TheInfoList

OR:

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 Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orient ...
of a real variable (usually $t$, in the ''
time domain Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the ca ...
'') to a function of a
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
variable $s$ (in the complex
frequency domain In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a sig ...
, 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 fo ...
and
engineering Engineering is the use of 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 range of more speciali ...
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 equations into
algebraic equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
s and
convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
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 addi ...
. For suitable functions ''f'', the Laplace transform is the
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
$\mathcal\(s) = \int_0^\infty f(t)e^ \, dt.$

# History 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 ...
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, moons, comets and galaxies – in either obs ...
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 ...
. Laplace wrote extensively about the use of generating functions 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 case which was discussed by
Niels Henrik Abel Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
. 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 ve ...
, 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 World War II by country, vast majority of the world's countries—including all of the great power ...
, replacing the earlier Heaviside operational calculus. The advantages of the Laplace transform had been emphasized by Gustav Doetsch, 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 m ...
investigated integrals of the form $z = \int X(x) e^\, dx \quad\text\quad z = \int X(x) x^A \, dx$ 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) ca ...
s, investigated expressions of the form $\int X(x) e^ a^x\, dx,$ 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 $\int x^s \varphi (x)\, dx,$ 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 mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analysis and har ...
'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 law ...
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 Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orient ...
, 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 ...
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 $s = \sigma + i \omega,$ with real numbers and . An alternate notation for the Laplace transform is $\mathcal\$ instead of . The meaning of the integral depends on types of functions of interest. A necessary condition for existence of the integral is that must be locally integrable on . For locally integrable functions that decay at infinity or are of exponential type ($, f\left(t\right), \le Ae^$), the integral can be understood to be a (proper)
Lebesgue integral In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
. 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 by the Lebesgue integral $\mathcal\(s) = \int_ e^\, d\mu(t).$ 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 gen ...
, 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 entir ...
. 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 $\mathcal\(s) = \int_^\infty f(t)e^ \, dt,$ where the lower limit of is shorthand notation for $\lim_\int_^\infty.$ 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 In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment generating function. Two-sided Laplace transforms are closely related to the Fourier transform, the Melli ...
, 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 argum ...
. The bilateral Laplace transform is defined as follows: An alternate notation for the bilateral Laplace transform is $\mathcal\$, instead of $F$.

## 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 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 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 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. 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 and applied probability, 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, this is referred to as the Laplace transform of the random variable itself. Here, replacing by gives the
moment generating function In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compare ...
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 app ...
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. 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. Ever ...
of a continuous random variable , by means of the Laplace transform as follows: $F_X(x) = \mathcal^\! \left\\! (x) = \mathcal^\! \left\\! (x).$

# 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 $\lim_\int_0^R f(t)e^\,dt$ exists. The Laplace transform converges absolutely if the integral $\int_0^\infty \left, f(t)e^\\,dt$ 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. 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 $F(s) = (s-s_0)\int_0^\infty e^\beta(t)\,dt, \quad \beta(u) = \int_0^u e^f(t)\,dt.$ 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 theorems 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 describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
s. The most significant advantage is that differentiation becomes multiplication, and integration 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 equation In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n) ...
s 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 equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
s, 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 , $\begin f(t) &= \mathcal^\(s),\\ g(t) &= \mathcal^\(s), \end$ the following table is a list of properties of unilateral Laplace transform: ; Initial value theorem :$f\left(0^+\right)=\lim_.$ ; Final value theorem :$f\left(\infty\right)=\lim_$, 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 ...
of $sF\left(s\right)$ 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 as ...
decompositions (or other difficult algebra). If has a pole in the right-hand plane or poles on the imaginary axis (e.g., if $f\left(t\right) = e^t$ or $f\left(t\right) = \sin\left(t\right)$), 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 $\sum_^ a(n) x^n$ where is a real variable (see Z transform). Replacing summation over with integration over , a continuous version of the power series becomes $\int_^ f(t) x^t\, dt$ where the discrete function is replaced by the continuous one . Changing the base of the power from to gives $\int_^ f(t) \left(e^\right)^t\, dt$ For this to converge for, say, all bounded functions , it is necessary to require that . Making the substitution gives just the Laplace transform: $\int_^ f(t) e^\, dt$ 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 $\mu_n = \int_0^\infty t^nf(t)\, dt$ are the ''moments'' of the function . If the first moments of converge absolutely, then by repeated differentiation under the integral, $(-1)^n(\mathcal L f)^(0) = \mu_n .$ This is of special significance in probability theory, where the moments of a random variable are given by the expectation values

## Computation of the Laplace 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: yielding $\mathcal \ = s\cdot\mathcal \-f(0^-),$ and in the bilateral case, $\mathcal \ = s \int_^\infty e^ f(t)\,dt = s \cdot \mathcal \.$ The general result $\mathcal \left\ = s^n \cdot \mathcal \ - s^ f(0^-) - \cdots - f^(0^-),$ where $f^$ denotes the th derivative of , can then be established with an inductive argument.

## Evaluating integrals over the positive real axis

A useful property of the Laplace transform is the following: $\int_0^\infty f(x)g(x)\,dx = \int_0^\infty(\mathcal f)(s)\cdot(\mathcal^g)(s)\,ds$ under suitable assumptions on the behaviour of $f,g$ in a right neighbourhood of $0$ and on the decay rate of $f,g$ in a left neighbourhood of $\infty$. The above formula is a variation of integration by parts, with the operators $\frac$ and $\int \,dx$ being replaced by $\mathcal$ and $\mathcal^$. Let us prove the equivalent formulation: $\int_0^\infty(\mathcal f)(x)g(x)\,dx = \int_0^\infty f(s)(\mathcalg)(s)\,ds.$ By plugging in $\left(\mathcalf\right)\left(x\right)=\int_0^\infty f\left(s\right)e^\,ds$ the left-hand side turns into: $\int_0^\infty\int_0^\infty f(s)g(x) e^\,ds\,dx,$ 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, $\int_0^\infty\fracdx = \int_0^\infty \mathcal(1)(x)\sin x dx = \int_0^\infty 1 \cdot \mathcal(\sin)(x)dx = \int_0^\infty \frac = \frac.$

# Relationship to other transforms

## Laplace–Stieltjes transform

The (unilateral) Laplace–Stieltjes transform of a function is defined by the Lebesgue–Stieltjes integral $\(s) = \int_0^\infty e^ \, d\,g(t) ~.$ The function is assumed to be of
bounded variation In mathematical analysis, a function of bounded variation, also known as ' function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a cont ...
. If is the
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolicall ...
of : $g(x) = \int_0^x f(t)\,d\,t$ then the Laplace–Stieltjes transform of and the Laplace transform of coincide. In general, the Laplace–Stieltjes transform is the Laplace transform of the Stieltjes measure associated to . 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 Laplace–Stieltjes transform is thought of as operating on its
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. Ever ...
.

## Fourier transform

The
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
is a special case (under certain conditions) of the bilateral Laplace transform. While the Fourier transform of a function is a complex function of a ''real'' variable (frequency), the Laplace transform of a function is a complex function of a ''complex'' variable. The Laplace transform is usually restricted to transformation of functions of with . A consequence of this restriction is that the Laplace transform of a function is a
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deri ...
of the variable . Unlike the Fourier transform, the Laplace transform of a distribution is generally a
well-behaved In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved. Th ...
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 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 ...
representation. This power series expresses a function as a linear superposition of moments of the function. This perspective has applications in probability theory. The Fourier transform is equivalent to evaluating the bilateral Laplace transform with imaginary argument or when the condition explained below is fulfilled, This convention of the Fourier transform ($\hat f_3\left(\omega\right)$ in ) requires a factor of on the inverse Fourier transform. This relationship between the Laplace and Fourier transforms is often used to determine the
frequency spectrum The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
of a
signal In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The '' IEEE Transactions on Signal Processing' ...
or dynamical system. The above relation is valid as stated
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicond ...
the region of convergence (ROC) of contains the imaginary axis, . For example, the function has a Laplace transform whose ROC is . As is a pole of , substituting in does not yield the Fourier transform of , which is proportional to the Dirac delta-function . However, a relation of the form $\lim_ F(\sigma+i\omega) = \hat(\omega)$ 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 Paley–Wiener theorems.

## Mellin transform

The Mellin transform and its inverse are related to the two-sided Laplace transform by a simple change of variables. If in the Mellin transform $G(s) = \mathcal\ = \int_0^\infty \theta^s g(\theta) \, \frac \theta$ we set we get a two-sided Laplace transform.

## Z-transform

The unilateral or one-sided Z-transform is simply the Laplace transform of an ideally sampled signal with the substitution of $z \stackrel e^ ,$ where is the sampling interval (in units of time e.g., seconds) and is the
sampling rate In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of "samples". A sample is a value of the signal at a point in time and/or sp ...
(in
samples per second In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of "samples". A sample is a value of the signal at a point in time and/or sp ...
or
hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that one he ...
). Let $\Delta_T(t) \ \stackrel\ \sum_^ \delta(t - n T)$ be a sampling impulse train (also called a Dirac comb) and be the sampled representation of the continuous-time The Laplace transform of the sampled signal is This is the precise definition of the unilateral Z-transform of the discrete function with the substitution of . Comparing the last two equations, we find the relationship between the unilateral Z-transform and the Laplace transform of the sampled signal, $X_q(s) = X(z) \Big, _.$ The similarity between the and Laplace transforms is expanded upon in the theory of time scale calculus.

## Borel transform

The integral form of the Borel transform $F(s) = \int_0^\infty f(z)e^\, dz$ is a special case of the Laplace transform for an
entire function In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fini ...
of exponential type, meaning that $, f(z), \le Ae^$ for some constants and . 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

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

The following table provides Laplace transforms for many common functions of a single variable. 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.$\mathcal\ = \mathcal\ + \mathcal\$ * The Laplace transform of a multiple of a function is that multiple times the Laplace transformation of that function.$\mathcal\ = a \mathcal\$ Using this linearity, and various trigonometric,
hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...
, 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 In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...
reals, which is why all of the time domain functions in the table below are multiples of the Heaviside step function, . The entries of the table that involve a time delay are required to be
causal Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
(meaning that ). A causal system is a system where the
impulse response In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse (). More generally, an impulse response is the reac ...
is zero for all time prior to . In general, the region of convergence for causal systems is not the same as that of anticausal systems.

# ''s''-domain equivalent circuits and impedances

The Laplace transform is often used in circuit analysis, and simple conversions to the -domain of circuit elements can be made. Circuit elements can be transformed into impedances, very similar to phasor impedances. Here is a summary of equivalents: : Note that the resistor is exactly the same in the time domain and the -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 -domain account for that. The equivalents for current and voltage sources are simply derived from the transformations in the table above.

# Examples and applications

The Laplace transform is used frequently in
engineering Engineering is the use of 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 range of more speciali ...
and
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
; the output of a
linear time-invariant In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly define ...
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 Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to ...
. The Laplace transform is invertible on a large class of functions. Given a simple mathematical or functional description of an input or output to a
system A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its boundaries, structure and purpose and express ...
, 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. The Laplace transform can also be used to solve differential equations and is used extensively in
mechanical engineering Mechanical engineering is the study of physical machines that may involve force and movement. It is an engineering branch that combines engineering physics and mathematics principles with materials science, to design, analyze, manufacture, a ...
and
electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
. 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 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 ve ...
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

Let $\mathcal\left\ = F\left(s\right)$. Then (see the table above) $\mathcal \left\ = \int_0^\infty \frace^\, dt = \int_s^\infty F(p)\, dp.$ In the limit $s \rightarrow 0$, one gets $\int_0^\infty \frac t \, dt = \int_0^\infty F(p)\, dp,$ provided that the interchange of limits can be justified. This is often possible as a consequence of the final value theorem. Even when the interchange cannot be justified the calculation can be suggestive. For example, with , proceeding formally one has The validity of this identity can be proved by other means. It is an example of a Frullani integral. Another example is
Dirichlet integral In mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of the sinc function over the positive real line: : \int_0^\inf ...
.

## Complex impedance of a capacitor

In the theory of
electrical circuit An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sources, c ...
s, the current flow in a
capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of a ...
is proportional to the capacitance and rate of change in the electrical potential (in SI units). Symbolically, this is expressed by the differential equation $i = C ,$ where is the capacitance (in
farad The farad (symbol: F) is the unit of electrical capacitance, the ability of a body to store an electrical charge, in the International System of Units (SI). It is named after the English physicist Michael Faraday (1791–1867). In SI base unit ...
s) of the capacitor, is the
electric current An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving par ...
(in
ampere The ampere (, ; symbol: A), often shortened to amp,SI supports only the use of symbols and deprecates the use of abbreviations for units. is the unit of electric current in the International System of Units (SI). One ampere is equal to elec ...
s) through the capacitor as a function of time, and is the
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge t ...
(in
volt The volt (symbol: V) is the unit of electric potential, electric potential difference (voltage), and electromotive force in the International System of Units (SI). It is named after the Italian physicist Alessandro Volta (1745–1827). Defin ...
s) across the terminals of the capacitor, also as a function of time. Taking the Laplace transform of this equation, we obtain $I(s) = C(s V(s) - V_0),$ where $\begin I(s) &= \mathcal \,\\ V(s) &= \mathcal \, \end$ and $V_0 = v(0).$ Solving for we have $V(s) = + .$ The definition of the complex impedance (in
ohm Ohm (symbol Ω) is a unit of electrical resistance named after Georg Ohm. Ohm or OHM may also refer to: People * Georg Ohm (1789–1854), German physicist and namesake of the term ''ohm'' * Germán Ohm (born 1936), Mexican boxer * Jörg Ohm ( ...
s) is the ratio of the complex voltage divided by the complex current while holding the initial state at zero: $Z(s) = \left. \_.$ Using this definition and the previous equation, we find: $Z(s) = \frac,$ which is the correct expression for the complex impedance of a capacitor. In addition, the Laplace transform has large applications in control theory.

## Partial fraction expansion

Consider a linear time-invariant system with
transfer function In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that theoretically models the system's output for each possible input. They are widely used ...
$H(s) = \frac.$ The
impulse response In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse (). More generally, an impulse response is the reac ...
is simply the inverse Laplace transform of this transfer function: $h(t) = \mathcal^\.$ To evaluate this inverse transform, we begin by expanding using the method of partial fraction expansion, $\frac = + .$ The unknown constants and 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 , we multiply both sides of the equation by to get $\frac = P + .$ Then by letting , the contribution from vanishes and all that is left is $P = \left.\_ = .$ Similarly, the residue is given by $R = \left.\_ = .$ Note that $R = = - P$ and so the substitution of and into the expanded expression for gives $H(s) = \left(\frac \right) \cdot \left( - \right).$ 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 to obtain $h(t) = \mathcal^\ = \frac\left(e^ - e^\right),$ 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 of and . That is, the inverse of $H(s) = \frac = \frac \cdot \frac$ is $\mathcal^\! \left\ * \mathcal^\! \left\ = e^ * e^ = \int_0^t e^e^\, dx = \frac.$

## Phase delay

Starting with the Laplace transform, $X(s) = \frac$ we find the inverse by first rearranging terms in the fraction: $\begin X(s) &= \frac + \frac \\ &= \sin(\varphi) \left(\frac \right) + \cos(\varphi) \left(\frac \right). \end$ We are now able to take the inverse Laplace transform of our terms: $\begin x(t) &= \sin(\varphi) \mathcal^\left\ + \cos(\varphi) \mathcal^\left\ \\ &= \sin(\varphi)\cos(\omega t) + \cos(\varphi)\sin(\omega t). \end$ This is just the sine of the sum of the arguments, yielding: $x(t) = \sin (\omega t + \varphi).$ We can apply similar logic to find that $\mathcal^ \left\ = \cos.$

## Statistical mechanics

In
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...
, the Laplace transform of the density of states $g\left(E\right)$ defines the partition function. That is, the canonical partition function $Z\left(\beta\right)$ is given by $Z(\beta) = \int_0^\infty e^g(E)\,dE$ and the inverse is given by $g(E) = \frac \int_^ e^Z(\beta) \, d\beta$

## 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 Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galax ...
source of
radiofrequency Radio frequency (RF) is the oscillation rate of an alternating electric current or voltage or of a magnetic, electric or electromagnetic field or mechanical system in the frequency range from around to around . This is roughly between the upper ...
thermal radiation Thermal radiation is electromagnetic radiation generated by the thermal motion of particles in matter. Thermal radiation is generated when heat from the movement of charges in the material (electrons and protons in common forms of matter) is ...
too distant to resolve as more than a point, given its
flux density Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport p ...
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
, 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., and
When independent information on the structure of an object is available, the inverse Laplace transform method has been found to be in good agreement.

# Gallery

*
Analog signal processing Analog signal processing is a type of signal processing conducted on continuous analog signals by some analog means (as opposed to the discrete digital signal processing where the signal processing is carried out by a digital process). "Analog" ind ...
* Bernstein's theorem on monotone functions * Continuous-repayment mortgage * Hamburger moment problem * Hardy–Littlewood tauberian theorem * Laplace–Carson transform * Moment-generating function * Nonlocal operator * Post's inversion formula * Signal-flow graph

# References

* * * * * * *

## Historical

* * * , Chapters 3–5 * * *

* . * * * * * Mathews, Jon; Walker, Robert L. (1970), ''Mathematical methods of physics'' (2nd ed.), New York: W. A. Benjamin, * * * - See Chapter VI. The Laplace transform. * *

*
Online Computation
of the transform or inverse transform, wims.unice.fr

at EqWorld: The World of Mathematical Equations. *
Good explanations of the initial and final value theorems

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. {{DEFAULTSORT:Laplace Transform Differential equations Fourier analysis Mathematical physics Integral transforms