In mathematics, the Laplace transform is an integral transform named after its inventor PierreSimon Laplace (/ləˈplɑːs/). It transforms a function of a real variable t (often time) to a function of a complex variable s (complex frequency). The transform has many applications in science and engineering.
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. Laplace transforms are usually restricted to functions of t with t ≥ 0. A consequence of this restriction is that the Laplace transform of a function is a holomorphic function of the variable s. Unlike the Fourier transform, the Laplace transform of a distribution is generally a wellbehaved 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 Laplace transform is invertible on a large class of functions. The inverse Laplace transform takes a function of a complex variable s (often frequency) and yields a function of a real variable t (often time). 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.^{[1]} So, for example, Laplace transformation from the time domain to the frequency domain transforms differential equations into algebraic equations and convolution into multiplication.
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 ^{[2]}.
The Laplace transform is named after mathematician and astronomer PierreSimon Laplace, who used a similar transform in his work on probability theory.^{[3]} Laplace's use of generating functions was similar to what is now known as the ztransform and he gave little attention to the continuous variable case which was discussed by Niels Henrik Abel.^{[4]} The theory was further developed in the 19th and early 20th centuries by Mathias Lerch,^{[5]} Oliver Heaviside,^{[6]} and Thomas Bromwich.^{[7]} The current widespread use of the transform (mainly in engineering) came about during and soon after World War II^{[8]} replacing the earlier Heaviside operational calculus. The advantages of the Laplace transform had been emphasized by Gustav Doetsch^{[9]} to whom the name Laplace Transform is apparently due.
The early history of methods having some similarity to Laplace transform is as follows. From 1744, Leonhard Euler investigated integrals of the form
as solutions of differential equations but did not pursue the matter very far.^{[10]}
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.^{[11]}^{[12]}^{[clarification needed]}
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.^{[13]} However, in 1785, Laplace took the critical step forward when, rather than just 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.^{[14]}
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.^{[15]}
The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the function F(s), which is a unilateral transform defined by


(Eq.1) 
where s is a complex number frequency parameter
An alternate notation for the Laplace transform is instead of F.
The meaning of the integral depends on types of functions of interest. A necessary condition for existence of the integral is that f must be locally integrable on [0, ∞). 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 μ by the Lebesgue integral^{[16]}
An important special case is where μ is a probability measure, for example, the Dirac delta function. In operational calculus, the Laplace transform of a measure is often treated as though the measure came from a probability density function f. In that case, to avoid potential confusion, one often writes
where the lower limit of 0^{−} is shorthand notation for
This limit emphasizes that any point mass located at 0 is entirely captured by the Laplace transform. Although with the Lebesgue integral, it is not necessary to take such a limit, it does appear more naturally in connection with the Laplace–Stieltjes transform.
When one says "the Laplace transform" without qualification, the unilateral or onesided transform is normally intended. The Laplace transform can be alternatively defined as the bilateral Laplace transform or twosided 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: F(s), which is a unilateral transform defined by


(Eq.2) 
An alternate notation for the bilateral Laplace transform is instead of .
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 onetoone mapping from one function space into another in many other function spaces as well, although there is usually no easy characterization of the range. Typical function spaces in which this is true include the spaces of bounded continuous functions, the space L^{∞}(0, ∞), or more generally tempered distributions on (0, ∞). 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):


(Eq.3) 
where γ is a real number so that the contour path of integration is in the region of convergence of F(s). An alternative formula for the inverse Laplace transform is given by Post's inversion formula. The limit here is interpreted in the weak* topology.
In practice, it is typically more convenient to decompose a Laplace transform into known transforms of functions obtained from a table, and construct the inverse by inspection.
In pure and applied probability, the Laplace transform is defined as an expected value. If X is a random variable with probability density function f, then the Laplace transform of f is given by the expectation
By convention, this is referred to as the Laplace transform of the random variable X itself. Replacing s by −t gives the moment generating function of X. The Laplace transform has applications throughout probability theory, including first passage times of stochastic processes such as Markov chains, and renewal theory.
Of particular use is the ability to recover the cumulative distribution function of a continuous random variable X by means of the Laplace transform as follows^{[17]}
If f is a locally integrable function (or more generally a Borel measure locally of bounded variation), then the Laplace transform F(s) of f converges provided that the limit
exists.
The Laplace transform converges absolutely if the integral
exists (as a proper Lebesgue integral). The Laplace transform is usually understood as conditionally convergent, meaning that it converges in the former instead of the latter sense.
The set of values for which F(s) converges absolutely is either of the form Re(s) > a or else Re(s) ≥ a, where a is an extended real constant, −∞ ≤ a ≤ ∞. (This follows from the dominated convergence theorem.) The constant a is known as the abscissa of absolute convergence, and depends on the growth behavior of f(t).^{[18]} Analogously, the twosided transform converges absolutely in a strip of the form a < Re(s) < b, and possibly including the lines Re(s) = a or Re(s) = b.^{[19]} The subset of values of s for which the Laplace transform converges absolutely is called the region of absolute convergence or the domain of absolute convergence. In the twosided case, it is sometimes called the strip of absolute convergence. The Laplace transform is analytic in the region of absolute convergence: this is a consequence of Fubini's theorem and Morera's theorem.
Similarly, the set of values for which F(s) converges (conditionally or absolutely) is known as the region of conditional convergence, or simply the region of convergence (ROC). If the Laplace transform converges (conditionally) at s = s_{0}, then it automatically converges for all s with Re(s) > Re(s_{0}). Therefore, the region of convergence is a halfplane of the form Re(s) > a, possibly including some points of the boundary line Re(s) = a.
In the region of convergence Re(s) > Re(s_{0}), the Laplace transform of f can be expressed by integrating by parts as the integral
That is, in the region of convergence F(s) can effectively be expressed 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 f and the properties of the Laplace transform within the region of convergence.
In engineering applications, a function corresponding to a linear timeinvariant (LTI) system is stable if every bounded input produces a bounded output. This is equivalent to the absolute convergence of the Laplace transform of the impulse response function in the region Re(s) ≥ 0. As a result, LTI systems are stable provided 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.
The Laplace transform has a number of properties that make it useful for analyzing linear dynamical systems. The most significant advantage is that differentiation and integration become multiplication and division, respectively, by s (similarly to logarithms changing multiplication of numbers to addition of their logarithms).
Because of this property, the Laplace variable s is also known as operator variable in the L domain: either derivative operator or (for s^{−1}) 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 f(t) and g(t), and their respective Laplace transforms F(s) and G(s),
The following table is a list of properties of unilateral Laplace transform:^{[20]}
Time domain  s domain  Comment  

Linearity  Can be proved using basic rules of integration.  
Frequencydomain derivative  F′ is the first derivative of F with respect to s.  
Frequencydomain general derivative  More general form, nth derivative of F(s).  
Derivative  f is assumed to be a differentiable function, and its derivative is assumed to be of exponential type. This can then be obtained by integration by parts  
Second derivative  f is assumed twice differentiable and the second derivative to be of exponential type. Follows by applying the Differentiation property to f′(t).  
General derivative  f is assumed to be ntimes differentiable, with nth derivative of exponential type. Follows by mathematical induction.  
Frequencydomain integration  This is deduced using the nature of frequency differentiation and conditional convergence.  
Timedomain integration  u(t) is the Heaviside step function and (u ∗ f)(t) is the convolution of u(t) and f(t).  
Frequency shifting  
Time shifting  u(t) is the Heaviside step function  
Time scaling  
Multiplication  The integration is done along the vertical line Re(σ) = c that lies entirely within the region of convergence of F.^{[21]}  
Convolution  
Complex conjugation  
Crosscorrelation  
Periodic function  f(t) is a periodic function of period T so that f(t) = f(t + T), for all t ≥ 0. This is the result of the time shifting property and the geometric series. 
The Laplace transform can be viewed as a continuous analogue of a power series.^{[22]} If a(n) is a discrete function of a positive integer n, then the power series associated to a(n) is the series
where x is a real variable (see Z transform). Replacing summation over n with integration over t, a continuous version of the power series becomes
where the discrete function a(n) is replaced by the continuous one f(t).
Changing the base of the power from x to e gives
For this to converge for, say, all bounded functions f, it is necessary to require that ln x < 0. Making the substitution −s = ln x gives just the Laplace transform:
In other words, the Laplace transform is a continuous analog of a power series in which the discrete parameter n is replaced by the continuous parameter t, and x is replaced by e^{−s}.
The quantities
are the moments of the function f. If the first n moments of f converge absolutely, then by repeated differentiation under the integral,
This is of special significance in probability theory, where the moments of a random variable X are given by the expectation values . Then, the relation holds
It is often convenient to use the differentiation property of the Laplace transform to find the transform of a function's derivative. This can be derived from the basic expression for a Laplace transform as follows:
yielding
and in the bilateral case,
The general result
where denotes the n^{th} derivative of f, can then be established with an inductive argument.
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 lefthand side turns into:
but assuming Fubini's theorem holds, by reversing the order of integration we get the wanted righthand side.
The (unilateral) Laplace–Stieltjes transform of a function g : R → R is defined by the Lebesgue–Stieltjes integral
The function g is assumed to be of bounded variation. If g is the antiderivative of f:
then the Laplace–Stieltjes transform of g and the Laplace transform of f coincide. In general, the Laplace–Stieltjes transform is the Laplace transform of the Stieltjes measure associated to g. So in practice, the only distinction between the two transforms is that the Laplace transform is thought of as operating on the density function of the measure, whereas the Laplace–Stieltjes transform is thought of as operating on its cumulative distribution function.^{[23]}
The continuous Fourier transform is equivalent to evaluating the bilateral Laplace transform with imaginary argument s = iω or s = 2πfi^{[24]} when the condition explained below is fulfilled,
This definition of the Fourier transform requires a prefactor of 1/(2π) 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 F(s) contains the imaginary axis, σ = 0.
For example, the function f(t) = cos(ω_{0}t) has a Laplace transform F(s) = s/(s^{2} + ω_{0}^{2}) whose ROC is Re(s) > 0. As s = iω is a pole of F(s), substituting s = iω in F(s) does not yield the Fourier transform of f(t)u(t), which is proportional to the Dirac deltafunction δ(ω − ω_{0}).
However, a relation of the form
holds under much weaker conditions. For instance, this holds for the above example provided that the limit is understood as a weak limit of measures (see vague topology). General conditions relating the limit of the Laplace transform of a function on the boundary to the Fourier transform take the form of Paley–Wiener theorems.
The Mellin transform and its inverse are related to the twosided Laplace transform by a simple change of variables.
If in the Mellin transform
we set θ = e^{−t} we get a twosided Laplace transform.
The unilateral or onesided Ztransform is simply the Laplace transform of an ideally sampled signal with the substitution of
where T = 1/f_{s} is the sampling period (in units of time e.g., seconds) and f_{s} is the sampling rate (in samples per second or hertz).
Let
be a sampling impulse train (also called a Dirac comb) and
be the sampled representation of the continuoustime x(t)
The Laplace transform of the sampled signal x_{q}(t) is
This is the precise definition of the unilateral Ztransform of the discrete function x[n]
with the substitution of z → e^{sT}.
Comparing the last two equations, we find the relationship between the unilateral Ztransform and the Laplace transform of the sampled signal,
The similarity between the Z and Laplace transforms is expanded upon in the theory of time scale calculus.
The integral form of the Borel transform
is a special case of the Laplace transform for f an entire function of exponential type, meaning that
for some constants A and B. The generalized Borel transform allows a different weighting function to be used, rather than the exponential function, to transform functions not of exponential type. Nachbin's theorem gives necessary and sufficient conditions for the Borel transform to be well defined.
Since an ordinary Laplace transform can be written as a special case of a twosided transform, and since the twosided transform can be written as the sum of two onesided transforms, the theory of the Laplace, Fourier, Mellin, and Ztransforms 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.
The following table provides Laplace transforms for many common functions of a single variable.^{[25]}^{[26]} 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 nonnegative reals, which is why all of the time domain functions in the table below are multiples of the Heaviside step function, u(t).
The entries of the table that involve a time delay τ are required to be causal (meaning that τ > 0). A causal system is a system where the impulse response h(t) is zero for all time t prior to t = 0. In general, the region of convergence for causal systems is not the same as that of anticausal systems.
Function  Time domain 
Laplace sdomain 
Region of convergence  Reference  

unit impulse  all s  inspection  
delayed impulse  time shift of unit impulse  
unit step  Re(s) > 0  integrate unit impulse  
delayed unit step  Re(s) > 0  time shift of unit step  
ramp  Re(s) > 0  integrate unit impulse twice  
nth power (for integer n) 
Re(s) > 0 (n > −1) 
Integrate unit step n times  
qth power (for complex q) 
Re(s) > 0 Re(q) > −1 
^{[27]}^{[28]}  
nth root  Re(s) > 0  Set q = 1/n above.  
nth power with frequency shift  Re(s) > −α  Integrate unit step, apply frequency shift  
delayed nth power with frequency shift 
Re(s) > −α  Integrate unit step, apply frequency shift, apply time shift  
exponential decay  Re(s) > −α  Frequency shift of unit step  
twosided exponential decay (only for bilateral transform) 
−α < Re(s) < α  Frequency shift of unit step  
exponential approach  Re(s) > 0  Unit step minus exponential decay  
sine  Re(s) > 0  Bracewell 1978, p. 227  
cosine  Re(s) > 0  Bracewell 1978, p. 227  
hyperbolic sine  Re(s) > α  Williams 1973, p. 88  
hyperbolic cosine  Re(s) > α  Williams 1973, p. 88  
exponentially decaying sine wave 
Re(s) > −α  Bracewell 1978, p. 227  
exponentially decaying cosine wave 
Re(s) > −α  Bracewell 1978, p. 227  
natural logarithm  Re(s) > 0  Williams 1973, p. 88  
Bessel function of the first kind, of order n 
Re(s) > 0 (n > −1) 
Williams 1973, p. 89  
Error function  Re(s) > 0  Williams 1973, p. 89  
Explanatory notes:

The Laplace transform is often used in circuit analysis, and simple conversions to the sdomain 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 sdomain. 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 sdomain account for that.
The equivalents for current and voltage sources are simply derived from the transformations in the table above.
The Laplace transform is used frequently in engineering and physics; the output of a linear timeinvariant 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 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. The 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.
Let , then (see the table above)
or
Letting s → 0, gives one the identity
provided that the interchange of limits can be justified. Even when the interchange cannot be justified the calculation can be suggestive. For example, with a ≠ 0 ≠ b, 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 nuclear physics, the following fundamental relationship governs radioactive decay: the number of radioactive atoms N in a sample of a radioactive isotope decays at a rate proportional to N. This leads to the first order linear differential equation
where λ is the decay constant. The Laplace transform can be used to solve this equation.
Rearranging the equation to one side, we have
Next, we take the Laplace transform of both sides of the equation:
where
and
Solving, we find
Finally, we take the inverse Laplace transform to find the general solution
which is indeed the correct form for radioactive decay.
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 C is the capacitance (in farads) of the capacitor, i = i(t) is the electric current (in amperes) through the capacitor as a function of time, and v = v(t) 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 V(s) we have
The definition of the complex impedance Z (in ohms) is the ratio of the complex voltage V divided by the complex current I while holding the initial state V_{0} at zero:
Using this definition and the previous equation, we find:
which is the correct expression for the complex impedance of a capacitor. In addition, the Laplace transform has large applications in control theory.
Consider a linear timeinvariant 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 H(s) using the method of partial fraction expansion,
The unknown constants P and R are the residues located at the corresponding poles of the transfer function. Each residue represents the relative contribution of that singularity to the transfer function's overall shape.
By the residue theorem, the inverse Laplace transform depends only upon the poles and their residues. To find the residue P, we multiply both sides of the equation by s + α to get
Then by letting s = −α, the contribution from R vanishes and all that is left is
Similarly, the residue R is given by
Note that
and so the substitution of R and P into the expanded expression for H(s) gives
Finally, using the linearity property and the known transform for exponential decay (see Item #3 in the Table of Laplace Transforms, above), we can take the inverse Laplace transform of H(s) to obtain
which is the impulse response of the system.
The same result can be achieved using the convolution property as if the system is a series of filters with transfer functions of 1/(s + a) and 1/(s + b). That is, the inverse of
is
Time function  Laplace transform 

Starting with the Laplace transform,
we find the inverse by first rearranging terms in the fraction:
We are now able to take the inverse Laplace transform of our terms:
This is just the sine of the sum of the arguments, yielding:
We can apply similar logic to find that
The wide and general applicability of the Laplace transform and its inverse is illustrated by an application in astronomy which provides some information on the spatial distribution of matter of an astronomical source of radiofrequency thermal radiation too distant to resolve as more than a point, given its flux density spectrum, rather than relating the time domain with the spectrum (frequency domain).
Assuming certain properties of the object, e.g. spherical shape and constant temperature, calculations based on carrying out an inverse Laplace transformation on the spectrum of the object can produce the only possible model of the distribution of matter in it (density as a function of distance from the center) consistent with the spectrum.^{[29]} When independent information on the structure of an object is available, the inverse Laplace transform method has been found to be in good agreement.
In statistical mechanics, the Laplace transform of the density of states defines the partition function.^{[30]} That is, the partition function is given by
and the inverse is given by
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