In mathematics, the Laplace transform is a widely used integral transform. It has many important applications in mathematics, physics, engineering, and probability theory.
The Laplace transform is related to the Fourier transform, but whereas the Fourier transform resolves a function or signal into its modes of vibration, the Laplace transform resolves a function into its moments. Like the Fourier transform, the Laplace transform is used for solving differential and integral equations. In physics and engineering, it is used for analysis of linear timeinvariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. In this analysis, the Laplace transform is often interpreted as a transformation from the timedomain, in which inputs and outputs are functions of time, to the frequencydomain, where the same inputs and outputs are functions of complex angular frequency, in radians per unit 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.
Denoted , it is a linear operator on a function f(t) (original) with a real argument t (t ≥ 0) that transforms it to a function F(s) (image) with a complex argument s. This transformation is essentially bijective for the majority of practical uses; the respective pairs of f(t) and F(s) are matched in tables. The Laplace transform has the useful property that many relationships and operations over the originals f(t) correspond to simpler relationships and operations over the images F(s).^{[1]}
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The Laplace transform is named in honor of mathematician and astronomer PierreSimon Laplace, who used the transform in his work on probability theory. From 1744, Leonhard Euler investigated integrals of the form
— as solutions of differential equations but did not pursue the matter very far.^{[2]} 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.^{[3]}^{[4]} 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.^{[5]} 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.^{[6]}
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 as the solutions were periodic. In 1809, Laplace applied his transform to find solutions that diffused indefinitely in space.^{[7]}
The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the function F(s), defined by:
The parameter s is a complex number:
The meaning of the integral depends on types of functions of interest. For functions that decay at infinity or are of exponential type, it can be understood as 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^{[8]}
An important special case is where μ is a probability measure or, even more specifically, the Dirac delta function. In operational calculus, the Laplace transform of a measure is often treated as though the measure came from a distribution function ƒ. In that case, to avoid potential confusion, one often writes
where the lower limit of 0^{−} is short notation to mean
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.
In pure and applied probability, the Laplace transform is defined by means of an expectation value. If X is a random variable with probability density function ƒ, then the Laplace transform of ƒ is given by the expectation
By abuse of language, one often refers instead to this as the Laplace transform of the random variable X. 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.
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:
The inverse Laplace transform is given by the following complex integral, which is known by various names (the Bromwich integral, the FourierMellin integral, and Mellin's inverse formula):
where γ is a real number so that the contour path of integration is in the region of convergence of F(s) normally requiring γ > Re(s_{p}) for every singularity s_{p} of F(s) and i^{2} = −1. If all singularities are in the left halfplane, that is Re(s_{p}) < 0 for every s_{p}, then γ can be set to zero and the above inverse integral formula becomes identical to the inverse Fourier transform.
An alternative formula for the inverse Laplace transform is given by Post's inversion formula.
If ƒ is a locally integrable function (or more generally a Borel measure locally of bounded variation), then the Laplace transform F(s) of ƒ converges provided that the limit
exists. The Laplace transform converges absolutely if the integral
exists (as 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 ƒ(t).^{[9]} 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.^{[10]} 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.
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 ƒ 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.
A variety of theorems, in the form of Paley–Wiener theorems, exist concerning the relationship between the decay properties of ƒ 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.
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. (This is similar to the way that logarithms change an operation of multiplication of numbers to addition of their logarithms.) This changes integral equations and differential equations to polynomial equations, which are much easier to solve. Once solved, use of the inverse Laplace transform reverts back to the time 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:
Time domain  's' domain  Comment  

Linearity  Can be proved using basic rules of integration.  
Frequency differentiation  is the first derivative of .  
Frequency differentiation  More general form, (n)th derivative of F(s).  
Differentiation  ƒ 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 Differentiation  ƒ is assumed twice differentiable and the second derivative to be of exponential type. Follows by applying the Differentiation property to .  
General Differentiation  ƒ is assumed to be ntimes differentiable, with n^{th} derivative of exponential type. Follow by mathematical induction.  
Frequency integration  
Integration  u(t) is the Heaviside step function. Note (u * f)(t) is the convolution of u(t) and f(t).  
Scaling  
Frequency shifting  
Time shifting  u(t) is the Heaviside step function  
Convolution  ƒ(t) and g(t) are extended by zero for t < 0 in the definition of the convolution.  
Periodic Function  f(t) is a periodic function of period T so that . This is the result of the time shifting property and the geometric series. 
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, we have
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 ƒ:
then the Laplace–Stieltjes transform of g and the Laplace transform of ƒ 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.^{[11]}
The continuous Fourier transform is equivalent to evaluating the bilateral Laplace transform with complex argument s = iω or s = 2πfi:
This expression excludes the scaling factor , which is often included in definitions of the 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)u(t) has a Laplace transform F(s) = s/(s^{2} + ω_{0}^{2}) whose ROC is Re(s) > 0. Therefore, substituting s = iω in F(s) does not yield the Fourier transform of f(t) = cos(ω_{0}t).
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 PaleyWiener 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
Let
be a sampling impulse train (also called a Dirac comb) and
be the continuoustime representation of the sampled
The Laplace transform of the sampled signal is
This is precisely the definition of the unilateral Ztransform of the discrete function
with the substitution of .
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 ƒ 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. For definitions and explanations, see the Explanatory Notes at the end of the table.
Because the Laplace transform is a linear operator:
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.
ID  Function  Time domain 
Laplace sdomain 
Region of convergence  

1  ideal delay  
1a  unit impulse  1  
2  delayed nth power with frequency shift 

2a  nth power ( for integer n ) 

2a.1  qth power ( for complex q ) 

2a.2  unit step  
2b  delayed unit step  
2c  ramp  
2d  nth power with frequency shift  
2d.1  exponential decay  
3  exponential approach  
4  sine  
5  cosine  
6  hyperbolic sine  
7  hyperbolic cosine  
8  Exponentiallydecaying sine wave 

9  Exponentiallydecaying cosine wave 

10  nth root  
11  natural logarithm  
12  Bessel function of the first kind, of order n 

13  Modified Bessel function of the first kind, of order n 

14  Bessel function of the second kind, of order 0 

15  Modified Bessel function of the second kind, of order 0 

16  Error function  
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 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 can also be used to solve differential equations and is used extensively in 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.
Consider the following firstorder, linear differential equation:
This equation is the fundamental relationship describing radioactive decay, where
represents the number of undecayed atoms remaining in a sample of a radioactive isotope at time t (in seconds), and is the decay constant.
We can use the Laplace transform 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.
The constitutive relation governing the dynamic behavior of a capacitor is the following 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
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_{o} at zero:
Using this definition and the previous equation, we find:
which is the correct expression for the complex impedance of a capacitor.
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.
Time function  Laplace transform 

Starting with the Laplace transform
we find the inverse transform by first adding and subtracting the same constant α to the numerator:
By the shiftinfrequency property, we have
Finally, using the Laplace transforms for sine and cosine (see the table, above), we have
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:
To simplify this answer, we must recall the trigonometric identity that
and apply it to our value for x(t):
We can apply similar logic to find that
