, M.; Seaton, M. J. (1974), "I. Continuum spectra and brightness contours", *[Monthly Notices of the Royal Astronomical Society](https://en.wikipedia.org/wiki/Monthly_Notices_of_the_Royal_Astronomical_Society "Monthly Notices of the Royal Astronomical Society")*, **167**: 493â510, [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1974MNRAS.167..493S](https://ui.adsabs.harvard.edu/abs/1974MNRAS.167..493S), [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1093/mnras/167.3.493](https://doi.org/10.1093%2Fmnras%2F167.3.493)
, and
Salem, M. (1974), "II. Three-dimensional models", *Monthly Notices of the Royal Astronomical Society*, **167**: 511â516, [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1974MNRAS.167..511S](https://ui.adsabs.harvard.edu/abs/1974MNRAS.167..511S), [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1093/mnras/167.3.511](https://doi.org/10.1093%2Fmnras%2F167.3.511)
45. ^ [***a***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-Feller_45-0) [***b***](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-Feller_45-1)
Feller, *Introduction to Probability Theory, volume II,pp=479-483*
46. **[^](https://en.wikipedia.org/wiki/Laplace_transform#cite_ref-46)**
[S. Ikehara](https://en.wikipedia.org/wiki/Shikao_Ikehara "Shikao Ikehara") (1931), "An extension of Landau's theorem in the analytic theory of numbers", *Journal of Mathematics and Physics*, **10** (1â4\): 1â12, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1002/sapm19311011](https://doi.org/10.1002%2Fsapm19311011), [Zbl](https://en.wikipedia.org/wiki/Zbl_\(identifier\) "Zbl (identifier)") [0001\.12902](https://zbmath.org/?format=complete&q=an:0001.12902)
- Bracewell, Ronald N. (1978), *The Fourier Transform and its Applications* (2nd ed.), McGraw-Hill Kogakusha, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-07-007013-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-007013-4 "Special:BookSources/978-0-07-007013-4")
- Bracewell, R. N. (2000), *The Fourier Transform and Its Applications* (3rd ed.), Boston: McGraw-Hill, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-07-116043-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-116043-8 "Special:BookSources/978-0-07-116043-8")
- [Feller, William](https://en.wikipedia.org/wiki/William_Feller "William Feller") (1971), *An introduction to probability theory and its applications. Vol. II.*, Second edition, New York: [John Wiley & Sons](https://en.wikipedia.org/wiki/John_Wiley_%26_Sons "John Wiley & Sons"), [MR](https://en.wikipedia.org/wiki/MR_\(identifier\) "MR (identifier)") [0270403](https://mathscinet.ams.org/mathscinet-getitem?mr=0270403)
- Korn, G. A.; [Korn, T. M.](https://en.wikipedia.org/wiki/Theresa_M._Korn "Theresa M. Korn") (1967), *Mathematical Handbook for Scientists and Engineers* (2nd ed.), McGraw-Hill Companies, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-07-035370-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-035370-1 "Special:BookSources/978-0-07-035370-1")
- Widder, David Vernon (1941), *The Laplace Transform*, Princeton Mathematical Series, v. 6, [Princeton University Press](https://en.wikipedia.org/wiki/Princeton_University_Press "Princeton University Press"), [MR](https://en.wikipedia.org/wiki/MR_\(identifier\) "MR (identifier)") [0005923](https://mathscinet.ams.org/mathscinet-getitem?mr=0005923)
- Williams, J. (1973), *Laplace Transforms*, Problem Solvers, George Allen & Unwin, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-04-512021-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-04-512021-5 "Special:BookSources/978-0-04-512021-5")
- Takacs, J. (1953), "Fourier amplitudok meghatarozasa operatorszamitassal", *Magyar Hiradastechnika* (in Hungarian), **IV** (7â8\): 93â96
- [Euler, L.](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler") (1744), "De constructione aequationum" \[The Construction of Equations\], *Opera Omnia*, 1st series (in Latin), **22**: 150â161
- [Euler, L.](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler") (1753), "Methodus aequationes differentiales" \[A Method for Solving Differential Equations\], *Opera Omnia*, 1st series (in Latin), **22**: 181â213
- [Euler, L.](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler") (1992) \[1769\], "Institutiones calculi integralis, Volume 2" \[Institutions of Integral Calculus\], *Opera Omnia*, 1st series (in Latin), **12**, Basel: BirkhÀuser, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-3764314743](https://en.wikipedia.org/wiki/Special:BookSources/978-3764314743 "Special:BookSources/978-3764314743")
`{{citation}}`: CS1 maint: work parameter with ISBN ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_work_parameter_with_ISBN "Category:CS1 maint: work parameter with ISBN")), Chapters 3â5
- [Euler, Leonhard](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler") (1769), [*Institutiones calculi integralis*](https://books.google.com/books?id=BFqWNwpfqo8C) \[*Institutions of Integral Calculus*\] (in Latin), vol. II, Paris: Petropoli, ch. 3â5, pp. 57â153
- [Grattan-Guinness, I](https://en.wikipedia.org/wiki/Ivor_Grattan-Guinness "Ivor Grattan-Guinness") (1997), "Laplace's integral solutions to partial differential equations", in Gillispie, C. C. (ed.), *Pierre Simon Laplace 1749â1827: A Life in Exact Science*, Princeton: Princeton University Press, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-691-01185-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-01185-1 "Special:BookSources/978-0-691-01185-1")
- [Lagrange, J. L.](https://en.wikipedia.org/wiki/Joseph_Louis_Lagrange "Joseph Louis Lagrange") (1773), *MĂ©moire sur l'utilitĂ© de la mĂ©thode*, Ćuvres de Lagrange, vol. 2, pp. 171â234
- Arendt, Wolfgang; Batty, Charles J.K.; Hieber, Matthias; Neubrander, Frank (2002), *Vector-Valued Laplace Transforms and Cauchy Problems*, BirkhÀuser Basel, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-3-7643-6549-3](https://en.wikipedia.org/wiki/Special:BookSources/978-3-7643-6549-3 "Special:BookSources/978-3-7643-6549-3")
- Davies, Brian (2002), *Integral transforms and their applications* (Third ed.), New York: Springer, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-387-95314-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-95314-4 "Special:BookSources/978-0-387-95314-4")
- Deakin, M. A. B. (1981), "The development of the Laplace transform", *Archive for History of Exact Sciences*, **25** (4): 343â390, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/BF01395660](https://doi.org/10.1007%2FBF01395660), [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [117913073](https://api.semanticscholar.org/CorpusID:117913073)
- Deakin, M. A. B. (1982), "The development of the Laplace transform", *Archive for History of Exact Sciences*, **26** (4): 351â381, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/BF00418754](https://doi.org/10.1007%2FBF00418754), [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [123071842](https://api.semanticscholar.org/CorpusID:123071842)
- [Doetsch, Gustav](https://en.wikipedia.org/wiki/Gustav_Doetsch "Gustav Doetsch") (1974), *Introduction to the Theory and Application of the Laplace Transformation*, Springer, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-387-06407-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-06407-9 "Special:BookSources/978-0-387-06407-9")
- Mathews, Jon; Walker, Robert L. (1970), *Mathematical methods of physics* (2nd ed.), New York: W. A. Benjamin, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-8053-7002-1](https://en.wikipedia.org/wiki/Special:BookSources/0-8053-7002-1 "Special:BookSources/0-8053-7002-1")
- Polyanin, A. D.; Manzhirov, A. V. (1998), *Handbook of Integral Equations*, Boca Raton: CRC Press, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-8493-2876-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8493-2876-3 "Special:BookSources/978-0-8493-2876-3")
- [Schwartz, Laurent](https://en.wikipedia.org/wiki/Laurent_Schwartz "Laurent Schwartz") (1952), "Transformation de Laplace des distributions", *Comm. SĂ©m. Math. Univ. Lund \[Medd. Lunds Univ. Mat. Sem.\]* (in French), **1952**: 196â206, [MR](https://en.wikipedia.org/wiki/MR_\(identifier\) "MR (identifier)") [0052555](https://mathscinet.ams.org/mathscinet-getitem?mr=0052555)
- [Schwartz, Laurent](https://en.wikipedia.org/wiki/Laurent_Schwartz "Laurent Schwartz") (2008) \[1966\], [*Mathematics for the Physical Sciences*](https://books.google.com/books?id=-_AuDQAAQBAJ&pg=PA215), Dover Books on Mathematics, New York: Dover Publications, pp. 215â241, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-486-46662-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-46662-0 "Special:BookSources/978-0-486-46662-0")
â see Chapter VI. The Laplace transform
- Siebert, William McC. (1986), *Circuits, Signals, and Systems*, Cambridge, Massachusetts: MIT Press, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-262-19229-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-262-19229-3 "Special:BookSources/978-0-262-19229-3")
- Widder, David Vernon (1945), "What is the Laplace transform?", *[The American Mathematical Monthly](https://en.wikipedia.org/wiki/American_Mathematical_Monthly "American Mathematical Monthly")*, **52** (8): 419â425, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.2307/2305640](https://doi.org/10.2307%2F2305640), [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0002-9890](https://search.worldcat.org/issn/0002-9890), [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [2305640](https://www.jstor.org/stable/2305640), [MR](https://en.wikipedia.org/wiki/MR_\(identifier\) "MR (identifier)") [0013447](https://mathscinet.ams.org/mathscinet-getitem?mr=0013447)
- Weidman, J.A.C.; Fornberg, Bengt (2023), "Fully numerical Laplace transform methods", *Numerical Algorithms*, **92**: 985â1006, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/s11075-022-01368-x](https://doi.org/10.1007%2Fs11075-022-01368-x)
- ["Laplace transform"](https://www.encyclopediaofmath.org/index.php?title=Laplace_transform), *[Encyclopedia of Mathematics](https://en.wikipedia.org/wiki/Encyclopedia_of_Mathematics "Encyclopedia of Mathematics")*, [EMS Press](https://en.wikipedia.org/wiki/European_Mathematical_Society "European Mathematical Society"), 2001 \[1994\]
- [Online Computation](http://wims.unice.fr/wims/wims.cgi?lang=en&+module=tool%2Fanalysis%2Ffourierlaplace) of the transform or inverse transform, wims.unice.fr
- [Tables of Integral Transforms](http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm) at EqWorld: The World of Mathematical Equations.
- [Weisstein, Eric W.](https://en.wikipedia.org/wiki/Eric_W._Weisstein "Eric W. Weisstein"), ["Laplace Transform"](https://mathworld.wolfram.com/LaplaceTransform.html), *[MathWorld](https://en.wikipedia.org/wiki/MathWorld "MathWorld")*
- [Good explanations of the initial and final value theorems](http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/) [Archived](https://web.archive.org/web/20090108132440/http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/) 2009-01-08 at the [Wayback Machine](https://en.wikipedia.org/wiki/Wayback_Machine "Wayback Machine")
- [Laplace Transforms](http://www.mathpages.com/home/kmath508/kmath508.htm) at MathPages
- [Computational Knowledge Engine](http://www.wolframalpha.com/input/?i=laplace+transform+example) allows to easily calculate Laplace Transforms and its inverse Transform.
- [Laplace Calculator](http://www.laplacetransformcalculator.com/easy-laplace-transform-calculator/) to calculate Laplace Transforms online easily.
- [Code to visualize Laplace Transforms](https://johnflux.com/2019/02/12/laplace-transform-visualized/) and many example videos. |
| Shard | 152 (laksa) |
| Root Hash | 17790707453426894952 |
| Unparsed URL | org,wikipedia!en,/wiki/Laplace_transform s443 |