Laplace transform

In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable formula_1 (often time) to a function of a complex variable formula_2 (complex frequency). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms differential equations into algebraic equations and convolution into multiplication.
History.
The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace, who used a similar transform in his work on probability theory. 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. The theory was further developed in the 19th and early 20th centuries by Mathias Lerch, Oliver Heaviside, and Thomas Bromwich.
The current widespread use of the transform (mainly in engineering) 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.
From 1744, Leonhard Euler 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 functions, 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 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.
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 number frequency parameter
An alternate notation for the Laplace transform is formula_7 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 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 is multiplied by the Heaviside step function.
The bilateral Laplace transform is defined as follows:
An alternate notation for the bilateral Laplace transform is formula_11, instead of formula_12.
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 (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 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
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 cumulative distribution function 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 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.
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 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 systems. The most significant advantage is that differentiation becomes multiplication, and integration becomes division, by (reminiscent of the way logarithms 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 equations 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:
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 (see Z transform). 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, 
This is of special significance in probability theory, where the moments of a random variable are given by the expectation values formula_30. Then, the relation holds
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
and in the bilateral case,
The general result
where formula_36 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 formula_38 in a right neighbourhood of formula_39 and on the decay rate of formula_38 in a left neighbourhood of formula_41. The above formula is a variation of integration by parts, with the operators 
formula_42 and formula_43 being replaced by formula_44 and formula_45. Let us prove the equivalent formulation:
By plugging in formula_47 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 (unilateral) 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.
The Laplace transform is similar to the Fourier 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 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. The continuous Fourier transform is equivalent to evaluating the bilateral Laplace transform with imaginary argument or when the condition explained below is fulfilled,
This definition of the Fourier transform requires a prefactor of on the reverse Fourier transform. This relationship between the Laplace and Fourier transforms is often used to determine the frequency spectrum of a signal or dynamical system.
The above relation is valid as stated if and only if the region of convergence (ROC) of 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
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
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 period (in units of time e.g., seconds) and is the sampling rate (in samples per second or hertz).
Let
be a sampling impulse train (also called a Dirac comb) and
be the sampled representation of the continuous-time 
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 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,
Using this linearity, and various trigonometric, hyperbolic, and complex number (etc.) properties and/or identities, some Laplace transforms can be obtained from others more quickly than by using the definition directly.
The unilateral Laplace transform takes as input a function whose time domain is the non-negative reals, which is why all of the time domain functions in the table below are multiples of the Heaviside step function, .
The entries of the table that involve a time delay are required to be causal (meaning that ). 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.
 \ </math> 
"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 formula_87. Then (see the table above)
In the limit formula_89, 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
The validity of this identity can be proved by other means. It is an example of a Frullani integral.
Another example is Dirichlet integral.
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 (in SI units). Symbolically, this is expressed by the differential equation
where is the capacitance (in farads) of the capacitor, is the electric current (in amperes) through the capacitor as a function of time, and is the voltage (in volts) 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 (in ohms) 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.
Partial fraction expansion.
Consider a linear time-invariant system with transfer function
The impulse response is simply the inverse Laplace transform of this transfer function:
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 (see "Item" #"3" in the "Table of Laplace Transforms", above), we can take the inverse Laplace transform of to obtain
which is the impulse response of the system.
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
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 formula_115 defines the partition function. That is, the canonical partition function formula_116 is given by
and the inverse is given by