In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Laplace transform, named after its discoverer
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
(), is an
integral transform that converts a
function of a
real variable
Variable may refer to:
* Variable (computer science), a symbolic name associated with a value and whose associated value may be changed
* Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
(usually
, in the ''
time domain'') to a function of a
complex variable
(in the complex
frequency domain, also known as ''s''-domain, or s-plane). The transform has many applications in
science
Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earliest archeological evidence for ...
and
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
because it is a tool for solving
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s.
In particular, it transforms
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
s into
algebraic equations and
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
into
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
.
For suitable functions ''f'', the Laplace transform is the
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
History
![Laplace, Pierre-Simon, marquis de](https://upload.wikimedia.org/wikipedia/commons/3/39/Laplace%2C_Pierre-Simon%2C_marquis_de.jpg)
The Laplace transform is named after
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
and
astronomer
An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, natural satellite, moons, comets and galaxy, g ...
Pierre-Simon, marquis de Laplace, who used a similar transform in his work on
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
. Laplace wrote extensively about the use of
generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...
s in ''Essai philosophique sur les probabilités'' (1814), and the integral form of the Laplace transform evolved naturally as a result.
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
In mathematics and statistics, a quantitative variable may be continuous or discrete if they are typically obtained by ''measuring'' or ''counting'', respectively. If it can take on two particular real values such that it can also take on all re ...
case which was discussed by
Niels Henrik Abel. The theory was further developed in the 19th and early 20th centuries by
Mathias Lerch,
Oliver Heaviside
Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vec ...
, and
Thomas Bromwich.
The current widespread use of the transform (mainly in engineering) came about during and soon after
World War II
World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposin ...
, replacing the earlier Heaviside
operational calculus. The advantages of the Laplace transform had been emphasized by
Gustav Doetsch
Gustav Doetsch (29 November 1892 – 9 June 1977) was a German mathematician, aviation researcher, decorated war veteran, and Nazi supporter. Early life
Doetsch was born into a strict Catholic family on 29 November 1892 in Cologne. , to whom the name Laplace transform is apparently due.
From 1744,
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
investigated integrals of the form
as solutions of differential equations, but did not pursue the matter very far.
Joseph Louis Lagrange was an admirer of Euler and, in his work on integrating
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
s, investigated expressions of the form
which some modern historians have interpreted within modern Laplace transform theory.
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. 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 mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
, 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.
Laplace also recognised that
Joseph Fourier
Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French people, French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier an ...
's method of
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
for solving the
diffusion equation
The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's la ...
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.
Formal definition
The Laplace transform of a
function , defined for all
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s , is the function , which is a unilateral transform defined by
where ''s'' is a
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
frequency parameter
with real numbers and .
An alternate notation for the Laplace transform is
instead of .
The meaning of the integral depends on types of functions of interest. A necessary condition for existence of the integral is that must be
locally integrable on . For locally integrable functions that decay at infinity or are of
exponential type (
), 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 . 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
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below.
F ...
by the Lebesgue integral
An important special case is where is a
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more gener ...
, for example, the
Dirac delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
. In
operational calculus, the Laplace transform of a measure is often treated as though the measure came from a probability density function . In that case, to avoid potential confusion, one often writes
where the lower limit of is shorthand notation for
This limit emphasizes that any point mass located at 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
Laplace–Stieltjes transform.
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, by extending the limits of integration to be the entire real axis. If that is done, the common unilateral transform simply becomes a special case of the bilateral transform, where the definition of the function being transformed is multiplied by the
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
.
The bilateral Laplace transform is defined as follows:
An alternate notation for the bilateral Laplace transform is
, instead of
.
Inverse Laplace transform
Two integrable functions have the same Laplace transform only if they differ on a set of
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
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 , or more generally
tempered distributions
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to derivative, differentiate functions whose de ...
on . 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 function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...
s in the
region of convergence
In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series co ...
. The
inverse Laplace transform is given by the following complex integral, which is known by various names (the Bromwich integral, the Fourier–Mellin integral, and Mellin's inverse formula):
where is a real number so that the contour path of integration is in the region of convergence of . 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
In mathematics, the inverse Laplace transform of a function ''F''(''s'') is the piecewise-continuous and exponentially-restricted real function ''f''(''t'') which has the property:
:\mathcal\(s) = \mathcal\(s) = F(s),
where \mathcal denotes the L ...
. The limit here is interpreted in the
weak-* topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
.
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
In
pure
Pure may refer to:
Computing
* A pure function
* A pure virtual function
* PureSystems, a family of computer systems introduced by IBM in 2012
* Pure Software, a company founded in 1991 by Reed Hastings to support the Purify tool
* Pure-FTPd, ...
and
applied probability
Applied probability is the application of probability theory to statistical problems and other scientific and engineering domains.
Scope
Much research involving probability is done under the auspices of applied probability. However, while such res ...
, the Laplace transform is defined as an
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
. If is a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
with probability density function , then the Laplace transform of is given by the expectation
By
convention
Convention may refer to:
* Convention (norm), a custom or tradition, a standard of presentation or conduct
** Treaty, an agreement in international law
* Convention (meeting), meeting of a (usually large) group of individuals and/or companies in a ...
, this is referred to as the Laplace transform of the random variable itself. Here, replacing by gives the
moment generating function of . The Laplace transform has applications throughout probability theory, including
first passage times of
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
es such as
Markov chain
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
s, and
renewal theory
Renewal theory is the branch of probability theory that generalizes the Poisson process for arbitrary holding times. Instead of exponentially distributed holding times, a renewal process may have any independent and identically distributed (IID) ho ...
.
Of particular use is the ability to recover the
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
of a continuous random variable , by means of the Laplace transform as follows:
Region of convergence
If is a locally integrable function (or more generally a Borel measure locally of bounded variation), then the Laplace transform of converges provided that the limit
exists.
The Laplace transform
converges absolutely
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said ...
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 converges absolutely is either of the form or , where is an
extended real constant with (a consequence of the
dominated convergence theorem). The constant is known as the abscissa of absolute convergence, and depends on the growth behavior of . Analogously, the two-sided transform converges absolutely in a strip of the form , and possibly including the lines or . The subset of values of 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
In complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic.
Morera's theorem states that a continuous, complex-valued function ''f'' defined ...
.
Similarly, the set of values for which 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 , then it automatically converges for all with . Therefore, the region of convergence is a half-plane of the form , possibly including some points of the boundary line .
In the region of convergence , the Laplace transform of can be expressed by
integrating by parts as the integral
That is, can effectively be expressed, in the region of convergence, as the absolutely convergent Laplace transform of some other function. In particular, it is analytic.
There are several
Paley–Wiener theorem In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. The theorem is named for Raymond Paley (1907–1933) and Norbert Wiener (189 ...
s concerning the relationship between the decay properties of , 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 . 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
The Laplace transform has a number of properties that make it useful for analyzing linear
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
s. The most significant advantage is that
differentiation becomes multiplication, and
integration
Integration may refer to:
Biology
*Multisensory integration
*Path integration
* Pre-integration complex, viral genetic material used to insert a viral genome into a host genome
*DNA integration, by means of site-specific recombinase technology, ...
becomes division, by (reminiscent of the way
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
s change multiplication to addition of logarithms).
Because of this property, the Laplace variable is also known as ''operator variable'' in the domain: either ''derivative operator'' or (for ''integration operator''. The transform turns
integral equations and
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s to
polynomial equations, which are much easier to solve. Once solved, use of the inverse Laplace transform reverts to the original domain.
Given the functions and , and their respective Laplace transforms and ,
the following table is a list of properties of unilateral Laplace transform:
;
Initial value theorem In mathematical analysis, the initial value theorem is a theorem used to relate frequency domain expressions to the time domain behavior as time approaches zero.
Let
: F(s) = \int_0^\infty f(t) e^\,dt
be the (one-sided) Laplace transform of ''& ...
:
;
Final value theorem
In mathematical analysis, the final value theorem (FVT) is one of several similar theorems used to relate frequency domain expressions to the time domain behavior as time approaches infinity.
Mathematically, if f(t) in continuous time has (unilate ...
:
, if all
poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in Ce ...
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
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction a ...
decompositions (or other difficult algebra). If has a pole in the right-hand plane or poles on the imaginary axis (e.g., if
or
), then the behaviour of this formula is undefined.
Relation to power series
The Laplace transform can be viewed as a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
analogue of a
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
. If is a discrete function of a positive integer , then the power series associated to is the series
where is a real variable (see
Z transform
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation.
It can be considered as a discrete-tim ...
). Replacing summation over with integration over , a continuous version of the power series becomes
where the discrete function is replaced by the continuous one .
Changing the base of the power from to gives
For this to converge for, say, all bounded functions , it is necessary to require that . Making the substitution gives just the Laplace transform:
In other words, the Laplace transform is a continuous analog of a power series, in which the discrete parameter is replaced by the continuous parameter , and is replaced by .
Relation to moments
The quantities
are the ''moments'' of the function . If the first moments of converge absolutely, then by repeated
differentiation under the integral
In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form
\int_^ f(x,t)\,dt,
where -\infty < a(x), b(x) < \infty and the integral are ,
This is of special significance in probability theory, where the moments of a random variable are given by the expectation values