Laplace transform
In mathematics, the Laplace transform, named after its discoverer PierreSimon Laplace (/ləˈplɑːs/), is an integral transform that converts a function of a real variable (usually [math]\displaystyle{ t }[/math], in the time domain) to a function of a complex variable [math]\displaystyle{ s }[/math] (in the complexvalued frequency domain, also known as sdomain, or splane).
The transform is useful for converting differentiation and integration in the time domain into much easier multiplication and division in the Laplace domain (analogous to how logarithms are useful for simplifying multiplication and division into addition and subtraction). This gives the transform many applications in science and engineering, mostly as a tool for solving linear differential equations^{[1]} and dynamical systems by simplifying ordinary differential equations and integral equations into algebraic polynomial equations, and by simplifying convolution into multiplication.^{[2]}^{[3]} Once solved, the inverse Laplace transform reverts back to the original domain.
The Laplace transform is defined (for suitable functions f) by the integral:[math]\displaystyle{ \mathcal{L}\{f\}(s) = \int_0^\infty f(t)e^{st} \, dt. }[/math]
History
The Laplace transform is named after mathematician and astronomer PierreSimon, Marquis de Laplace, who used a similar transform in his work on probability theory.^{[4]} Laplace wrote extensively about the use of generating functions (1814), and the integral form of the Laplace transform evolved naturally as a result.^{[5]}
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.^{[6]} The theory was further developed in the 19th and early 20th centuries by Mathias Lerch,^{[7]} Oliver Heaviside,^{[8]} and Thomas Bromwich.^{[9]}
The current widespread use of the transform (mainly in engineering) came about during and soon after World War II,^{[10]} replacing the earlier Heaviside operational calculus. The advantages of the Laplace transform had been emphasized by Gustav Doetsch,^{[11]} to whom the name Laplace transform is apparently due.
From 1744, Leonhard Euler investigated integrals of the form [math]\displaystyle{ z = \int X(x) e^{ax}\, dx \quad\text{ and }\quad z = \int X(x) x^A \, dx }[/math] as solutions of differential equations, but did not pursue the matter very far.^{[12]} JosephLouis Lagrange was an admirer of Euler and, in his work on integrating probability density functions, investigated expressions of the form [math]\displaystyle{ \int X(x) e^{ a x } a^x\, dx, }[/math] which some modern historians have interpreted within modern Laplace transform theory.^{[13]}^{[14]}^{[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.^{[15]} 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 [math]\displaystyle{ \int x^s \varphi (x)\, dx, }[/math] 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.^{[16]}
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.^{[17]}
Formal definition
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
[math]\displaystyle{ F(s) =\int_0^\infty f(t)e^{st} \, dt }[/math] 

( ) 
where s is a complex frequency domain parameter [math]\displaystyle{ s = \sigma + i \omega, }[/math] with real numbers σ and ω.
An alternate notation for the Laplace transform is [math]\displaystyle{ \mathcal{L}\{f\} }[/math] instead of F.^{[3]}
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 ([math]\displaystyle{ f(t)\le Ae^{Bt} }[/math]), 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^{[18]} [math]\displaystyle{ \mathcal{L}\{\mu\}(s) = \int_{[0,\infty)} e^{st}\, d\mu(t). }[/math]
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 [math]\displaystyle{ \mathcal{L}\{f\}(s) = \int_{0^}^\infty f(t)e^{st} \, dt, }[/math] where the lower limit of 0^{−} is shorthand notation for [math]\displaystyle{ \lim_{\varepsilon\rightarrow 0^+}\int_{\varepsilon}^\infty. }[/math]
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.
Bilateral Laplace transform
When one says "the Laplace transform" without qualification, the unilateral or onesided transform is usually 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 F(s) is defined as follows:
[math]\displaystyle{ F(s) = \int_{\infty}^\infty e^{st} f(t)\, dt }[/math] 

( ) 
An alternate notation for the bilateral Laplace transform is [math]\displaystyle{ \mathcal{B}\{f\} }[/math], instead of F.
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 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):
[math]\displaystyle{ f(t) = \mathcal{L}^{1}\{F\}(t) = \frac{1}{2 \pi i} \lim_{T\to\infty}\int_{\gamma  i T}^{\gamma + i T} e^{st} F(s)\, ds }[/math] 

( ) 
where γ is a real number so that the contour path of integration is in the region of convergence of F(s). 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 X is a random variable with probability density function f, then the Laplace transform of f is given by the expectation [math]\displaystyle{ \mathcal{L}\{f\}(s) = \operatorname{E}\! \left[e^{sX} \right]\! . }[/math]
By convention, this is referred to as the Laplace transform of the random variable X itself. Here, 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:^{[19]} [math]\displaystyle{ F_X(x) = \mathcal{L}^{1}\! \left\{\frac{1}{s}\operatorname{E}\left[e^{sX}\right]\right\}\! (x) = \mathcal{L}^{1}\! \left\{\frac{1}{s}\mathcal{L}\{f\}(s)\right\}\! (x). }[/math]
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 halfline. 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 (avoiding the related difficulties with proving convergence).^{[20]}
Region of convergence
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 [math]\displaystyle{ \lim_{R\to\infty}\int_0^R f(t)e^{st}\,dt }[/math] exists.
The Laplace transform converges absolutely if the integral [math]\displaystyle{ \int_0^\infty \leftf(t)e^{st}\right\,dt }[/math] 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 F(s) converges absolutely is either of the form Re(s) > a or Re(s) ≥ a, where a is an extended real constant with −∞ ≤ a ≤ ∞ (a consequence of the dominated convergence theorem). The constant a is known as the abscissa of absolute convergence, and depends on the growth behavior of f(t).^{[21]} 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.^{[22]} 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 [math]\displaystyle{ F(s) = (ss_0)\int_0^\infty e^{(ss_0)t}\beta(t)\,dt, \quad \beta(u) = \int_0^u e^{s_0t}f(t)\,dt. }[/math]
That is, F(s) 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 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 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 is converts differentiation and integration in the time domain into multiplication and division s in the Laplace domain. Thus, the Laplace variable s is also known as operator variable in the Laplace domain: either the derivative operator or (for s^{−1}) the integration operator.
Given the functions f(t) and g(t), and their respective Laplace transforms F(s) and G(s), [math]\displaystyle{ \begin{align} f(t) &= \mathcal{L}^{1}\{F\}(s),\\ g(t) &= \mathcal{L}^{1}\{G\}(s), \end{align} }[/math]
the following table is a list of properties of unilateral Laplace transform:^{[23]}
Property  Time domain  s domain  Comment 

Linearity  [math]\displaystyle{ a f(t) + b g(t) \ }[/math]  [math]\displaystyle{ a F(s) + b G(s) \ }[/math]  Can be proved using basic rules of integration. 
Frequencydomain derivative  [math]\displaystyle{ t f(t) \ }[/math]  [math]\displaystyle{ F'(s) \ }[/math]  F′ is the first derivative of F with respect to s. 
Frequencydomain general derivative  [math]\displaystyle{ t^{n} f(t) \ }[/math]  [math]\displaystyle{ (1)^{n} F^{(n)}(s) \ }[/math]  More general form, nth derivative of F(s). 
Derivative  [math]\displaystyle{ f'(t) \ }[/math]  [math]\displaystyle{ s F(s)  f(0^{}) \ }[/math]  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  [math]\displaystyle{ f''(t) \ }[/math]  [math]\displaystyle{ s^2 F(s)  s f(0^{})  f'(0^{}) \ }[/math]  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  [math]\displaystyle{ f^{(n)}(t) \ }[/math]  [math]\displaystyle{ s^n F(s)  \sum_{k=1}^{n} s^{nk} f^{(k1)}(0^{}) \ }[/math]  f is assumed to be ntimes differentiable, with nth derivative of exponential type. Follows by mathematical induction. 
Frequencydomain integration  [math]\displaystyle{ \frac{1}{t}f(t) \ }[/math]  [math]\displaystyle{ \int_s^\infty F(\sigma)\, d\sigma \ }[/math]  This is deduced using the nature of frequency differentiation and conditional convergence. 
Timedomain integration  [math]\displaystyle{ \int_0^t f(\tau)\, d\tau = (u * f)(t) }[/math]  [math]\displaystyle{ {1 \over s} F(s) }[/math]  u(t) is the Heaviside step function and (u ∗ f)(t) is the convolution of u(t) and f(t). 
Frequency shifting  [math]\displaystyle{ e^{at} f(t) }[/math]  [math]\displaystyle{ F(s  a) \ }[/math]  
Time shifting  [math]\displaystyle{ f(t  a) u(t  a) }[/math]  [math]\displaystyle{ e^{as} F(s) \ }[/math]  a > 0, u(t) is the Heaviside step function 
Time scaling  [math]\displaystyle{ f(at) }[/math]  [math]\displaystyle{ \frac{1}{a} F \left ({s \over a} \right) }[/math]  a > 0 
Multiplication  [math]\displaystyle{ f(t)g(t) }[/math]  [math]\displaystyle{ \frac{1}{2\pi i}\lim_{T\to\infty}\int_{c  iT}^{c + iT}F(\sigma)G(s  \sigma)\,d\sigma \ }[/math]  The integration is done along the vertical line Re(σ) = c that lies entirely within the region of convergence of F.^{[24]} 
Convolution  [math]\displaystyle{ (f * g)(t) = \int_{0}^{t} f(\tau)g(t  \tau)\,d\tau }[/math]  [math]\displaystyle{ F(s) \cdot G(s) \ }[/math]  
Circular convolution  [math]\displaystyle{ (f * g)(t) = \int_{0}^T f(\tau)g(t  \tau)\,d\tau }[/math]  [math]\displaystyle{ F(s) \cdot G(s) \ }[/math]  For periodic functions with period T. 
Complex conjugation  [math]\displaystyle{ f^*(t) }[/math]  [math]\displaystyle{ F^*(s^*) }[/math]  
Crosscorrelation  [math]\displaystyle{ (f \star g)(t) = \int_0^{\infty} f(\tau)^* \, g(t+\tau)\,d\tau }[/math]  [math]\displaystyle{ F^*(s^*)\cdot G(s) }[/math]  
Periodic function  [math]\displaystyle{ f(t) }[/math]  [math]\displaystyle{ {1 \over 1  e^{Ts}} \int_0^T e^{st} f(t)\,dt }[/math]  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. 
Periodic summation  [math]\displaystyle{ f_P(t) = \sum_{n=0}^{\infty} f(tTn) }[/math]
[math]\displaystyle{ f_P(t) = \sum_{n=0}^{\infty} (1)^n f(tTn) }[/math] 
[math]\displaystyle{ F_P(s) = \frac{1}{1e^{Ts}} F(s) }[/math]
[math]\displaystyle{ F_P(s) = \frac{1}{1+e^{Ts}} F(s) }[/math] 
 Initial value theorem
 [math]\displaystyle{ f(0^+)=\lim_{s\to \infty}{sF(s)}. }[/math]
 Final value theorem
 [math]\displaystyle{ f(\infty)=\lim_{s\to 0}{sF(s)} }[/math], if all poles of [math]\displaystyle{ sF(s) }[/math] are in the left halfplane.
 The final value theorem is useful because it gives the longterm behaviour without having to perform partial fraction decompositions (or other difficult algebra). If F(s) has a pole in the righthand plane or poles on the imaginary axis (e.g., if [math]\displaystyle{ f(t) = e^t }[/math] or [math]\displaystyle{ f(t) = \sin(t) }[/math]), then the behaviour of this formula is undefined.
Relation to power series
The Laplace transform can be viewed as a continuous analogue of a power series.^{[25]} If a(n) is a discrete function of a positive integer n, then the power series associated to a(n) is the series [math]\displaystyle{ \sum_{n=0}^{\infty} a(n) x^n }[/math] where x is a real variable (see Ztransform). Replacing summation over n with integration over t, a continuous version of the power series becomes [math]\displaystyle{ \int_{0}^{\infty} f(t) x^t\, dt }[/math] 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 [math]\displaystyle{ \int_{0}^{\infty} f(t) \left(e^{\ln{x}}\right)^t\, dt }[/math]
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: [math]\displaystyle{ \int_{0}^{\infty} f(t) e^{st}\, dt }[/math]
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}.
Relation to moments
The quantities [math]\displaystyle{ \mu_n = \int_0^\infty t^nf(t)\, dt }[/math]
are the moments of the function f. If the first n moments of f converge absolutely, then by repeated differentiation under the integral, [math]\displaystyle{ (1)^n(\mathcal L f)^{(n)}(0) = \mu_n . }[/math] This is of special significance in probability theory, where the moments of a random variable X are given by the expectation values [math]\displaystyle{ \mu_n=\operatorname{E}[X^n] }[/math]. Then, the relation holds [math]\displaystyle{ \mu_n = (1)^n\frac{d^n}{ds^n}\operatorname{E}\left[e^{sX}\right](0). }[/math]
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: [math]\displaystyle{ \begin{align} \mathcal{L} \left\{f(t)\right\} &= \int_{0^}^\infty e^{st} f(t)\, dt \\[6pt] &= \left[\frac{f(t)e^{st}}{s} \right]_{0^}^\infty  \int_{0^}^\infty \frac{e^{st}}{s} f'(t) \, dt\quad \text{(by parts)} \\[6pt] &= \left[\frac{f(0^)}{s}\right] + \frac 1 s \mathcal{L} \left\{f'(t)\right\}, \end{align} }[/math] yielding [math]\displaystyle{ \mathcal{L} \{ f'(t) \} = s\cdot\mathcal{L} \{ f(t) \}f(0^), }[/math] and in the bilateral case, [math]\displaystyle{ \mathcal{L} \{ f'(t) \} = s \int_{\infty}^\infty e^{st} f(t)\,dt = s \cdot \mathcal{L} \{ f(t) \}. }[/math]
The general result [math]\displaystyle{ \mathcal{L} \left\{ f^{(n)}(t) \right\} = s^n \cdot \mathcal{L} \{ f(t) \}  s^{n  1} f(0^)  \cdots  f^{(n  1)}(0^), }[/math] where [math]\displaystyle{ f^{(n)} }[/math] denotes the nth derivative of f, 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: [math]\displaystyle{ \int_0^\infty f(x)g(x)\,dx = \int_0^\infty(\mathcal{L} f)(s)\cdot(\mathcal{L}^{1}g)(s)\,ds }[/math] under suitable assumptions on the behaviour of [math]\displaystyle{ f,g }[/math] in a right neighbourhood of [math]\displaystyle{ 0 }[/math] and on the decay rate of [math]\displaystyle{ f,g }[/math] in a left neighbourhood of [math]\displaystyle{ \infty }[/math]. The above formula is a variation of integration by parts, with the operators [math]\displaystyle{ \frac{d}{dx} }[/math] and [math]\displaystyle{ \int \,dx }[/math] being replaced by [math]\displaystyle{ \mathcal{L} }[/math] and [math]\displaystyle{ \mathcal{L}^{1} }[/math]. Let us prove the equivalent formulation: [math]\displaystyle{ \int_0^\infty(\mathcal{L} f)(x)g(x)\,dx = \int_0^\infty f(s)(\mathcal{L}g)(s)\,ds. }[/math]
By plugging in [math]\displaystyle{ (\mathcal{L}f)(x)=\int_0^\infty f(s)e^{sx}\,ds }[/math] the lefthand side turns into: [math]\displaystyle{ \int_0^\infty\int_0^\infty f(s)g(x) e^{sx}\,ds\,dx, }[/math] but assuming Fubini's theorem holds, by reversing the order of integration we get the wanted righthand side.
This method can be used to compute integrals that would otherwise be difficult to compute using elementary methods of real calculus. For example, [math]\displaystyle{ \int_0^\infty\frac{\sin x}{x}dx = \int_0^\infty \mathcal{L}(1)(x)\sin x dx = \int_0^\infty 1 \cdot \mathcal{L}(\sin)(x)dx = \int_0^\infty \frac{dx}{x^2 + 1} = \frac{\pi}{2}. }[/math]
Relationship to other transforms
Laplace–Stieltjes transform
The (unilateral) Laplace–Stieltjes transform of a function g : ℝ → ℝ is defined by the Lebesgue–Stieltjes integral
[math]\displaystyle{ \{ \mathcal{L}^*g \}(s) = \int_0^\infty e^{st} \, d\,g(t) ~. }[/math]
The function g is assumed to be of bounded variation. If g is the antiderivative of f:
[math]\displaystyle{ g(x) = \int_0^x f(t)\,d\,t }[/math]
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.^{[26]}
Fourier transform
The Fourier transform is a special case (under certain conditions) of the bilateral Laplace transform. While the Fourier transform of a function is a complex function of a real variable (frequency), the Laplace transform of a function is a complex function of a complex variable. The Laplace transform is usually restricted to transformation of functions of 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 Fourier transform is equivalent to evaluating the bilateral Laplace transform with imaginary argument s = iω or s = 2πiξ^{[27]} when the condition explained below is fulfilled,
[math]\displaystyle{ \begin{align} \hat{f}(\omega) &= \mathcal{F}\{f(t)\} \\[4pt] &= \mathcal{L}\{f(t)\}_{s = i \omega} = F(s)_{s = i \omega} \\[4pt] &= \int_{\infty}^\infty e^{i \omega t} f(t)\,dt~. \end{align} }[/math]
This convention of the Fourier transform ([math]\displaystyle{ \hat f_3(\omega) }[/math] in Fourier transform § Other conventions) requires a factor of 1/2π 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 (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_{0} is a pole of F(s), substituting s = iω in F(s) does not yield the Fourier transform of f(t)u(t), which contains terms proportional to the Dirac delta functions δ(ω ± ω_{0}).
However, a relation of the form [math]\displaystyle{ \lim_{\sigma\to 0^+} F(\sigma+i\omega) = \hat{f}(\omega) }[/math] 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 twosided Laplace transform by a simple change of variables.
If in the Mellin transform [math]\displaystyle{ G(s) = \mathcal{M}\{g(\theta)\} = \int_0^\infty \theta^s g(\theta) \, \frac{d\theta} \theta }[/math] we set θ = e^{−t} we get a twosided Laplace transform.
Ztransform
The unilateral or onesided Ztransform is simply the Laplace transform of an ideally sampled signal with the substitution of [math]\displaystyle{ z \stackrel{\mathrm{def} }{ {}={} } e^{sT} , }[/math] where T = 1/f_{s} is the sampling interval (in units of time e.g., seconds) and f_{s} is the sampling rate (in samples per second or hertz).
Let [math]\displaystyle{ \Delta_T(t) \ \stackrel{\mathrm{def}}{=}\ \sum_{n=0}^{\infty} \delta(t  n T) }[/math] be a sampling impulse train (also called a Dirac comb) and [math]\displaystyle{ \begin{align} x_q(t) &\stackrel{\mathrm{def} }{ {}={} } x(t) \Delta_T(t) = x(t) \sum_{n=0}^{\infty} \delta(t  n T) \\ &= \sum_{n=0}^{\infty} x(n T) \delta(t  n T) = \sum_{n=0}^{\infty} x[n] \delta(t  n T) \end{align} }[/math] be the sampled representation of the continuoustime x(t) [math]\displaystyle{ x[n] \stackrel{\mathrm{def} }{ {}={} } x(nT) ~. }[/math]
The Laplace transform of the sampled signal x_{q}(t) is [math]\displaystyle{ \begin{align} X_q(s) &= \int_{0^}^\infty x_q(t) e^{s t} \,dt \\ &= \int_{0^}^\infty \sum_{n=0}^\infty x[n] \delta(t  n T) e^{s t} \, dt \\ &= \sum_{n=0}^\infty x[n] \int_{0^}^\infty \delta(t  n T) e^{s t} \, dt \\ &= \sum_{n=0}^\infty x[n] e^{n s T}~. \end{align} }[/math]
This is the precise definition of the unilateral Ztransform of the discrete function x[n]
[math]\displaystyle{ X(z) = \sum_{n=0}^{\infty} x[n] z^{n} }[/math] 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, [math]\displaystyle{ X_q(s) = X(z) \Big_{z=e^{sT}}. }[/math]
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 [math]\displaystyle{ F(s) = \int_0^\infty f(z)e^{sz}\, dz }[/math] is a special case of the Laplace transform for f an entire function of exponential type, meaning that [math]\displaystyle{ f(z)\le Ae^{Bz} }[/math] 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.
Fundamental relationships
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.
Table of selected Laplace transforms
The following table provides Laplace transforms for many common functions of a single variable.^{[28]}^{[29]} 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.[math]\displaystyle{ \mathcal{L}\{f(t) + g(t)\} = \mathcal{L}\{f(t)\} + \mathcal{L}\{ g(t)\} }[/math]
 The Laplace transform of a multiple of a function is that multiple times the Laplace transformation of that function.[math]\displaystyle{ \mathcal{L}\{a f(t)\} = a \mathcal{L}\{ f(t)\} }[/math]
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 [math]\displaystyle{ f(t) = \mathcal{L}^{1}\{F(s)\} }[/math] 
Laplace sdomain [math]\displaystyle{ F(s) = \mathcal{L}\{f(t)\} }[/math] 
Region of convergence  Reference  

unit impulse  [math]\displaystyle{ \delta(t) \ }[/math]  [math]\displaystyle{ 1 }[/math]  all s  inspection  
delayed impulse  [math]\displaystyle{ \delta(t  \tau) \ }[/math]  [math]\displaystyle{ e^{\tau s} \ }[/math]  time shift of unit impulse  
unit step  [math]\displaystyle{ u(t) \ }[/math]  [math]\displaystyle{ { 1 \over s } }[/math]  [math]\displaystyle{ \operatorname{Re}(s) \gt 0 }[/math]  integrate unit impulse  
delayed unit step  [math]\displaystyle{ u(t  \tau) \ }[/math]  [math]\displaystyle{ \frac 1 s e^{\tau s} }[/math]  [math]\displaystyle{ \operatorname{Re}(s) \gt 0 }[/math]  time shift of unit step  
product of delayed function and delayed step  [math]\displaystyle{ f(t\tau)u(t\tau) }[/math]  [math]\displaystyle{ e^{s\tau}\mathcal{L}\{f(t)\} }[/math]  usubstitution, [math]\displaystyle{ u=t\tau }[/math]  
rectangular impulse  [math]\displaystyle{ u (t)  u(t  \tau) }[/math]  [math]\displaystyle{ \frac{1}{s}(1  e^{\tau s}) }[/math]  [math]\displaystyle{ \operatorname{Re}(s) \gt 0 }[/math]  
ramp  [math]\displaystyle{ t \cdot u(t)\ }[/math]  [math]\displaystyle{ \frac 1 {s^2} }[/math]  [math]\displaystyle{ \operatorname{Re}(s) \gt 0 }[/math]  integrate unit impulse twice  
nth power (for integer n) 
[math]\displaystyle{ t^n \cdot u(t) }[/math]  [math]\displaystyle{ { n! \over s^{n + 1} } }[/math]  [math]\displaystyle{ \operatorname{Re}(s) \gt 0 }[/math] (n > −1) 
integrate unit step n times  
qth power (for complex q) 
[math]\displaystyle{ t^q \cdot u(t) }[/math]  [math]\displaystyle{ { \operatorname{\Gamma}(q + 1) \over s^{q + 1} } }[/math]  [math]\displaystyle{ \operatorname{Re}(s) \gt 0 }[/math] [math]\displaystyle{ \operatorname{Re}(q) \gt 1 }[/math] 
^{[30]}^{[31]}  
nth root  [math]\displaystyle{ \sqrt[n]{t} \cdot u(t) }[/math]  [math]\displaystyle{ { 1 \over s^{\frac 1 n + 1} } \operatorname{\Gamma}\left(\frac 1 n + 1\right) }[/math]  [math]\displaystyle{ \operatorname{Re}(s) \gt 0 }[/math]  Set q = 1/n above.  
nth power with frequency shift  [math]\displaystyle{ t^{n} e^{\alpha t} \cdot u(t) }[/math]  [math]\displaystyle{ \frac{n!}{(s+\alpha)^{n+1}} }[/math]  [math]\displaystyle{ \operatorname{Re}(s) \gt \alpha }[/math]  Integrate unit step, apply frequency shift  
delayed nth power with frequency shift 
[math]\displaystyle{ (t\tau)^n e^{\alpha (t\tau)} \cdot u(t\tau) }[/math]  [math]\displaystyle{ \frac{n! \cdot e^{\tau s}}{(s+\alpha)^{n+1}} }[/math]  [math]\displaystyle{ \operatorname{Re}(s) \gt \alpha }[/math]  integrate unit step, apply frequency shift, apply time shift  
exponential decay  [math]\displaystyle{ e^{\alpha t} \cdot u(t) }[/math]  [math]\displaystyle{ { 1 \over s+\alpha } }[/math]  [math]\displaystyle{ \operatorname{Re}(s) \gt \alpha }[/math]  Frequency shift of unit step  
twosided exponential decay (only for bilateral transform) 
[math]\displaystyle{ e^{\alphat} \ }[/math]  [math]\displaystyle{ { 2\alpha \over \alpha^2  s^2 } }[/math]  [math]\displaystyle{ \alpha \lt \operatorname{Re}(s) \lt \alpha }[/math]  Frequency shift of unit step  
exponential approach  [math]\displaystyle{ (1e^{\alpha t}) \cdot u(t) \ }[/math]  [math]\displaystyle{ \frac{\alpha}{s(s+\alpha)} }[/math]  [math]\displaystyle{ \operatorname{Re}(s) \gt 0 }[/math]  unit step minus exponential decay  
sine  [math]\displaystyle{ \sin(\omega t) \cdot u(t) \ }[/math]  [math]\displaystyle{ { \omega \over s^2 + \omega^2 } }[/math]  [math]\displaystyle{ \operatorname{Re}(s) \gt 0 }[/math]  ^{[32]}  
cosine  [math]\displaystyle{ \cos(\omega t) \cdot u(t) \ }[/math]  [math]\displaystyle{ { s \over s^2 + \omega^2 } }[/math]  [math]\displaystyle{ \operatorname{Re}(s) \gt 0 }[/math]  ^{[32]}  
hyperbolic sine  [math]\displaystyle{ \sinh(\alpha t) \cdot u(t) \ }[/math]  [math]\displaystyle{ { \alpha \over s^2  \alpha^2 } }[/math]  [math]\displaystyle{ \operatorname{Re}(s) \gt \left \alpha \right }[/math]  ^{[33]}  
hyperbolic cosine  [math]\displaystyle{ \cosh(\alpha t) \cdot u(t) \ }[/math]  [math]\displaystyle{ { s \over s^2  \alpha^2 } }[/math]  [math]\displaystyle{ \operatorname{Re}(s) \gt \left \alpha \right }[/math]  ^{[33]}  
exponentially decaying sine wave 
[math]\displaystyle{ e^{\alpha t} \sin(\omega t) \cdot u(t) \ }[/math]  [math]\displaystyle{ { \omega \over (s+\alpha)^2 + \omega^2 } }[/math]  [math]\displaystyle{ \operatorname{Re}(s) \gt  \alpha }[/math]  ^{[32]}  
exponentially decaying cosine wave 
[math]\displaystyle{ e^{\alpha t} \cos(\omega t) \cdot u(t) \ }[/math]  [math]\displaystyle{ { s+\alpha \over (s+\alpha)^2 + \omega^2 } }[/math]  [math]\displaystyle{ \operatorname{Re}(s) \gt  \alpha }[/math]  ^{[32]}  
natural logarithm  [math]\displaystyle{ \ln(t) \cdot u(t) }[/math]  [math]\displaystyle{ {1 \over s} \left[ \ln(s)+\gamma \right] }[/math]  [math]\displaystyle{ \operatorname{Re}(s) \gt 0 }[/math]  ^{[33]}  
Bessel function of the first kind, of order n 
[math]\displaystyle{ J_n(\omega t) \cdot u(t) }[/math]  [math]\displaystyle{ \frac{ \left(\sqrt{s^2+ \omega^2}s\right)^{\!n}}{\omega^n \sqrt{s^2 + \omega^2}} }[/math]  [math]\displaystyle{ \operatorname{Re}(s) \gt 0 }[/math] (n > −1) 
^{[34]}  
Error function  [math]\displaystyle{ \operatorname{erf}(t) \cdot u(t) }[/math]  [math]\displaystyle{ \frac{1}{s} e^{(1/4)s^2} \!\left(1  \operatorname{erf} \frac{s}{2} \right) }[/math]  [math]\displaystyle{ \operatorname{Re}(s) \gt 0 }[/math]  ^{[34]}  
Explanatory notes:

sdomain equivalent circuits and impedances
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.
Examples and applications
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 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.^{[35]}
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 [math]\displaystyle{ \mathcal{L}\left\{f(t)\right\} = F(s) }[/math]. Then (see the table above)
[math]\displaystyle{ \partial_s\mathcal{L} \left\{\frac{f(t)} t \right\} = \partial_s\int_0^\infty \frac{f(t)}{t}e^{st}\, dt = \int_0^\infty f(t)e^{st} =  F(s) }[/math]
From which one gets:
[math]\displaystyle{ \mathcal{L} \left\{\frac{f(t)} t \right\} = \int_s^\infty F(p)\, dp. }[/math]
In the limit [math]\displaystyle{ s \rightarrow 0 }[/math], one gets [math]\displaystyle{ \int_0^\infty \frac{f(t)} t \, dt = \int_0^\infty F(p)\, dp, }[/math] 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 a ≠ 0 ≠ b, proceeding formally one has [math]\displaystyle{ \begin{align} \int_0^\infty \frac{ \cos(at)  \cos(bt) }{t} \, dt &=\int_0^\infty \left(\frac p {p^2 + a^2}  \frac{p}{p^2 + b^2}\right)\, dp \\[6pt] &=\left[ \frac{1}{2} \ln\frac{p^2 + a^2}{p^2 + b^2} \right]_0^\infty = \frac{1}{2} \ln \frac{b^2}{a^2} = \ln \left \frac {b}{a} \right. \end{align} }[/math]
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 (with equations as for the SI unit system). Symbolically, this is expressed by the differential equation [math]\displaystyle{ i = C { dv \over dt} , }[/math] where C is the capacitance of the capacitor, i = i(t) is the electric current through the capacitor as a function of time, and v = v(t) is the voltage across the terminals of the capacitor, also as a function of time.
Taking the Laplace transform of this equation, we obtain [math]\displaystyle{ I(s) = C(s V(s)  V_0), }[/math] where [math]\displaystyle{ \begin{align} I(s) &= \mathcal{L} \{ i(t) \},\\ V(s) &= \mathcal{L} \{ v(t) \}, \end{align} }[/math] and [math]\displaystyle{ V_0 = v(0). }[/math]
Solving for V(s) we have [math]\displaystyle{ V(s) = { I(s) \over sC } + { V_0 \over s }. }[/math]
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: [math]\displaystyle{ Z(s) = \left. { V(s) \over I(s) } \right_{V_0 = 0}. }[/math]
Using this definition and the previous equation, we find: [math]\displaystyle{ Z(s) = \frac{1}{sC}, }[/math] 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 timeinvariant system with transfer function [math]\displaystyle{ H(s) = \frac{1}{(s + \alpha)(s + \beta)}. }[/math]
The impulse response is simply the inverse Laplace transform of this transfer function: [math]\displaystyle{ h(t) = \mathcal{L}^{1}\{H(s)\}. }[/math]
 Partial fraction expansion
To evaluate this inverse transform, we begin by expanding H(s) using the method of partial fraction expansion, [math]\displaystyle{ \frac{1}{(s + \alpha)(s + \beta)} = { P \over s + \alpha } + { R \over s+\beta }. }[/math]
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 [math]\displaystyle{ \frac{1}{s + \beta} = P + { R (s + \alpha) \over s + \beta }. }[/math]
Then by letting s = −α, the contribution from R vanishes and all that is left is [math]\displaystyle{ P = \left.{1 \over s+\beta}\right_{s=\alpha} = {1 \over \beta  \alpha}. }[/math]
Similarly, the residue R is given by [math]\displaystyle{ R = \left.{1 \over s + \alpha}\right_{s=\beta} = {1 \over \alpha  \beta}. }[/math]
Note that [math]\displaystyle{ R = {1 \over \beta  \alpha} =  P }[/math] and so the substitution of R and P into the expanded expression for H(s) gives [math]\displaystyle{ H(s) = \left(\frac{1}{\beta  \alpha} \right) \cdot \left( { 1 \over s + \alpha }  { 1 \over s + \beta } \right). }[/math]
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 [math]\displaystyle{ h(t) = \mathcal{L}^{1}\{H(s)\} = \frac{1}{\beta  \alpha}\left(e^{\alpha t}  e^{\beta t}\right), }[/math] 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 1/(s + α) and 1/(s + β). That is, the inverse of [math]\displaystyle{ H(s) = \frac{1}{(s + \alpha)(s + \beta)} = \frac{1}{s+\alpha} \cdot \frac{1}{s + \beta} }[/math] is [math]\displaystyle{ \mathcal{L}^{1}\! \left\{ \frac{1}{s + \alpha} \right\} * \mathcal{L}^{1}\! \left\{\frac{1}{s + \beta} \right\} = e^{\alpha t} * e^{\beta t} = \int_0^t e^{\alpha x}e^{\beta (t  x)}\, dx = \frac{e^{\alpha t}e^{\beta t}}{\beta  \alpha}. }[/math]
Phase delay
Time function  Laplace transform 

[math]\displaystyle{ \sin{(\omega t + \varphi)} }[/math]  [math]\displaystyle{ \frac{s\sin(\varphi) + \omega \cos(\varphi)}{s^2 + \omega^2} }[/math] 
[math]\displaystyle{ \cos{(\omega t + \varphi)} }[/math]  [math]\displaystyle{ \frac{s\cos(\varphi)  \omega \sin(\varphi)}{s^2 + \omega^2}. }[/math] 
Starting with the Laplace transform, [math]\displaystyle{ X(s) = \frac{s\sin(\varphi) + \omega \cos(\varphi)}{s^2 + \omega^2} }[/math] we find the inverse by first rearranging terms in the fraction: [math]\displaystyle{ \begin{align} X(s) &= \frac{s \sin(\varphi)}{s^2 + \omega^2} + \frac{\omega \cos(\varphi)}{s^2 + \omega^2} \\ &= \sin(\varphi) \left(\frac{s}{s^2 + \omega^2} \right) + \cos(\varphi) \left(\frac{\omega}{s^2 + \omega^2} \right). \end{align} }[/math]
We are now able to take the inverse Laplace transform of our terms: [math]\displaystyle{ \begin{align} x(t) &= \sin(\varphi) \mathcal{L}^{1}\left\{\frac{s}{s^2 + \omega^2} \right\} + \cos(\varphi) \mathcal{L}^{1}\left\{\frac{\omega}{s^2 + \omega^2} \right\} \\ &= \sin(\varphi)\cos(\omega t) + \cos(\varphi)\sin(\omega t). \end{align} }[/math]
This is just the sine of the sum of the arguments, yielding: [math]\displaystyle{ x(t) = \sin (\omega t + \varphi). }[/math]
We can apply similar logic to find that [math]\displaystyle{ \mathcal{L}^{1} \left\{ \frac{s\cos\varphi  \omega \sin\varphi}{s^2 + \omega^2} \right\} = \cos{(\omega t + \varphi)}. }[/math]
Statistical mechanics
In statistical mechanics, the Laplace transform of the density of states [math]\displaystyle{ g(E) }[/math] defines the partition function.^{[36]} That is, the canonical partition function [math]\displaystyle{ Z(\beta) }[/math] is given by [math]\displaystyle{ Z(\beta) = \int_0^\infty e^{\beta E}g(E)\,dE }[/math] and the inverse is given by [math]\displaystyle{ g(E) = \frac{1}{2\pi i} \int_{\beta_0i\infty}^{\beta_0+i\infty} e^{\beta E}Z(\beta) \, d\beta }[/math]
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 (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.^{[37]} When independent information on the structure of an object is available, the inverse Laplace transform method has been found to be in good agreement.
Gallery
See also
 Analog signal processing
 Bernstein's theorem on monotone functions
 Continuousrepayment mortgage
 Hamburger moment problem
 Hardy–Littlewood Tauberian theorem
 Laplace–Carson transform
 Momentgenerating function
 Nonlocal operator
 Post's inversion formula
 Signalflow graph
 Transfer function
Notes
 ↑ Lynn, Paul A. (1986). "The Laplace Transform and the ztransform". Electronic Signals and Systems. London: Macmillan Education UK. pp. 225–272. doi:10.1007/9781349184613_6. ISBN 9780333391648. "Laplace Transform and the ztransform are closely related to the Fourier Transform. Laplace Transform is somewhat more general in scope than the Fourier Transform, and is widely used by engineers for describing continuous circuits and systems, including automatic control systems."
 ↑ "Differential Equations  Laplace Transforms". https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx.
 ↑ ^{3.0} ^{3.1} Weisstein, Eric W.. "Laplace Transform" (in en). https://mathworld.wolfram.com/LaplaceTransform.html.
 ↑ "Des Fonctions génératrices" (in fr), Théorie analytique des Probabilités (2nd ed.), Paris, 1814, chap.I sect.220, https://archive.org/details/thorieanalytiqu01laplgoog
 ↑ Jaynes, E. T. (Edwin T.) (2003). Probability theory : the logic of science. Bretthorst, G. Larry. Cambridge, UK: Cambridge University Press. ISBN 0511065892. OCLC 57254076.
 ↑ Abel, Niels H. (1820), "Sur les fonctions génératrices et leurs déterminantes" (in fr), Œuvres Complètes, II (published 1839), pp. 77–88 1881 edition
 ↑ Lerch, Mathias (1903), "Sur un point de la théorie des fonctions génératrices d'Abel" (in fr), Acta Mathematica 27: 339–351, doi:10.1007/BF02421315
 ↑ Heaviside, Oliver (January 2008), "The solution of definite integrals by differential transformation", Electromagnetic Theory, III, London, section 526, ISBN 9781605206189, https://books.google.com/books?id=y9auR0L6ZRcC&pg=PA234
 ↑ Bromwich, Thomas J. (1916), "Normal coordinates in dynamical systems", Proceedings of the London Mathematical Society 15: 401–448, doi:10.1112/plms/s215.1.401, https://zenodo.org/record/2319588
 ↑ An influential book was: Gardner, Murray F.; Barnes, John L. (1942), Transients in Linear Systems studied by the Laplace Transform, New York: Wiley
 ↑ Doetsch, Gustav (1937) (in de), Theorie und Anwendung der Laplacesche Transformation, Berlin: Springer translation 1943
 ↑ Euler 1744, Euler 1753, Euler 1769
 ↑ Lagrange 1773
 ↑ GrattanGuinness 1997, p. 260
 ↑ GrattanGuinness 1997, p. 261
 ↑ GrattanGuinness 1997, pp. 261–262
 ↑ GrattanGuinness 1997, pp. 262–266
 ↑ Feller 1971, §XIII.1
 ↑ The cumulative distribution function is the integral of the probability density function.
 ↑ Mikusiński, Jan (14 July 2014). Operational Calculus. Elsevier. ISBN 9781483278933. https://books.google.com/books?id=e8LSBQAAQBAJ.
 ↑ Widder 1941, Chapter II, §1
 ↑ Widder 1941, Chapter VI, §2
 ↑ Korn & Korn 1967, pp. 226–227
 ↑ Bracewell 2000, Table 14.1, p. 385
 ↑ Archived at Ghostarchive and the Wayback Machine: Mattuck, Arthur. "Where the Laplace Transform comes from". https://www.youtube.com/watch?v=zvbdoSeGAgI.
 ↑ Feller 1971, p. 432
 ↑ Takacs 1953, p. 93
 ↑ Riley, K. F.; Hobson, M. P.; Bence, S. J. (2010), Mathematical methods for physics and engineering (3rd ed.), Cambridge University Press, p. 455, ISBN 9780521861533
 ↑ Distefano, J. J.; Stubberud, A. R.; Williams, I. J. (1995), Feedback systems and control, Schaum's outlines (2nd ed.), McGrawHill, p. 78, ISBN 9780070170520
 ↑ Lipschutz, S.; Spiegel, M. R.; Liu, J. (2009). Mathematical Handbook of Formulas and Tables. Schaum's Outline Series (3rd ed.). McGrawHill. p. 183. ISBN 9780071548557. – provides the case for real q.
 ↑ http://mathworld.wolfram.com/LaplaceTransform.html – Wolfram Mathword provides case for complex q
 ↑ ^{32.0} ^{32.1} ^{32.2} ^{32.3} Bracewell 1978, p. 227.
 ↑ ^{33.0} ^{33.1} ^{33.2} Williams 1973, p. 88.
 ↑ ^{34.0} ^{34.1} Williams 1973, p. 89.
 ↑ Korn & Korn 1967, §8.1
 ↑ RK Pathria; Paul Beal (1996). Statistical mechanics (2nd ed.). ButterworthHeinemann. p. 56. ISBN 9780750624695. https://archive.org/details/statisticalmecha00path_911.
 ↑ Salem, M.; Seaton, M. J. (1974), "I. Continuum spectra and brightness contours", Monthly Notices of the Royal Astronomical Society 167: 493–510, doi:10.1093/mnras/167.3.493, Bibcode: 1974MNRAS.167..493S, and
Salem, M. (1974), "II. Threedimensional models", Monthly Notices of the Royal Astronomical Society 167: 511–516, doi:10.1093/mnras/167.3.511, Bibcode: 1974MNRAS.167..511S
References
Modern
 Bracewell, Ronald N. (1978), The Fourier Transform and its Applications (2nd ed.), McGrawHill Kogakusha, ISBN 9780070070134
 Bracewell, R. N. (2000), The Fourier Transform and Its Applications (3rd ed.), Boston: McGrawHill, ISBN 9780071160438
 Feller, William (1971), An introduction to probability theory and its applications. Vol. II., Second edition, New York: John Wiley & Sons
 Korn, G. A.; Korn, T. M. (1967), Mathematical Handbook for Scientists and Engineers (2nd ed.), McGrawHill Companies, ISBN 9780070353701
 Widder, David Vernon (1941), The Laplace Transform, Princeton Mathematical Series, v. 6, Princeton University Press
 Williams, J. (1973), Laplace Transforms, Problem Solvers, George Allen & Unwin, ISBN 9780045120215
 Takacs, J. (1953), "Fourier amplitudok meghatarozasa operatorszamitassal" (in hu), Magyar Hiradastechnika IV (7–8): 93–96
Historical
 Euler, L. (1744), "De constructione aequationum" (in la), Opera Omnia, 1st series 22: 150–161
 Euler, L. (1753), "Methodus aequationes differentiales" (in la), Opera Omnia, 1st series 22: 181–213
 Euler, L. (1992), "Institutiones calculi integralis, Volume 2" (in la), Opera Omnia, 1st series (Basel: Birkhäuser) 12, ISBN 9783764314743, Chapters 3–5
 Euler, Leonhard (1769) (in la), Institutiones calculi integralis, II, Paris: Petropoli, ch. 3–5, pp. 57–153, https://books.google.com/books?id=BFqWNwpfqo8C
 GrattanGuinness, I (1997), "Laplace's integral solutions to partial differential equations", in Gillispie, C. C., Pierre Simon Laplace 1749–1827: A Life in Exact Science, Princeton: Princeton University Press, ISBN 9780691011851
 Lagrange, J. L. (1773), Mémoire sur l'utilité de la méthode, Œuvres de Lagrange, 2, pp. 171–234
Further reading
 Arendt, Wolfgang; Batty, Charles J.K.; Hieber, Matthias; Neubrander, Frank (2002), VectorValued Laplace Transforms and Cauchy Problems, Birkhäuser Basel, ISBN 9783764365493.
 Davies, Brian (2002), Integral transforms and their applications (Third ed.), New York: Springer, ISBN 9780387953144
 Deakin, M. A. B. (1981), "The development of the Laplace transform", Archive for History of Exact Sciences 25 (4): 343–390, doi:10.1007/BF01395660
 Deakin, M. A. B. (1982), "The development of the Laplace transform", Archive for History of Exact Sciences 26 (4): 351–381, doi:10.1007/BF00418754
 Doetsch, Gustav (1974), Introduction to the Theory and Application of the Laplace Transformation, Springer, ISBN 9780387064079
 Mathews, Jon; Walker, Robert L. (1970), Mathematical methods of physics (2nd ed.), New York: W. A. Benjamin, ISBN 0805370021
 Polyanin, A. D.; Manzhirov, A. V. (1998), Handbook of Integral Equations, Boca Raton: CRC Press, ISBN 9780849328763
 Schwartz, Laurent (1952), "Transformation de Laplace des distributions" (in fr), Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952: 196–206
 Schwartz, Laurent (2008), Mathematics for the Physical Sciences, Dover Books on Mathematics, New York: Dover Publications, pp. 215–241, ISBN 9780486466620, https://books.google.com/books?id=_AuDQAAQBAJ&pg=PA215  See Chapter VI. The Laplace transform.
 Siebert, William McC. (1986), Circuits, Signals, and Systems, Cambridge, Massachusetts: MIT Press, ISBN 9780262192293
 Widder, David Vernon (1945), "What is the Laplace transform?", The American Mathematical Monthly 52 (8): 419–425, doi:10.2307/2305640, ISSN 00029890
 J.A.C.Weidman and Bengt Fornberg: "Fully numerical Laplace transform methods", Numerical Algorithms, vol.92 (2023), pp. 985–1006. https://doi.org/10.1007/s1107502201368x .
External links
 Hazewinkel, Michiel, ed. (2001), "Laplace transform", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104, https://www.encyclopediaofmath.org/index.php?title=p/l057540
 Online Computation of the transform or inverse transform, wims.unice.fr
 Tables of Integral Transforms at EqWorld: The World of Mathematical Equations.
 Weisstein, Eric W.. "Laplace Transform". http://mathworld.wolfram.com/LaplaceTransform.html.
 Good explanations of the initial and final value theorems
 Laplace Transforms at MathPages
 Computational Knowledge Engine allows to easily calculate Laplace Transforms and its inverse Transform.
 Laplace Calculator to calculate Laplace Transforms online easily.
 Code to visualize Laplace Transforms and many example videos.
Original source: https://en.wikipedia.org/wiki/Laplace transform.
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