Laplace transform

Template:Short description Template:CS1 config In mathematics, the Laplace transform, named after Pierre-Simon Laplace (Template:IPAc-en), is an integral transform that converts a function of a real variable (usually Template:Tmath, in the time domain) to a function of a complex variable $s$ (in the complex-valued frequency domain, also known as s-domain or s-plane). The functions are often denoted using a lowercase symbol for the time-domain function and the corresponding uppercase symbol for the frequency-domain function, e.g. $x(t)$ and Template:Tmath.

The transform is useful for converting differentiation and integration in the time domain into the algebraic operations 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 and dynamical systems by replacing ordinary differential equations and integral equations with algebraic polynomial equations, and by replacing convolution with multiplication.

For example, through the Laplace transform, the equation of the simple harmonic oscillator (Hooke's law) $$x^{″}(t)+kx(t)=0$$ is converted into the algebraic equation $$s^{2}X(s)-sx(0)-x^{\prime }(0)+kX(s)=0,$$ which incorporates the initial conditions $x(0)$ and Template:Tmath, and can be solved for the unknown function Template:Tmath. Once solved, the inverse Laplace transform can be used to transform it to the original domain. This is often aided by referencing tables such as that given below.

The Laplace transform is defined (for suitable functions Template:Tmath) by the integral $$ℒ\{f\}(s)={\int }_0^{\mathrm{\infty }}f(t)e^{-st}dt,$$ where Template:Tmath is a complex number.

The Laplace transform is related to many other transforms. It is essentially the same as the Mellin transform and is closely related to the Fourier transform. Unlike for the Fourier transform, the Laplace transform of a function is often an analytic function, meaning that it can be expressed as a power series that converges locally, the coefficients of which represent the moments of the original function. Moreover, the techniques of complex analysis, especially contour integrals, can be used for simplifying calculations.

History

Pierre-Simon, marquis de Laplace

The Laplace transform is named after mathematician and astronomer Pierre-Simon, Marquis de Laplace, who used a similar transform in his work on probability theory. Laplace wrote extensively about the use of generating functions (1814), and the integral form of the Laplace transform evolved naturally as a result.

Laplace's use of generating functions was similar to what is now known as the z-transform, and he gave little attention to the continuous variable case which was discussed by Niels Henrik Abel.

From 1744, Leonhard Euler investigated integrals of the form $$z=\int X(x)e^{ax}dx\text{ and }z=\int X(x)x^{A}dx$$ as solutions of differential equations, introducing in particular the gamma function. Joseph-Louis Lagrange was an admirer of Euler and, in his work on integrating probability density functions, investigated expressions of the form $$\int X(x)e^{-ax}{a}^{x}dx,$$ which resembles a Laplace transform.

These types of integrals seem first to have attracted Laplace's attention in 1782, where he was following in the spirit of Euler in using the integrals themselves as solutions of equations. However, in 1785, Laplace took the critical step forward when, rather than simply looking for a solution in the form of an integral, he started to apply the transforms in the sense that was later to become popular. He used an integral of the form $$\int x^s\phi (x)dx,$$ akin to a Mellin transform, to transform the whole of a difference equation, in order to look for solutions of the transformed equation. He then went on to apply the Laplace transform in the same way and started to derive some of its properties, beginning to appreciate its potential power.

Laplace also recognised that Joseph Fourier's method of Fourier series for solving the diffusion equation could only apply to a limited region of space, because those solutions were periodic. In 1809, Laplace applied his transform to find solutions that diffused indefinitely in space. In 1821, Cauchy developed an operational calculus for the Laplace transform that could be used to study linear differential equations in much the same way the transform is now used in basic engineering. This method was popularized, and perhaps rediscovered, by Oliver Heaviside around the turn of the century.

Bernhard Riemann used the Laplace transform in his 1859 paper On the number of primes less than a given magnitude, in which he also developed the inversion theorem. Riemann used the Laplace transform to develop the functional equation of the Riemann zeta function, and his method is still used to relate the modular transformation law of the Jacobi theta function, which is readily proved via Poisson summation, to the functional equation.

Hjalmar Mellin was among the first to study the Laplace transform rigorously in the Karl Weierstrass school of analysis, and apply it to the study of differential equations and special functions, at the turn of the 20th century. At around the same time, Heaviside was busy with his operational calculus. Thomas Joannes Stieltjes considered a generalization of the Laplace transform connected to his work on moments. Other contributors in this time period included Mathias Lerch, Oliver Heaviside, and Thomas Bromwich.

In 1929, Vannevar Bush and Norbert Wiener published Operational Circuit Analysis as a text for engineering analysis of electrical circuits, applying both Fourier transforms and operational calculus, and in which they included one of the first predecessors of the modern table of Laplace transforms. In 1934, Raymond Paley and Norbert Wiener published the important work Fourier transforms in the complex domain, about what is now called the Laplace transform (see below). Also during the 30s, the Laplace transform was instrumental in Godfrey Harold Hardy and John Edensor Littlewood's study of tauberian theorems, and this application was later expounded on by Template:Harvtxt, who developed other aspects of the theory such as a new method for inversion. Edward Charles Titchmarsh wrote the influential Introduction to the theory of the Fourier integral (1937).

The current widespread use of the transform (mainly in engineering) came about during and soon after World War II, replacing the earlier Heaviside operational calculus. The advantages of the Laplace transform had been emphasized by Gustav Doetsch.

Formal definition

$\mathrm{\Re }(e^{-st})$ for various complex frequencies in the Template:Math-domain Template:Tmath, which can be expressed as Template:Tmath. The axis at $\sigma =0$ contains pure cosines. Positive $\sigma$ contains damped cosines. Negative $\sigma$ contains exponentially growing cosines.

The Laplace transform of a function Template:Math, defined for all real numbers Template:Math, is the function Template:Math defined by Template:Equation box 1 where $s$ is a complex frequency-domain parameter $s=\sigma +i\omega$ with real numbers Template:Mvar and Template:Mvar.

An alternate notation for the Laplace transform is Template:Anchor$ℒ\{f\}$ instead of Template:Math. Thus $F(s)=ℒ\{f\}(s)$ in functional notation. This is often written, especially in engineering settings, as Template:Tmath, with the understanding that the dummy variable $t$ does not appear in the function Template:Tmath.

The meaning of the integral depends on types of functions of interest. A necessary condition for existence of the integral is that Template:Mvar must be locally integrable on Template:Closed-open. For locally integrable functions that decay at infinity or are of exponential type (Template:Tmath), 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 Template:Math. 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 Template:Mvar by the Lebesgue integral $$ℒ\{\mu \}(s)={\int }_{[0,\mathrm{\infty })}{e}^{-st}d\mu (t).$$

An important special case is where Template:Mvar 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 Template:Mvar. In that case, to avoid potential confusion, one often writes $$ℒ\{f\}(s)={\int }_{0^{-}}^{\mathrm{\infty }}f(t)e^{-st}dt,$$ where the lower limit of Template:Math is shorthand notation for $$\underset{\epsilon \to 0^{+}}{lim}{\int }_{-\epsilon }^{\mathrm{\infty }}.$$

This limit emphasizes that any point mass located at Template:Math is entirely captured by the Laplace transform. Although with the Lebesgue integral, it is not necessary to take such a limit, it does appear more naturally in connection with the Laplace–Stieltjes transform.

Bilateral Laplace transform

When one says "the Laplace transform" without qualification, the unilateral or one-sided transform is usually intended. The Laplace transform can be alternatively defined as the bilateral Laplace transform, or two-sided Laplace transform, by extending the limits of integration to be the entire real axis. If that is done, the common unilateral transform becomes a special case of the bilateral transform, where the definition of the function being transformed includes being multiplied by the Heaviside step function.

The bilateral Laplace transform Template:Math is defined as follows: Template:Equation box 1 An alternate notation for the bilateral Laplace transform is Template:Tmath, instead of Template:Mvar.

Inverse Laplace transform

Two integrable functions have the same Laplace transform only if they differ on a set of Lebesgue measure zero. This means that, on the range of the transform, there is an inverse transform. In fact, besides integrable functions, the Laplace transform is a one-to-one mapping from one function space into another in many other function spaces as well, although there is usually no easy characterization of the range.

Typical function spaces in which this is true include the spaces of bounded continuous functions, the space Template:Math, or more generally tempered distributions on Template:Open-open. 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): Template:Equation box 1 where Template:Mvar is a real number so that the contour path of integration is in the region of convergence of Template:Math. 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 Template:Mvar is a random variable with probability density function Template:Mvar, then the Laplace transform of Template:Mvar is given by the expectation $$ℒ\{f\}(s)=\mathrm{E}\left[e^{-sX}\right],$$ where $\mathrm{E}[r]$ is the expectation of random variable Template:Tmath.

By convention, this is referred to as the Laplace transform of the random variable Template:Mvar itself. Here, replacing Template:Mvar by Template:Math gives the moment generating function of Template:Mvar. 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 Template:Mvar by means of the Laplace transform as follows: $$F_X(x)={ℒ}^{-1}\{\frac{1}{s}\mathrm{E}\left[e^{-sX}\right]\}(x)={ℒ}^{-1}\{\frac{1}{s}ℒ\{f\}(s)\}(x).$$

Algebraic construction

The Laplace transform can be alternatively defined in a purely algebraic manner by applying a field of fractions construction to the convolution ring of functions on the positive half-line. The resulting space of abstract operators is exactly equivalent to Laplace space, but in this construction the forward and reverse transforms never need to be explicitly defined (avoiding the related difficulties with proving convergence).

Region of convergence

Template:See also If Template:Math is a locally integrable function (or more generally a Borel measure locally of bounded variation), then the Laplace transform Template:Math of Template:Math converges provided that the limit $$\underset{R\to \mathrm{\infty }}{lim}{\int }_0^{R}f(t)e^{-st}dt$$ exists.

The Laplace transform converges absolutely if the integral $${\int }_0^{\mathrm{\infty }}|f(t)e^{-st}|dt$$ 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 Template:Math converges absolutely is either of the form Template:Math or Template:Math, where Template:Math is an extended real constant with Template:Math (a consequence of the dominated convergence theorem). The constant Template:Math is known as the abscissa of absolute convergence, and depends on the growth behavior of Template:Math. Analogously, the two-sided transform converges absolutely in a strip of the form Template:Math, and possibly including the lines Template:Math or Template:Math. The subset of values of Template:Math for which the Laplace transform converges absolutely is called the region of absolute convergence, or the domain of absolute convergence. In the two-sided case, it is sometimes called the strip of absolute convergence. The Laplace transform is analytic in the region of absolute convergence: this is a consequence of Fubini's theorem and Morera's theorem.

Similarly, the set of values for which Template:Math 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 Template:Math, then it automatically converges for all Template:Math with Template:Math. Therefore, the region of convergence is a half-plane of the form Template:Math, possibly including some points of the boundary line Template:Math.

In the region of convergence Template:Math, the Laplace transform of Template:Math can be expressed by integrating by parts as the integral $$F(s)=(s-s_0){\int }_0^{\mathrm{\infty }}{e}^{-(s-s_0)t}\beta (t)dt,\beta (u)={\int }_0^u{e}^{-s_{0}t}f(t)dt.$$

That is, Template:Math can effectively be expressed, in the region of convergence, as the absolutely convergent Laplace transform of some other function. In particular, it is analytic. In its most general form, the Laplace transform gives a one-to-one correspondence between the holomorphic functions which, for some Template:Tmath, are defined on $\{s\in ℂ|\mathrm{R}\mathrm{e}(s)>\sigma \}$ and are bounded there in absolute value by a polynomial, and the distributions on the real line supported on Template:Tmath which become tempered distributions after multiplied by $e^{-\sigma t}$ for some Template:Tmath.

There are several Paley–Wiener theorems concerning the relationship between the decay properties of Template:Math, and the properties of the Laplace transform within the region of convergence.

In engineering applications, a function corresponding to a linear time-invariant (LTI) system is stable if every bounded input produces a bounded output. This is equivalent to the absolute convergence of the Laplace transform of the impulse response function in the region Template:Math. As a result, LTI systems are stable, provided that the poles of the Laplace transform of the impulse response function have negative real part.

This ROC is used in knowing about the causality and stability of a system.

Properties and theorems

The Laplace transform's key property is that it converts differentiation and integration in the time domain into multiplication and division by Template:Math in the Laplace domain. Thus, the Laplace variable Template:Math is also known as an operator variable in the Laplace domain: either the derivative operator or (for Template:Math the integration operator.

Given the functions Template:Math and Template:Math, and their respective Laplace transforms Template:Math and Template:Math, $$\begin{array}{cc}f(t)& ={ℒ}^{-1}\{F(s)\},\\ g(t)& ={ℒ}^{-1}\{G(s)\},\end{array}$$

the following table is a list of properties of unilateral Laplace transform:

Properties of the unilateral Laplace transform

Property	Time domain	Template:Math domain	Comment
Linearity	$af(t)+bg(t)$	$aF(s)+bG(s)$	Can be proved using basic rules of integration.
Frequency-domain derivative	$tf(t)$	$-F^{\prime }(s)$	Template:Math is the first derivative of Template:Math with respect to Template:Math.
Frequency-domain general derivative	$t^{n}f(t)$	$(-1{)}^n{F}^{(n)}(s)$	More general form, Template:Mathth derivative of Template:Math.
Derivative	$f^{\prime }(t)$	$sF(s)-f(0^{-})$	Template:Math 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	$f^{″}(t)$	$s^{2}F(s)-sf(0^{-})-f^{\prime }(0^{-})$	Template:Math is assumed twice differentiable and the second derivative to be of exponential type. Follows by applying the Differentiation property to Template:Math.
General derivative	$f^{(n)}(t)$	$s^{n}F(s)-\sum _{k=1}^n{s}^{n-k}{f}^{(k-1)}(0^{-})$	Template:Math is assumed to be Template:Math-times differentiable, with Template:Mathth derivative of exponential type. Follows by mathematical induction.
Frequency-domain integration	$\frac{1}{t}f(t)$	${\int }_s^{\mathrm{\infty }}F(\sigma )d\sigma$	This is deduced using the nature of frequency differentiation and conditional convergence.
Time-domain integration	${\int }_0^{t}f(\tau )d\tau =(u\ast f)(t)$	$\frac{1}{s}F(s)$	Template:Math is the Heaviside step function and Template:Math is the convolution of Template:Math and Template:Math.
Frequency shifting	$e^{at}f(t)$	$F(s-a)$
Time shifting	$f(t-a)u(t-a)$ $f(t)u(t-a)$	$e^{-as}F(s)$ $e^{-as}ℒ\{f(t+a)\}$	Template:Math, Template:Math is the Heaviside step function
Time scaling	$f(at)$	$\frac{1}{a}F\left(\frac{s}{a}\right)$	Template:Math
Multiplication	$f(t)g(t)$	$\frac{1}{2\pi i}\underset{T\to \mathrm{\infty }}{lim}{\int }_{c-iT}^{c+iT}F(\sigma )G(s-\sigma )d\sigma$	The integration is done along the vertical line Template:Math that lies entirely within the region of convergence of Template:Math.
Convolution	$(f\ast g)(t)={\int }_0^{t}f(\tau )g(t-\tau )d\tau$	$F(s)\cdot G(s)$
Circular convolution	$(f\ast g)(t)={\int }_0^{T}f(\tau )g(t-\tau )d\tau$	$F(s)\cdot G(s)$	For periodic functions with period Template:Math.
Complex conjugation	$f^{\ast }(t)$	$F^{\ast }(s^{\ast })$
Periodic function	$f(t)$	$\frac{1}{1-e^{-Ts}}{\int }_0^T{e}^{-st}f(t)dt$	Template:Math is a periodic function of period Template:Math so that Template:Math, for all Template:Math. This is the result of the time shifting property and the geometric series.
Periodic summation	$f_P(t)=\sum _{n=0}^{\mathrm{\infty }}f(t-Tn)$ $f_P(t)=\sum _{n=0}^{\mathrm{\infty }}(-1{)}^{n}f(t-Tn)$	$F_P(s)=\frac{1}{1-e^{-Ts}}F(s)$ $F_P(s)=\frac{1}{1+e^{-Ts}}F(s)$

Initial value theorem

$f(0^{+})=\underset{s\to \mathrm{\infty }}{lim}sF(s).$

Final value theorem

Template:Tmath, if all poles of $sF(s)$ are in the left half-plane.

The final value theorem is useful because it gives the long-term behaviour without having to perform partial fraction decompositions (or other difficult algebra). If Template:Math has a pole in the right-hand plane or poles on the imaginary axis (e.g., if $f(t)=e^t$ or Template:Tmath), 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. If Template:Math is a discrete function of a positive integer Template:Math, then the power series associated to Template:Math is the series $$\sum _{n=0}^{\mathrm{\infty }}a(n)x^n$$ where Template:Math is a real variable (see Z-transform). Replacing summation over Template:Math with integration over Template:Math, a continuous version of the power series becomes $${\int }_0^{\mathrm{\infty }}f(t)x^{t}dt$$ where the discrete function Template:Math is replaced by the continuous one Template:Math.

Changing the base of the power from Template:Math to Template:Math gives $${\int }_0^{\mathrm{\infty }}f(t){\left(e^{\ln x}\right)}^{t}dt$$

For this to converge for, say, all bounded functions Template:Math, it is necessary to require that Template:Math. Making the substitution Template:Math gives just the Laplace transform: $${\int }_0^{\mathrm{\infty }}f(t)e^{-st}dt$$

In other words, the Laplace transform is a continuous analog of a power series, in which the discrete parameter Template:Math is replaced by the continuous parameter Template:Math, and Template:Math is replaced by Template:Math.

Analogously to a power series, if Template:Tmath, then the power series converges to an analytic function in Template:Tmath, if Template:Tmath, the Laplace transform converges to an analytic function for Template:Tmath.Template:Sfn

Relation to moments

The quantities $${\mu }_n={\int }_0^{\mathrm{\infty }}{t}^{n}f(t)dt$$ are the moments of the function Template:Math. If the first Template:Math moments of Template:Math converge absolutely, then by repeated differentiation under the integral, $$(-1{)}^n(ℒf{)}^{(n)}(0)={\mu }_n.$$ This is of special significance in probability theory, where the moments of a random variable Template:Math are given by the expectation values Template:Tmath. Then, the relation holds $${\mu }_n=(-1{)}^n\frac{d^n}{d{s}^n}\mathrm{E}\left[e^{-sX}\right](0).$$

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: $$\begin{array}{cc}ℒ\{f(t)\}& ={\int }_{0^{-}}^{\mathrm{\infty }}{e}^{-st}f(t)dt\\ & ={\left[\frac{f(t)e^{-st}}{-s}\right]}_{0^{-}}^{\mathrm{\infty }}-{\int }_{0^{-}}^{\mathrm{\infty }}\frac{e^{-st}}{-s}{f}^{\prime }(t)dt\text{(by parts)}\\ & =[-\frac{f(0^{-})}{-s}]+\frac{1}{s}ℒ\{f^{\prime }(t)\},\end{array}$$ yielding $$ℒ\{f^{\prime }(t)\}=s\cdot ℒ\{f(t)\}-f(0^{-}),$$ and in the bilateral case, $$ℒ\{f^{\prime }(t)\}=s{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-st}f(t)dt=s\cdot ℒ\{f(t)\}.$$

The general result $$ℒ\{f^{(n)}(t)\}=s^n\cdot ℒ\{f(t)\}-s^{n-1}f(0^{-})-\cdots -f^{(n-1)}(0^{-}),$$ where $f^{(n)}$ denotes the Template:Mathth derivative of Template:Math, 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: $${\int }_0^{\mathrm{\infty }}f(x)g(x)dx={\int }_0^{\mathrm{\infty }}(ℒf)(s)\cdot ({ℒ}^{-1}g)(s)ds$$ under suitable assumptions on the behaviour of Template:Tmath and Template:Tmath in a right neighbourhood of $0$ and on the decay rate of Template:Tmath and Template:Tmath in a left neighbourhood of Template:Tmath. The above formula is a variation of integration by parts, with the operators $\frac{d}{dx}$ and $\int dx$ being replaced by $ℒ$ and Template:Tmath. Let us prove the equivalent formulation: $${\int }_0^{\mathrm{\infty }}(ℒf)(x)g(x)dx={\int }_0^{\mathrm{\infty }}f(s)(ℒg)(s)ds.$$

By plugging in $(ℒf)(x)={\int }_0^{\mathrm{\infty }}f(s)e^{-sx}ds$ the left-hand side turns into: $${\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}f(s)g(x)e^{-sx}dsdx,$$ but assuming Fubini's theorem holds, by reversing the order of integration we get the wanted right-hand side.

This method can be used to compute integrals that would otherwise be difficult to compute using elementary methods of real calculus. For example, the Dirichlet integral can be evaluated using the Laplace transform: $${\int }_0^{\mathrm{\infty }}\frac{\sin x}{x}dx={\int }_0^{\mathrm{\infty }}ℒ(1)(x)\sin xdx={\int }_0^{\mathrm{\infty }}1\cdot ℒ(\sin )(x)dx={\int }_0^{\mathrm{\infty }}\frac{dx}{x^2+1}=\frac{\pi }{2}.$$

Relationship to other transforms

Laplace–Stieltjes transform

The (unilateral) Laplace–Stieltjes transform of a function Template:Math is defined by the Lebesgue–Stieltjes integral $$\{{ℒ}^{\ast }g\}(s)={\int }_0^{\mathrm{\infty }}{e}^{-st}dg(t).$$

The function Template:Math is assumed to be of bounded variation. If Template:Math is the antiderivative of Template:Math: $$g(x)={\int }_0^{x}f(t)dt$$

then the Laplace–Stieltjes transform of Template:Mvar and the Laplace transform of Template:Mvar coincide. In general, the Laplace–Stieltjes transform is the Laplace transform of the Stieltjes measure associated to Template:Mvar. So in practice, the only distinction between the two transforms is that the Laplace transform is thought of as operating on the density function of the measure, whereas the Laplace–Stieltjes transform is thought of as operating on its cumulative distribution function.

Fourier transform

Let $f$ be a complex-valued Lebesgue integrable function supported on Template:Tmath, and let $F(s)=ℒf(s)$ be its Laplace transform. Then, within the region of convergence, we have $$F(\sigma +i\tau )={\int }_0^{\mathrm{\infty }}f(t)e^{-\sigma t}{e}^{-i\tau t}dt,$$ which is the Fourier transform of the function Template:Tmath.

Indeed, the Fourier transform is a special case (under certain conditions) of the bilateral Laplace transform. The main difference is that the Fourier transform of a function is a complex function of a real variable (frequency Template:Tmath), the Laplace transform of a function is a complex function of a complex variable (damping factor $\sigma$ and frequency Template:Tmath). The Laplace transform is usually restricted to transformation of functions of Template:Math with Template:Math. A consequence of this restriction is that the Laplace transform of a function is a holomorphic function of the variable Template:Math. Unlike the Fourier transform, the Laplace transform of a distribution is generally a well-behaved function. Techniques of complex variables can also be used to directly study Laplace transforms. As a holomorphic function, the Laplace transform has a power series representation. This power series expresses a function as a linear superposition of moments of the function. This perspective has applications in probability theory.

Formally, the Fourier transform is equivalent to evaluating the bilateral Laplace transform with imaginary argument Template:Math when the condition explained below is fulfilled, $$\begin{array}{cc}\hat{f}(\omega )& =ℱ\{f(t)\}\\ & =ℒ\{f(t)\}{|}_{s=i\omega }=F(s){|}_{s=i\omega }\\ & ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-i\omega t}f(t)dt.\end{array}$$

This convention of the Fourier transform (Template:Tmath in Template:Section link) requires a factor of Template:Math 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 Template:Math contains the imaginary axis, Template:Math.

For example, the function Template:Math has a Laplace transform Template:Math whose ROC is Template:Math. As Template:Math is a pole of Template:Math, substituting Template:Math in Template:Math does not yield the Fourier transform of Template:Math, which contains terms proportional to the Dirac delta functions Template:Math.

However, a relation of the form $$\underset{\sigma \to 0^{+}}{lim}F(\sigma +i\omega )=\hat{f}(\omega )$$ holds under much weaker conditions. For instance, this holds for the above example provided that the limit is understood as a weak limit of measures (see vague topology). General conditions relating the limit of the Laplace transform of a function on the boundary to the Fourier transform take the form of Paley–Wiener theorems.

Mellin transform

The Mellin transform and its inverse are related to the two-sided Laplace transform by a change of variables.

If in the Mellin transform $$G(s)=ℳ\{g(\theta )\}={\int }_0^{\mathrm{\infty }}{\theta }^{s}g(\theta )\frac{d\theta }{\theta }$$ we set Template:Math we get a two-sided Laplace transform.

Z-transform

The unilateral or one-sided Z-transform is the Laplace transform of an ideally sampled signal with the substitution of $$z\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{e}^{sT},$$ where Template:Math is the sampling interval (in units of time e.g., seconds) and Template:Math is the sampling rate (in samples per second or hertz).

Let $${\mathrm{\Delta }}_T(t)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\sum _{n=0}^{\mathrm{\infty }}\delta (t-nT)$$ be a sampling impulse train (also called a Dirac comb) and $$\begin{array}{cc}{x}_q(t)& \stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}x(t){\mathrm{\Delta }}_T(t)=x(t)\sum _{n=0}^{\mathrm{\infty }}\delta (t-nT)\\ & =\sum _{n=0}^{\mathrm{\infty }}x(nT)\delta (t-nT)=\sum _{n=0}^{\mathrm{\infty }}x[n]\delta (t-nT)\end{array}$$ be the sampled representation of the continuous-time Template:Math $$x[n]\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}x(nT).$$

The Laplace transform of the sampled signal Template:Math is $$\begin{array}{cc}{X}_q(s)& ={\int }_{0^{-}}^{\mathrm{\infty }}{x}_q(t)e^{-st}dt\\ & ={\int }_{0^{-}}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}x[n]\delta (t-nT)e^{-st}dt\\ & =\sum _{n=0}^{\mathrm{\infty }}x[n]{\int }_{0^{-}}^{\mathrm{\infty }}\delta (t-nT)e^{-st}dt\\ & =\sum _{n=0}^{\mathrm{\infty }}x[n]e^{-nsT}.\end{array}$$

This is the precise definition of the unilateral Z-transform of the discrete function Template:Math $$X(z)=\sum _{n=0}^{\mathrm{\infty }}x[n]z^{-n}$$ with the substitution of Template:Math.

Comparing the last two equations, we find the relationship between the unilateral Z-transform and the Laplace transform of the sampled signal, $$X_q(s)=X(z){|}_{z=e^{sT}}.$$

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 $$F(s)={\int }_0^{\mathrm{\infty }}f(z)e^{-sz}dz$$ is a special case of the Laplace transform for Template:Math an entire function of exponential type, meaning that $$|f(z)|\le A{e}^{B|z|}$$ for some constants Template:Math and Template:Math. The generalized Borel transform allows a different weighting function to be used, rather than the exponential function, to transform functions not of exponential type. Nachbin's theorem gives necessary and sufficient conditions for the Borel transform to be well defined.

Fundamental relationships

Since an ordinary Laplace transform can be written as a special case of a two-sided transform, and since the two-sided transform can be written as the sum of two one-sided transforms, the theory of the Laplace-, Fourier-, Mellin-, and Z-transforms are at bottom the same subject. However, a different point of view and different characteristic problems are associated with each of these four major integral transforms.

Table of selected Laplace transforms

The following table provides Laplace transforms for many common functions of a single variable. For definitions and explanations, see the Explanatory Notes at the end of the table.

Because the Laplace transform is a linear operator,

• The Laplace transform of a sum is the sum of Laplace transforms of each term.$$ℒ\{f(t)+g(t)\}=ℒ\{f(t)\}+ℒ\{g(t)\}$$
• The Laplace transform of a multiple of a function is that multiple times the Laplace transformation of that function.$$ℒ\{af(t)\}=aℒ\{f(t)\}$$

Using this linearity, and various trigonometric, hyperbolic, and complex number (etc.) properties and/or identities, some Laplace transforms can be obtained from others more quickly than by using the definition directly.

The unilateral Laplace transform takes as input a function whose time domain is the non-negative reals, which is why all of the time domain functions in the table below are multiples of the Heaviside step function, Template:Math.

The entries of the table that involve a time delay Template:Math are required to be causal (meaning that Template:Math). A causal system is a system where the impulse response Template:Math is zero for all time Template:Mvar prior to Template:Math. In general, the region of convergence for causal systems is not the same as that of anticausal systems.

Selected Laplace transforms

Function	Time domain Template:Br $f(t)={ℒ}^{-1}\{F(s)\}$	Laplace Template:Math-domain $F(s)=ℒ\{f(t)\}$	Region of convergence	Reference
unit impulse	$\delta (t)$	$1$	all Template:Math	inspection
delayed impulse	$\delta (t-\tau )$	$e^{-\tau s}$	all Template:Math	time shift ofTemplate:Brunit impulse
unit step	$u(t)$	$\frac{1}{s}$	$\mathrm{Re}(s)>0$	integrate unit impulse
delayed unit step	$u(t-\tau )$	$\frac{1}{s}{e}^{-\tau s}$	$\mathrm{Re}(s)>0$	time shift ofTemplate:Brunit step
product of delayed function and delayed step	$f(t-\tau )u(t-\tau )$	$e^{-s\tau }ℒ\{f(t)\}$		u-substitution, $u=t-\tau$
rectangular impulse	$u(t)-u(t-\tau )$	$\frac{1}{s}(1-e^{-\tau s})$	$\mathrm{Re}(s)>0$
ramp	$t\cdot u(t)$	$\frac{1}{s^2}$	$\mathrm{Re}(s)>0$	integrate unitTemplate:Brimpulse twice
Template:Mathth power (for integer Template:Math)	$t^n\cdot u(t)$	$\frac{n!}{s^{n+1}}$	$\mathrm{Re}(s)>0$ (Template:Math)	integrate unitTemplate:Brstep Template:Math times
Template:Mathth power (for complex Template:Math)	$t^q\cdot u(t)$	$\frac{\mathrm{\Gamma }(q+1)}{s^{q+1}}$	$\mathrm{Re}(s)>0$ $\mathrm{Re}(q)>-1$
Template:Mathth root	$\sqrt[n]{t}\cdot u(t)$	$\frac{1}{s^{\frac{1}{n}+1}}\mathrm{\Gamma }(\frac{1}{n}+1)$	$\mathrm{Re}(s)>0$	Set Template:Math above.
Template:Mathth power with frequency shift	$t^n{e}^{-\alpha t}\cdot u(t)$	$\frac{n!}{(s+\alpha {)}^{n+1}}$	$\mathrm{Re}(s)>-\alpha$	Integrate unit step, apply frequency shift
delayed Template:Mathth power with frequency shift	$(t-\tau {)}^n{e}^{-\alpha (t-\tau )}\cdot u(t-\tau )$	$\frac{n!\cdot e^{-\tau s}}{(s+\alpha {)}^{n+1}}$	$\mathrm{Re}(s)>-\alpha$	integrate unit step,Template:Brapply frequency shift,Template:Brapply time shift
exponential decay	$e^{-\alpha t}\cdot u(t)$	$\frac{1}{s+\alpha }$	$\mathrm{Re}(s)>-\alpha$	Frequency shift ofTemplate:Brunit step
two-sided exponential decay Template:Br(only for bilateral transform)	$e^{-\alpha |t|}$	$\frac{2\alpha }{{\alpha }^2-s^2}$	$-\alpha <\mathrm{Re}(s)<\alpha$	Frequency shift ofTemplate:Brunit step
exponential approach	$(1-e^{-\alpha t})\cdot u(t)$	$\frac{\alpha }{s(s+\alpha )}$	$\mathrm{Re}(s)>0$	unit step minus exponential decay
sine	$\sin (\omega t)\cdot u(t)$	$\frac{\omega }{s^2+{\omega }^2}$	$\mathrm{Re}(s)>0$	Template:Sfn
cosine	$\cos (\omega t)\cdot u(t)$	$\frac{s}{s^2+{\omega }^2}$	$\mathrm{Re}(s)>0$	Template:Sfn
hyperbolic sine	$\sinh (\alpha t)\cdot u(t)$	$\frac{\alpha }{s^2-{\alpha }^2}$	$\mathrm{Re}(s)>\left|\alpha \right|$	Template:Sfn
hyperbolic cosine	$\cosh (\alpha t)\cdot u(t)$	$\frac{s}{s^2-{\alpha }^2}$	$\mathrm{Re}(s)>\left|\alpha \right|$	Template:Sfn
exponentially decaying sine wave	$e^{-\alpha t}\sin (\omega t)\cdot u(t)$	$\frac{\omega }{(s+\alpha {)}^2+{\omega }^2}$	$\mathrm{Re}(s)>-\alpha$	Template:Sfn
exponentially decaying cosine wave	$e^{-\alpha t}\cos (\omega t)\cdot u(t)$	$\frac{s+\alpha }{(s+\alpha {)}^2+{\omega }^2}$	$\mathrm{Re}(s)>-\alpha$	Template:Sfn
natural logarithm	$\ln (t)\cdot u(t)$	$-\frac{1}{s}[\ln (s)+\gamma ]$	$\mathrm{Re}(s)>0$	Template:Sfn
Bessel function Template:Br of the first kind, of order Template:Math	$J_n(\omega t)\cdot u(t)$	$\frac{{(\sqrt{s^2+{\omega }^2}-s)}^n}{{\omega }^n\sqrt{s^2+{\omega }^2}}$	$\mathrm{Re}(s)>0$ (Template:Math)	Template:Sfn
Error function	$\mathrm{erf}(t)\cdot u(t)$	$\frac{1}{s}{e}^{s^2/4}(1-\mathrm{erf}\frac{s}{2})$	$\mathrm{Re}(s)>0$	Template:Sfn
Explanatory notes: Template:Math represents the Heaviside step function. Template:Math represents the Dirac delta function. Template:Math represents the gamma function. Template:Math is the Euler–Mascheroni constant. Template:Math, a real number, typically represents time, although it can represent any independent dimension. Template:Math is the complex frequency domain parameter, and Template:Math is its real part. Template:Math are real numbers. Template:Math is an integer.

s-domain equivalent circuits and impedances

The Laplace transform is often used in circuit analysis by conversions to the Template:Math-domain of circuit elements. Circuit elements can be transformed into impedances, very similar to phasor impedances.

Here is a summary of equivalents:

Template:Math-domain equivalent circuits

Note that the resistor is exactly the same in the time domain and the Template:Math-domain. The sources are put in if there are initial conditions on the circuit elements. For example, if a capacitor has an initial voltage across it, or if the inductor has an initial current through it, the sources inserted in the Template:Math-domain account for that.

The equivalents for current and voltage sources are derived from the transformations in the table above.

Examples and applications

The Laplace transform is used frequently in engineering and physics; the output of a linear time-invariant system can be calculated by convolving its unit impulse response with the input signal. Performing this calculation in Laplace space turns the convolution into a multiplication; the latter being easier to solve because of its algebraic form. For more information, see control theory. The Laplace transform is invertible on a large class of functions. Given a mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.

The Laplace transform can also be used to solve differential equations and is used extensively in mechanical engineering and electrical engineering. The Laplace transform reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of algebra. The original differential equation can then be solved by applying the inverse Laplace transform. English electrical engineer Oliver Heaviside first proposed a similar scheme, although without using the Laplace transform; and the resulting operational calculus is credited as the Heaviside calculus.

Evaluating improper integrals

Let Template:Tmath. Then (see the table above) $${\partial }_sℒ\left\{\frac{f(t)}{t}\right\}={\partial }_s{\int }_0^{\mathrm{\infty }}\frac{f(t)}{t}{e}^{-st}dt=-{\int }_0^{\mathrm{\infty }}f(t)e^{-st}dt=-F(s)$$

From which one gets: $$ℒ\left\{\frac{f(t)}{t}\right\}={\int }_s^{\mathrm{\infty }}F(p)dp.$$

In the limit Template:Tmath, one gets $${\int }_0^{\mathrm{\infty }}\frac{f(t)}{t}dt={\int }_0^{\mathrm{\infty }}F(p)dp,$$ 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 Template:Math, proceeding formally one has $$\begin{array}{cc}{\int }_0^{\mathrm{\infty }}\frac{\cos (at)-\cos (bt)}{t}dt& ={\int }_0^{\mathrm{\infty }}(\frac{p}{p^2+a^2}-\frac{p}{p^2+b^2})dp\\ & ={[\frac{1}{2}\ln \frac{p^2+a^2}{p^2+b^2}]}_0^{\mathrm{\infty }}=\frac{1}{2}\ln \frac{b^2}{a^2}=\ln \left|\frac{b}{a}\right|.\end{array}$$

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 $$i=C\frac{dv}{dt},$$ where Template:Math is the capacitance of the capacitor, Template:Math is the electric current through the capacitor as a function of time, and Template:Math is the voltage across the terminals of the capacitor, also as a function of time.

Taking the Laplace transform of this equation, we obtain $$I(s)=C(sV(s)-V_0),$$ where $$\begin{array}{cc}I(s)& =ℒ\{i(t)\},\\ V(s)& =ℒ\{v(t)\},\end{array}$$ and $$V_0=v(0).$$

Solving for Template:Math we have $$V(s)=\frac{I(s)}{sC}+\frac{V_0}{s}.$$

The definition of the complex impedance Template:Math (in ohms) is the ratio of the complex voltage Template:Math divided by the complex current Template:Math while holding the initial state Template:Math at zero: $$Z(s)={\frac{V(s)}{I(s)}|}_{V_0=0}.$$

Using this definition and the previous equation, we find: $$Z(s)=\frac{1}{sC},$$ which is the correct expression for the complex impedance of a capacitor. In addition, the Laplace transform has large applications in control theory.

Impulse response

Consider a linear time-invariant system with transfer function $$H(s)=\frac{1}{(s+\alpha )(s+\beta )}.$$

The impulse response is the inverse Laplace transform of this transfer function: $$h(t)={ℒ}^{-1}\{H(s)\}.$$

Partial fraction expansion

To evaluate this inverse transform, we begin by expanding Template:Math using the method of partial fraction expansion, $$\frac{1}{(s+\alpha )(s+\beta )}=\frac{P}{s+\alpha }+\frac{R}{s+\beta }.$$

The unknown constants Template:Math and Template:Math 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 Template:Math, we multiply both sides of the equation by Template:Math to get $$\frac{1}{s+\beta }=P+\frac{R(s+\alpha )}{s+\beta }.$$

Then by letting Template:Math, the contribution from Template:Math vanishes and all that is left is $$P={\frac{1}{s+\beta }|}_{s=-\alpha }=\frac{1}{\beta -\alpha }.$$

Similarly, the residue Template:Math is given by $$R={\frac{1}{s+\alpha }|}_{s=-\beta }=\frac{1}{\alpha -\beta }.$$

Note that $$R=\frac{-1}{\beta -\alpha }=-P$$ and so the substitution of Template:Math and Template:Math into the expanded expression for Template:Math gives $$H(s)=\left(\frac{1}{\beta -\alpha }\right)\cdot (\frac{1}{s+\alpha }-\frac{1}{s+\beta }).$$

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 Template:Math to obtain $$h(t)={ℒ}^{-1}\{H(s)\}=\frac{1}{\beta -\alpha }(e^{-\alpha t}-e^{-\beta t}),$$ 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 Template:Math and Template:Math. That is, the inverse of $$H(s)=\frac{1}{(s+\alpha )(s+\beta )}=\frac{1}{s+\alpha }\cdot \frac{1}{s+\beta }$$ is $${ℒ}^{-1}\left\{\frac{1}{s+\alpha }\right\}\ast {ℒ}^{-1}\left\{\frac{1}{s+\beta }\right\}=e^{-\alpha t}\ast e^{-\beta t}={\int }_0^t{e}^{-\alpha x}{e}^{-\beta (t-x)}dx=\frac{e^{-\alpha t}-e^{-\beta t}}{\beta -\alpha }.$$

Phase delay

Time function	Laplace transform
$\sin (\omega t+\phi )$	$\frac{s\sin (\phi )+\omega \cos (\phi )}{s^2+{\omega }^2}$
$\cos (\omega t+\phi )$	$\frac{s\cos (\phi )-\omega \sin (\phi )}{s^2+{\omega }^2}.$

Starting with the Laplace transform, $$X(s)=\frac{s\sin (\phi )+\omega \cos (\phi )}{s^2+{\omega }^2}$$ we find the inverse by first rearranging terms in the fraction: $$\begin{array}{cc}X(s)& =\frac{s\sin (\phi )}{s^2+{\omega }^2}+\frac{\omega \cos (\phi )}{s^2+{\omega }^2}\\ & =\sin (\phi )\left(\frac{s}{s^2+{\omega }^2}\right)+\cos (\phi )\left(\frac{\omega }{s^2+{\omega }^2}\right).\end{array}$$

We are now able to take the inverse Laplace transform of our terms: $$\begin{array}{cc}x(t)& =\sin (\phi ){ℒ}^{-1}\left\{\frac{s}{s^2+{\omega }^2}\right\}+\cos (\phi ){ℒ}^{-1}\left\{\frac{\omega }{s^2+{\omega }^2}\right\}\\ & =\sin (\phi )\cos (\omega t)+\cos (\phi )\sin (\omega t).\end{array}$$

This is just the sine of the sum of the arguments, yielding: $$x(t)=\sin (\omega t+\phi ).$$

We can apply similar logic to find that $${ℒ}^{-1}\left\{\frac{s\cos \phi -\omega \sin \phi }{s^2+{\omega }^2}\right\}=\cos (\omega t+\phi ).$$

Statistical mechanics

In statistical mechanics, the Laplace transform of the density of states $g(E)$ defines the partition function. That is, the canonical partition function $Z(\beta )$ is given by $$Z(\beta )={\int }_0^{\mathrm{\infty }}{e}^{-\beta E}g(E)dE$$ and the inverse is given by $$g(E)=\frac{1}{2\pi i}{\int }_{{\beta }_0-i\mathrm{\infty }}^{{\beta }_0+i\mathrm{\infty }}{e}^{\beta E}Z(\beta )d\beta$$

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. When independent information on the structure of an object is available, the inverse Laplace transform method has been found to be in good agreement.

Birth and death processes

Consider a random walk, with steps $\{+1,-1\}$ occurring with probabilities Template:Tmath. Suppose also that the time step is a Poisson process, with parameter Template:Tmath. Then the probability of the walk being at the lattice point $n$ at time $t$ is $$P_n(t)={\int }_0^t\lambda e^{-\lambda (t-s)}(p{P}_{n-1}(s)+q{P}_{n+1}(s))ds(+e^{-\lambda t}\text{when}n=0).$$ This leads to a system of integral equations (or equivalently a system of differential equations). However, because it is a system of convolution equations, the Laplace transform converts it into a system of linear equations for $${\pi }_n(s)=ℒ(P_n)(s),$$ namely: $${\pi }_n(s)=\frac{\lambda }{\lambda +s}(p{\pi }_{n-1}(s)+q{\pi }_{n+1}(s))(+\frac{1}{\lambda +s}\text{when}n=0)$$ which may now be solved by standard methods.

Tauberian theory

The Laplace transform of the measure $\mu$ on $[0,\mathrm{\infty })$ is given by $$ℒ\mu (s)={\int }_0^{\mathrm{\infty }}{e}^{-st}d\mu (t).$$ It is intuitively clear that, for small Template:Tmath, the exponentially decaying integrand will become more sensitive to the concentration of the measure $\mu$ on larger subsets of the domain. To make this more precise, introduce the distribution function: $$M(t)=\mu ([0,t)).$$ Formally, we expect a limit of the following kind: $$\underset{s\to 0^{+}}{lim}ℒ\mu (s)=\underset{t\to \mathrm{\infty }}{lim}M(t).$$ Tauberian theorems are theorems relating the asymptotics of the Laplace transform, as Template:Tmath, to those of the distribution of $\mu$ as Template:Tmath. They are thus of importance in asymptotic formulae of probability and statistics, where often the spectral side has asymptotics that are simpler to infer.

Two Tauberian theorems of note are the Hardy–Littlewood Tauberian theorem and Wiener's Tauberian theorem. The Wiener theorem generalizes the Ikehara Tauberian theorem, which is the following statement:

Let Template:Tmath be a non-negative, monotonic nondecreasing function of Template:Tmath, defined for Template:Tmath. Suppose that $$f(s)={\int }_0^{\mathrm{\infty }}A(x)e^{-xs}dx$$ converges for Template:Tmath to the function Template:Tmath and that, for some non-negative number Template:Tmath, $$f(s)-\frac{c}{s-1}$$ has an extension as a continuous function for Template:Tmath. Then the limit as Template:Tmath goes to infinity of Template:Tmath is equal to Template:Tmath.

This statement can be applied in particular to the logarithmic derivative of Riemann zeta function, and thus provides an extremely short way to prove the prime number theorem.

See also

• Analog signal processing
• Bernstein's theorem on monotone functions
• Continuous-repayment mortgage
• Dirichlet integral
• Differential equation
• Generating function
• Hamburger moment problem
• Hardy–Littlewood Tauberian theorem
• Laplace–Carson transform
• Moment-generating function
• Nonlocal operator
• Partial fraction decomposition
• Post's inversion formula
• Signal-flow graph
• Transfer function
• Z-transform

References

Modern

• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation

Historical

• Template:Citation
• Template:Citation
• Template:Citation, Chapters 3–5
• Template:Citation
• Template:Citation
• Template:Citation

Further reading

• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation – see Chapter VI. The Laplace transform
• Template:Citation
• Template:Citation
• Template:Citation

External links

• Template:Springer
• Online Computation of the transform or inverse transform, wims.unice.fr
• Tables of Integral Transforms at EqWorld: The World of Mathematical Equations.
• Template:MathWorld
• Good explanations of the initial and final value theorems Template:Webarchive
• 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.