Laplace transform

In mathematics, the Laplace transform, named after Pierre-Simon Laplace, is an integral transform that converts a function of a real variable to a function of a complex variable . 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.
The transform is useful for converting differentiation and integration in the time domain into much easier multiplication and division in the Laplace domain. This gives the transform many applications in science and engineering, mostly as a tool for solving linear differential equations and dynamical systems by simplifying ordinary differential equations and integral equations into algebraic polynomial equations, and by simplifying convolution into multiplication.
For example, through the Laplace transform, the equation of the simple harmonic oscillator is converted into the algebraic equation which incorporates the initial conditions and, and can be solved for the unknown function 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 by the integral
where s 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 has a convergent power series, 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.

History

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. Laplace wrote extensively about the use of generating functions, 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.
From 1744, Leonhard Euler investigated integrals of the form
as solutions of differential equations, introducing in particular the gamma function. 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.
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 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'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. 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.
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 simple to prove via Poisson summation, to the functional equation.
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. 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, Oliver Heaviside, and Thomas Bromwich.
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. Also during the 30s, the Laplace transform was instrumental in G H Hardy and John Edensor Littlewood's study of tauberian theorems, and this application was later expounded on by, 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.
The current widespread use of the transform came about during and soon after World War II, 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.

Formal definition

The Laplace transform of a function, defined for all real numbers, is the function, which is a unilateral transform defined by
where s is a complex frequency-domain parameter
with real numbers and.
An alternate notation for the Laplace transform is instead of. Thus in functional notation. This is often written, especially in engineering settings, as, with the understanding that the dummy variable does not appear in the function.
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 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 by the Lebesgue integral
An important special case is where 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. 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 includes being multiplied by the Heaviside step function.
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 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 :
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 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. If is a random variable with probability density function, then the Laplace transform of is given by the expectation
where is the expectation of random variable.
By convention, 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 processes such as Markov chains, and renewal theory.
Of particular use is the ability to recover the distribution function">distribution (mathematics)">distribution function of a continuous random variable by means of the Laplace transform as follows:

Algebraic construction

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.

Region of convergence

If is a locally integrable function, then the Laplace transform of 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 converges absolutely is either of the form or, where is an extended real constant with . 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 is known as the region of conditional convergence, or simply the region of convergence. If the Laplace transform converges 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. In its most general form, the Laplace transform gives a one-to-one correspondence between the holomorphic functions which, for some, are defined on and are bounded there in absolute value by a polynomial, and the distributions on the real line supported on which become tempered distributions after multiplied by for some.
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's key property is that it converts differentiation and integration in the time domain into multiplication and division by in the Laplace domain. Thus, the Laplace variable is also known as an operator variable in the Laplace domain: either the derivative operator or (for the integration operator.
Given the functions and, and their respective Laplace transforms and,
the following table is a list of properties of unilateral Laplace transform:

Property	Time domain	domain	Comment
Linearity			Can be proved using basic rules of integration.
Frequency-domain derivative			is the first derivative of with respect to.
Frequency-domain general derivative			More general form, th derivative of.
Derivative			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			is assumed twice differentiable and the second derivative to be of exponential type. Follows by applying the Differentiation property to.
General derivative			is assumed to be -times differentiable, with 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			is the Heaviside step function and is the convolution of and.
Frequency shifting
Time shifting			, is the Heaviside step function
Time scaling
Multiplication			The integration is done along the vertical line that lies entirely within the region of convergence of.
Convolution
Circular convolution			For periodic functions with period.
Complex conjugation
Periodic function			is a periodic function of period so that, for all. This is the result of the time shifting property and the geometric series.
Periodic summation

; Initial value theorem :
; Final value theorem :

Relation to power series

The Laplace transform can be viewed as a continuous analogue of a power series. If is a discrete function of a positive integer, then the power series associated to is the series
where is a real variable. 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.
Analogously to a power series, if, then the power series converges to an analytic function in, if, the Laplace transform converges to an analytic function for.

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,
This is of special significance in probability theory, where the moments of a random variable are given by the expectation values. Then, the relation holds

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
and in the bilateral case,
The general result
where 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:
under suitable assumptions on the behaviour of in a right neighbourhood of and on the decay rate of in a left neighbourhood of. The above formula is a variation of integration by parts, with the operators
and being replaced by and. 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

Laplace–Stieltjes transform

The Laplace–Stieltjes transform of a function is defined by the Lebesgue–Stieltjes integral
The function is assumed to be of bounded variation. If is the antiderivative of :
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.

Fourier transform

Let be a complex-valued Lebesgue integrable function supported on, and let be its Laplace transform. Then, within the region of convergence, we have
which is the Fourier transform of the function.
Indeed, the Fourier transform is a special case of the bilateral Laplace transform. The main difference is that the Fourier transform of a function is a complex function of a real variable, 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 of the variable. 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 when the condition explained below is fulfilled,
This convention of the Fourier transform 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 of a signal or dynamical system.
The above relation is valid as stated if and only if the region of convergence 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 contains terms proportional to the Dirac delta functions.
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. 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
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
where is the sampling interval and is the sampling rate.
Let
be a sampling impulse train 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,
The similarity between the Z- and Laplace transforms is expanded upon in the theory of time scale calculus.

Borel transform

The integral form of the Borel transform
is a special case of the Laplace transform for an entire function of exponential type, meaning that
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.
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 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,.
The entries of the table that involve a time delay are required to be causal. A causal system is a system where the impulse response 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 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 simple 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.
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

Let. Then
From which one gets:
In the limit, 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, proceeding formally one has

Complex impedance of a capacitor

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. Symbolically, this is expressed by the differential equation
where is the capacitance of the capacitor, is the electric current through the capacitor as a function of time, and 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 we have
The definition of the complex impedance is the ratio of the complex voltage divided by the complex current while holding the initial state 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.

Impulse response

Consider a linear time-invariant system with transfer function
The impulse response is simply the inverse Laplace transform of this transfer function:
; Partial fraction expansion :
To evaluate this inverse transform, we begin by expanding using the method of partial fraction expansion,
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
Then by letting, the contribution from vanishes and all that is left is
Similarly, the residue is given by
Note that
and so the substitution of and into the expanded expression for gives
Finally, using the linearity property and the known transform for exponential decay, we can take the inverse Laplace transform of 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 and. That is, the inverse of
is

Phase delay

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

In statistical mechanics, the Laplace transform of the density of states defines the partition function. That is, the canonical partition function is given by
and the inverse is given by

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 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.
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 consistent with the spectrum. 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

Consider a random walk, with steps occurring with probabilities. Suppose also that the time step is a Poisson process, with parameter. Then the probability of the walk being at the lattice point at time is
This leads to a system of integral 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.

Tauberian theory

The Laplace transform of the measure on is given by
It is intuitively clear that, for small, 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, to those of the distribution of as. They are thus of importance in asymptotic formulae of probability and statistics, where often the spectral side has asymptotics that are simpler to infer.
Two Tauberian theorems of note are the Hardy–Littlewood Tauberian theorem and Wiener's Tauberian theorem. The Wiener theorem generalizes the Ikehara Tauberian theorem, which is the following statement:
Let A be a non-negative, monotonic nondecreasing function of x, defined for. Suppose that
converges for ℜ > 1 to the function f and that, for some non-negative number c,
has an extension as a continuous function for.
Then the limit as x goes to infinity of is equal to c.
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.

Modern

Historical

, Chapters 3–5