Laplace transform

Inmathematics,theLaplace transform,named afterPierre-Simon Laplace(/ləˈplɑːs/), is anintegral transformthat converts afunctionof arealvariable(usually$t$,in thetime domain) to a function of acomplexvariable$s$(in the complex-valuedfrequency domain,also known ass-domain,ors-plane).

The transform is useful for convertingdifferentiationandintegrationin the time domain into much easiermultiplicationanddivisionin the Laplace domain (analogous to howlogarithmsare useful for simplifying multiplication and division into addition and subtraction). This gives the transform many applications inscienceandengineering,mostly as a tool for solving lineardifferential equationsanddynamical systemsby simplifyingordinary differential equationsandintegral equationsintoalgebraic polynomial equations,and by simplifyingconvolutionintomultiplication.Once solved, the inverse Laplace transform reverts to the original domain.

The Laplace transform is defined (for suitable functions$f$) by theintegral $${\mathcal {L}}\{f\}(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt,$$ wheresis acomplex number.It is related to many other transforms, most notably theFourier transformand theMellin transform.Formally,the Laplace transform is converted into a Fourier transform by the substitution$s=i\ Omega$where$\ Omega$is real. However, unlike the Fourier transform, which gives the decomposition of a function into its components in each frequency, the Laplace transform of a function with suitable decay is ananalytic function,and so has a convergentpower series,the coefficients of which give the decomposition of a function into itsmoments.Also unlike the Fourier transform, when regarded in this way as an analytic function, the techniques ofcomplex analysis,and especiallycontour integrals,can be used for calculations.

History

Pierre-Simon, marquis de Laplace

The Laplace transform is named aftermathematicianandastronomerPierre-Simon, Marquis de Laplace,who used a similar transform in his work onprobability theory.Laplace wrote extensively about the use ofgenerating 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 thez-transform,and he gave little attention to thecontinuous variablecase which was discussed byNiels Henrik Abel.

From 1744,Leonhard Eulerinvestigated integrals of the form $$z=\int X(x)e^{ax}\,dx\quad {\text{ and }}\quad z=\int X(x)x^{A}\,dx$$ as solutions of differential equations, introducing in particular thegamma function.Joseph-Louis Lagrangewas an admirer of Euler and, in his work on integratingprobability 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 $$\int x^{s}\varphi (x)\,dx,$$ akin to aMellin transform,to transform the whole of adifference 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 thatJoseph Fourier's method ofFourier seriesfor solving thediffusion equationcould only apply to a limited region of space, because those solutions wereperiodic.In 1809, Laplace applied his transform to find solutions that diffused indefinitely in space.In 1821,Cauchydeveloped anoperational calculusfor 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, byOliver Heavisidearound the turn of the century.

Bernhard Riemannused the Laplace transform in his 1859 paperOn 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 theRiemann zeta function,and this method is still used to related themodular transformation lawof theJacobi theta function,which is simple to prove viaPoisson summation,to the functional equation.

Hjalmar Mellinwas among the first to study the Laplace transform, rigorously in theKarl Weierstrassschool of analysis, and apply it to the study ofdifferential equationsandspecial functions,at the turn of the 20th century.At around the same time, Heaviside was busy with his operational calculus.Thomas Joannes Stieltjesconsidered a generalization of the Laplace transform connected to hiswork on moments.Other contributors in this time period includedMathias Lerch,Oliver Heaviside,andThomas Bromwich.

In 1934,Raymond PaleyandNorbert Wienerpublished the important workFourier transforms in the complex domain,about what is now called the Laplace transform (see below). Also during the 30s, the Laplace transform was instrumental inG H HardyandJohn Edensor Littlewood's study oftauberian theorems,and this application was later expounded on by Widder (1941), who developed other aspects of the theory such as a new method for inversion.Edward Charles Titchmarshwrote the influentialIntroduction to the theory of the Fourier integral(1937).

The current widespread use of the transform (mainly in engineering) came about during and soon afterWorld War II,replacing the earlier Heavisideoperational calculus.The advantages of the Laplace transform had been emphasized byGustav Doetsch,to whom the name Laplace transform is apparently due.

Formal definition

$\Re (e^{-st})$for various complex frequencies in thes-domain$(s=\sigma +i\ Omega ),$which can be expressed as$e^{-\sigma t}\cos(\ Omega t).$The$\sigma =0$axis contains pure cosines. Positive$\sigma$containsdamped cosines.Negative$\sigma$containsexponentially growingcosines.

The Laplace transform of afunctionf(t),defined for allreal numberst≥ 0,is the functionF(s),which is a unilateral transform defined by

$F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt,$(Eq. 1)

wheresis acomplexfrequency-domain parameter $$s=\sigma +i\ Omega$$ with real numbersσandω.

An alternate notation for the Laplace transform is${\mathcal {L}}\{f\}$instead ofF.

The meaning of the integral depends on types of functions of interest. A necessary condition for existence of the integral is thatfmust belocally integrableon[0, ∞).For locally integrable functions that decay at infinity or are ofexponential type($|f(t)|\leq Ae^{B|t|}$), the integral can be understood to be a (proper)Lebesgue integral.However, for many applications it is necessary to regard it as aconditionally convergentimproper integralat∞.Still more generally, the integral can be understood in aweak sense,and this is dealt with below.

One can define the Laplace transform of a finiteBorel measureμby the Lebesgue integral $${\mathcal {L}}\{\mu \}(s)=\int _{[0,\infty )}e^{-st}\,d\mu (t).$$

An important special case is whereμis aprobability measure,for example, theDirac delta function.Inoperational calculus,the Laplace transform of a measure is often treated as though the measure came from a probability density functionf.In that case, to avoid potential confusion, one often writes $${\mathcal {L}}\{f\}(s)=\int _{0^{-}}^{\infty }f(t)e^{-st}\,dt,$$ where the lower limit of0⁻is shorthand notation for $$\lim _{\varepsilon \to 0^{+}}\int _{-\varepsilon }^{\infty }.$$

This limit emphasizes that any point mass located at0is 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 theLaplace–Stieltjes transform.

Bilateral Laplace transform

Main article:Two-sided 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 thebilateral Laplace transform,ortwo-sided Laplace transform,by extending the limits of integration to be the entire real axis. If that is done, the common unilateral transform simply becomes a special case of the bilateral transform, where the definition of the function being transformed is multiplied by theHeaviside step function.

The bilateral Laplace transformF(s)is defined as follows:

$F(s)=\int _{-\infty }^{\infty }e^{-st}f(t)\,dt.$(Eq. 2)

An alternate notation for the bilateral Laplace transform is${\mathcal {B}}\{f\}$,instead ofF.

Inverse Laplace transform

Main article:Inverse Laplace transform

Two integrable functions have the same Laplace transform only if they differ on a set ofLebesgue measurezero. This means that, on the range of the transform, there is an inverse transform. In fact, besides integrable functions, the Laplace transform is aone-to-one mappingfrom 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 spaceL^∞(0, ∞),or more generallytempered distributionson(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 ofanalytic functionsin theregion of convergence.Theinverse Laplace transformis given by the following complex integral, which is known by various names (theBromwich integral,theFourier–Mellin integral,andMellin's inverse formula):

$f(t)={\mathcal {L}}^{-1}\{F\}(t)={\frac {1}{2\pi i}}\lim _{T\to \infty }\int _{\gamma -iT}^{\gamma +iT}e^{st}F(s)\,ds,$(Eq. 3)

whereγis a real number so that the contour path of integration is in the region of convergence ofF(s).In most applications, the contour can be closed, allowing the use of theresidue theorem.An alternative formula for the inverse Laplace transform is given byPost's inversion formula.The limit here is interpreted in theweak-* 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

Inpureandapplied probability,the Laplace transform is defined as anexpected value.IfXis arandom variablewith probability density functionf,then the Laplace transform offis given by the expectation $${\mathcal {L}}\{f\}(s)=\operatorname {E} \left[e^{-sX}\right],$$ where$\operatorname {E} [r]$is theexpectationofrandom variable$r$.

Byconvention,this is referred to as the Laplace transform of the random variableXitself. Here, replacingsby−tgives themoment generating functionofX.The Laplace transform has applications throughout probability theory, includingfirst passage timesofstochastic processessuch asMarkov chains,andrenewal theory.

Of particular use is the ability to recover thecumulative distribution functionof a continuous random variableXby means of the Laplace transform as follows: $$F_{X}(x)={\mathcal {L}}^{-1}\left\{{\frac {1}{s}}\operatorname {E} \left[e^{-sX}\right]\right\}(x)={\mathcal {L}}^{-1}\left\{{\frac {1}{s}}{\mathcal {L}}\{f\}(s)\right\}(x).$$

Algebraic construction

The Laplace transform can be alternatively defined in a purely algebraic manner by applying afield of fractionsconstruction to the convolutionringof functions on the positive half-line. The resultingspace of abstract operatorsis 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

See also:Pole–zero plot § Continuous-time systems

Iffis a locally integrable function (or more generally a Borel measure locally of bounded variation), then the Laplace transformF(s)offconverges provided that the limit $$\lim _{R\to \infty }\int _{0}^{R}f(t)e^{-st}\,dt$$ exists.

The Laplace transformconverges absolutelyif the integral $$\int _{0}^{\infty }\left|f(t)e^{-st}\right|\,dt$$ exists as a proper Lebesgue integral. The Laplace transform is usually understood asconditionally convergent,meaning that it converges in the former but not in the latter sense.

The set of values for whichF(s)converges absolutely is either of the formRe(s) >aorRe(s) ≥a,whereais anextended real constantwith−∞ ≤a≤ ∞(a consequence of thedominated convergence theorem). The constantais known as the abscissa of absolute convergence, and depends on the growth behavior off(t).Analogously, the two-sided transform converges absolutely in a strip of the forma< Re(s) <b,and possibly including the linesRe(s) =aorRe(s) =b.The subset of values ofsfor 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 ofFubini's theoremandMorera's theorem.

Similarly, the set of values for whichF(s)converges (conditionally or absolutely) is known as the region of conditional convergence, or simply theregion of convergence(ROC). If the Laplace transform converges (conditionally) ats=s₀,then it automatically converges for allswithRe(s) > Re(s₀).Therefore, the region of convergence is a half-plane of the formRe(s) >a,possibly including some points of the boundary lineRe(s) =a.

In the region of convergenceRe(s) > Re(s₀),the Laplace transform offcan be expressed byintegrating by partsas the integral $$F(s)=(s-s_{0})\int _{0}^{\infty }e^{-(s-s_{0})t}\beta (t)\,dt,\quad \beta (u)=\int _{0}^{u}e^{-s_{0}t}f(t)\,dt.$$

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 severalPaley–Wiener theoremsconcerning the relationship between the decay properties off,and the properties of the Laplace transform within the region of convergence.

In engineering applications, a function corresponding to alinear time-invariant (LTI) systemisstableif 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 regionRe(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 convertsdifferentiationandintegrationin the time domain into multiplication and division bysin the Laplace domain. Thus, the Laplace variablesis also known asoperator variablein the Laplace domain: either thederivative operatoror (fors⁻¹)theintegration operator.

Given the functionsf(t)andg(t),and their respective Laplace transformsF(s)andG(s), $${\begin{aligned}f(t)&={\mathcal {L}}^{-1}\{F\}(s),\\g(t)&={\mathcal {L}}^{-1}\{G\}(s),\end{aligned}}$$

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

Properties of the unilateral Laplace transform

Property	Time domain	sdomain	Comment
Linearity	$af(t)+bg(t)\$	$aF(s)+bG(s)\$	Can be proved using basic rules of integration.
Frequency-domain derivative	$tf(t)\$	$-F'(s)\$	F′is the first derivative ofFwith respect tos.
Frequency-domain general derivative	$t^{n}f(t)\$	$(-1)^{n}F^{(n)}(s)\$	More general form,nth derivative ofF(s).
Derivative	$f'(t)\$	$sF(s)-f(0^{-})\$	fis assumed to be adifferentiable function,and its derivative is assumed to be of exponential type. This can then be obtained by integration by parts
Second derivative	$f''(t)\$	${\textstyle s^{2}F(s)-sf(0^{-})-f'(0^{-})\ }$	fis assumed twice differentiable and the second derivative to be of exponential type. Follows by applying the Differentiation property tof′(t).
General derivative	$f^{(n)}(t)\$	$s^{n}F(s)-\sum _{k=1}^{n}s^{n-k}f^{(k-1)}(0^{-})\$	fis assumed to ben-times differentiable, withnth derivative of exponential type. Follows bymathematical induction.
Frequency-domainintegration	${\frac {1}{t}}f(t)\$	$\int _{s}^{\infty }F(\sigma )\,d\sigma \$	This is deduced using the nature of frequency differentiation and conditional convergence.
Time-domain integration	$\int _{0}^{t}f(\tau )\,d\tau =(u*f)(t)$	${1 \over s}F(s)$	u(t)is the Heaviside step function and(u∗f)(t)is theconvolutionofu(t)andf(t).
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)\$ $e^{-as}{\mathcal {L}}\{f(t+a)\}$	a> 0,u(t)is the Heaviside step function
Time scaling	$f(at)$	${\frac {1}{a}}F\left({s \over a}\right)$	a> 0
Multiplication	$f(t)g(t)$	${\frac {1}{2\pi i}}\lim _{T\to \infty }\int _{c-iT}^{c+iT}F(\sigma )G(s-\sigma )\,d\sigma \$	The integration is done along the vertical lineRe(σ) =cthat lies entirely within the region of convergence ofF.
Convolution	$(f*g)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau$	$F(s)\cdot G(s)\$
Circular convolution	$(f*g)(t)=\int _{0}^{T}f(\tau )g(t-\tau )\,d\tau$	$F(s)\cdot G(s)\$	For periodic functions with periodT.
Complex conjugation	$f^{*}(t)$	$F^{*}(s^{*})$
Periodic function	$f(t)$	${1 \over 1-e^{-Ts}}\int _{0}^{T}e^{-st}f(t)\,dt$	f(t)is a periodic function of periodTso thatf(t) =f(t+T),for allt≥ 0.This is the result of the time shifting property and thegeometric series.
Periodic summation	$f_{P}(t)=\sum _{n=0}^{\infty }f(t-Tn)$ $f_{P}(t)=\sum _{n=0}^{\infty }(-1)^{n}f(t-Tn)$	$F_{P}(s)={\frac {1}{1-e^{-Ts}}}F(s)$ $F_{P}(s)={\frac {1}{1+e^{-Ts}}}F(s)$

Initial value theorem

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

Final value theorem

$f(\infty )=\lim _{s\to 0}{sF(s)}$,if allpolesof$sF(s)$are in the left half-plane.

The final value theorem is useful because it gives the long-term behaviour without having to performpartial fractiondecompositions (or other difficult algebra). IfF(s)has a pole in the right-hand plane or poles on the imaginary axis (e.g., if$f(t)=e^{t}$or$f(t)=\sin(t)$), then the behaviour of this formula is undefined.

Relation to power series

The Laplace transform can be viewed as acontinuousanalogue of apower series.Ifa(n)is a discrete function of a positive integern,then the power series associated toa(n)is the series $$\sum _{n=0}^{\infty }a(n)x^{n}$$ wherexis a real variable (seeZ-transform). Replacing summation overnwith integration overt,a continuous version of the power series becomes $$\int _{0}^{\infty }f(t)x^{t}\,dt$$ where the discrete functiona(n)is replaced by the continuous onef(t).

Changing the base of the power fromxtoegives $$\int _{0}^{\infty }f(t)\left(e^{\ln {x}}\right)^{t}\,dt$$

For this to converge for, say, all bounded functionsf,it is necessary to require thatlnx< 0.Making the substitution−s= lnxgives just the Laplace transform: $$\int _{0}^{\infty }f(t)e^{-st}\,dt$$

In other words, the Laplace transform is a continuous analog of a power series, in which the discrete parameternis replaced by the continuous parametert,andxis replaced bye^−s.

Relation to moments

Main article:Moment-generating function

The quantities $$\mu _{n}=\int _{0}^{\infty }t^{n}f(t)\,dt$$

are themomentsof the functionf.If the firstnmoments offconverge absolutely, then by repeateddifferentiation under the integral, $$(-1)^{n}({\mathcal {L}}f)^{(n)}(0)=\mu _{n}.$$ This is of special significance in probability theory, where the moments of a random variableXare given by the expectation values$\mu _{n}=\operatorname {E} [X^{n}]$.Then, the relation holds $$\mu _{n}=(-1)^{n}{\frac {d^{n}}{ds^{n}}}\operatorname {E} \left[e^{-sX}\right](0).$$

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: $${\begin{aligned}{\mathcal {L}}\left\{f(t)\right\}&=\int _{0^{-}}^{\infty }e^{-st}f(t)\,dt\\[6pt]&=\left[{\frac {f(t)e^{-st}}{-s}}\right]_{0^{-}}^{\infty }-\int _{0^{-}}^{\infty }{\frac {e^{-st}}{-s}}f'(t)\,dt\quad {\text{(by parts)}}\\[6pt]&=\left[-{\frac {f(0^{-})}{-s}}\right]+{\frac {1}{s}}{\mathcal {L}}\left\{f'(t)\right\},\end{aligned}}$$ yielding $${\mathcal {L}}\{f'(t)\}=s\cdot {\mathcal {L}}\{f(t)\}-f(0^{-}),$$ and in the bilateral case, $${\mathcal {L}}\{f'(t)\}=s\int _{-\infty }^{\infty }e^{-st}f(t)\,dt=s\cdot {\mathcal {L}}\{f(t)\}.$$

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

By plugging in$({\mathcal {L}}f)(x)=\int _{0}^{\infty }f(s)e^{-sx}\,ds$the left-hand side turns into: $$\int _{0}^{\infty }\int _{0}^{\infty }f(s)g(x)e^{-sx}\,ds\,dx,$$ 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, $$\int _{0}^{\infty }{\frac {\sin x}{x}}dx=\int _{0}^{\infty }{\mathcal {L}}(1)(x)\sin xdx=\int _{0}^{\infty }1\cdot {\mathcal {L}}(\sin )(x)dx=\int _{0}^{\infty }{\frac {dx}{x^{2}+1}}={\frac {\pi }{2}}.$$

Relationship to other transforms

Laplace–Stieltjes transform

The (unilateral) Laplace–Stieltjes transform of a functiong:ℝ → ℝis defined by theLebesgue–Stieltjes integral

$$\{{\mathcal {L}}^{*}g\}(s)=\int _{0}^{\infty }e^{-st}\,d\,g(t)~.$$

The functiongis assumed to be ofbounded variation.Ifgis theantiderivativeoff:

$$g(x)=\int _{0}^{x}f(t)\,d\,t$$

then the Laplace–Stieltjes transform ofgand the Laplace transform offcoincide. In general, the Laplace–Stieltjes transform is the Laplace transform of theStieltjes measureassociated tog.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 itscumulative distribution function.

Fourier transform

Further information:Fourier transform § Laplace transform

Let$f$be a complex-valued Lebesgue integrable function supported on$[0,\infty )$,and let$F(s)={\mathcal {L}}f(s)$be its Laplace transform. Then, within the region of convergence, we have

$F(\sigma +i\tau )=\int _{0}^{\infty }f(t)e^{-\sigma t}e^{-i\tau t}\,dt,$

which is the Fourier transform of the function$f(t)e^{-\sigma t}$.

Indeed, theFourier transformis 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 arealvariable (frequency), the Laplace transform of a function is a complex function of acomplexvariable. The Laplace transform is usually restricted to transformation of functions oftwitht≥ 0.A consequence of this restriction is that the Laplace transform of a function is aholomorphic functionof the variables.Unlike the Fourier transform, the Laplace transform of adistributionis generally awell-behavedfunction. Techniques of complex variables can also be used to directly study Laplace transforms. As a holomorphic function, the Laplace transform has apower seriesrepresentation. This power series expresses a function as a linear superposition ofmomentsof the function. This perspective has applications in probability theory.

Formally, the Fourier transform is equivalent to evaluating the bilateral Laplace transform with imaginary arguments=iωwhen the condition explained below is fulfilled,

$${\begin{aligned}{\hat {f}}(\ Omega )&={\mathcal {F}}\{f(t)\}\\[4pt]&={\mathcal {L}}\{f(t)\}|_{s=i\ Omega }=F(s)|_{s=i\ Omega }\\[4pt]&=\int _{-\infty }^{\infty }e^{-i\ Omega t}f(t)\,dt~.\end{aligned}}$$

This convention of the Fourier transform (${\hat {f}}_{3}(\ Omega )$inFourier transform § Other conventions) requires a factor of⁠1/2π⁠on the inverse Fourier transform. This relationship between the Laplace and Fourier transforms is often used to determine thefrequency spectrumof asignalor dynamical system.

The above relation is valid as statedif and only ifthe region of convergence (ROC) ofF(s)contains the imaginary axis,σ= 0.

For example, the functionf(t) = cos(ω₀t)has a Laplace transformF(s) =s/(s²+ω₀²)whose ROC isRe(s) > 0.Ass=iω₀is a pole ofF(s),substitutings=iωinF(s)does not yield the Fourier transform off(t)u(t),which contains terms proportional to theDirac delta functionsδ(ω±ω₀).

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

Mellin transform

Main article:Mellin transform

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

If in the Mellin transform $$G(s)={\mathcal {M}}\{g(\theta )\}=\int _{0}^{\infty }\theta ^{s}g(\theta )\,{\frac {d\theta }{\theta }}$$ we setθ=e^−twe get a two-sided Laplace transform.

Z-transform

Further information:Z-transform § Relationship to Laplace transform

The unilateral or one-sided Z-transform is simply the Laplace transform of an ideally sampled signal with the substitution of $$z{\stackrel {\mathrm {def} }{{}={}}}e^{sT},$$ whereT= 1/f_sis thesampling interval(in units of time e.g., seconds) andf_sis thesampling rate(insamples per secondorhertz).

Let $$\Delta _{T}(t)\ {\stackrel {\mathrm {def} }{=}}\ \sum _{n=0}^{\infty }\delta (t-nT)$$ be a sampling impulse train (also called aDirac comb) and $${\begin{aligned}x_{q}(t)&{\stackrel {\mathrm {def} }{{}={}}}x(t)\Delta _{T}(t)=x(t)\sum _{n=0}^{\infty }\delta (t-nT)\\&=\sum _{n=0}^{\infty }x(nT)\delta (t-nT)=\sum _{n=0}^{\infty }x[n]\delta (t-nT)\end{aligned}}$$ be the sampled representation of the continuous-timex(t) $$x[n]{\stackrel {\mathrm {def} }{{}={}}}x(nT)~.$$

The Laplace transform of the sampled signalx_q(t)is $${\begin{aligned}X_{q}(s)&=\int _{0^{-}}^{\infty }x_{q}(t)e^{-st}\,dt\\&=\int _{0^{-}}^{\infty }\sum _{n=0}^{\infty }x[n]\delta (t-nT)e^{-st}\,dt\\&=\sum _{n=0}^{\infty }x[n]\int _{0^{-}}^{\infty }\delta (t-nT)e^{-st}\,dt\\&=\sum _{n=0}^{\infty }x[n]e^{-nsT}~.\end{aligned}}$$

This is the precise definition of the unilateral Z-transform of the discrete functionx[n]

$$X(z)=\sum _{n=0}^{\infty }x[n]z^{-n}$$ with the substitution ofz→e^sT.

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

The similarity between the Z- and Laplace transforms is expanded upon in the theory oftime scale calculus.

Borel transform

The integral form of theBorel transform $$F(s)=\int _{0}^{\infty }f(z)e^{-sz}\,dz$$ is a special case of the Laplace transform forfanentire functionof exponential type, meaning that $$|f(z)|\leq Ae^{B|z|}$$ for some constantsAandB.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 theoremgives 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

Main article:List of Laplace transforms

The following table provides Laplace transforms for many common functions of a single variable.For definitions and explanations, see theExplanatory Notesat 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.$${\mathcal {L}}\{f(t)+g(t)\}={\mathcal {L}}\{f(t)\}+{\mathcal {L}}\{g(t)\}$$
• The Laplace transform of a multiple of a function is that multiple times the Laplace transformation of that function.$${\mathcal {L}}\{af(t)\}=a{\mathcal {L}}\{f(t)\}$$

Using this linearity, and varioustrigonometric,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 thenon-negativereals, 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 delayτare required to becausal(meaning thatτ> 0). A causal system is a system where theimpulse responseh(t)is zero for all timetprior tot= 0.In general, the region of convergence for causal systems is not the same as that ofanticausal systems.

Selected Laplace transforms

Function	Time domain $f(t)={\mathcal {L}}^{-1}\{F(s)\}$	Laplaces-domain $F(s)={\mathcal {L}}\{f(t)\}$	Region of convergence	Reference
unit impulse	$\delta (t)\$	$1$	alls	inspection
delayed impulse	$\delta (t-\tau )\$	$e^{-\tau s}\$	alls	time shift of unit impulse
unit step	$u(t)\$	${1 \over s}$	$\operatorname {Re} (s)>0$	integrate unit impulse
delayed unit step	$u(t-\tau )\$	${\frac {1}{s}}e^{-\tau s}$	$\operatorname {Re} (s)>0$	time shift of unit step
product of delayed function and delayed step	$f(t-\tau )u(t-\tau )$	$e^{-s\tau }{\mathcal {L}}\{f(t)\}$		u-substitution,$u=t-\tau$
rectangular impulse	$u(t)-u(t-\tau )$	${\frac {1}{s}}(1-e^{-\tau s})$	$\operatorname {Re} (s)>0$
ramp	$t\cdot u(t)\$	${\frac {1}{s^{2}}}$	$\operatorname {Re} (s)>0$	integrate unit impulse twice
nth power (for integern)	$t^{n}\cdot u(t)$	${n! \over s^{n+1}}$	$\operatorname {Re} (s)>0$ (n> −1)	integrate unit stepntimes
qth power (for complexq)	$t^{q}\cdot u(t)$	${\operatorname {\Gamma } (q+1) \over s^{q+1}}$	$\operatorname {Re} (s)>0$ $\operatorname {Re} (q)>-1$
nth root	${\sqrt[{n}]{t}}\cdot u(t)$	${1 \over s^{{\frac {1}{n}}+1}}\operatorname {\Gamma } \left({\frac {1}{n}}+1\right)$	$\operatorname {Re} (s)>0$	Setq= 1/nabove.
nth power with frequency shift	$t^{n}e^{-\ Alpha t}\cdot u(t)$	${\frac {n!}{(s+\ Alpha )^{n+1}}}$	$\operatorname {Re} (s)>-\ Alpha$	Integrate unit step, apply frequency shift
delayednth power with frequency shift	$(t-\tau )^{n}e^{-\ Alpha (t-\tau )}\cdot u(t-\tau )$	${\frac {n!\cdot e^{-\tau s}}{(s+\ Alpha )^{n+1}}}$	$\operatorname {Re} (s)>-\ Alpha$	integrate unit step, apply frequency shift, apply time shift
exponential decay	$e^{-\ Alpha t}\cdot u(t)$	${1 \over s+\ Alpha }$	$\operatorname {Re} (s)>-\ Alpha$	Frequency shift of unit step
two-sidedexponential decay (only for bilateral transform)	$e^{-\ Alpha |t|}\$	${2\ Alpha \over \ Alpha ^{2}-s^{2}}$	$-\ Alpha <\operatorname {Re} (s)<\ Alpha$	Frequency shift of unit step
exponential approach	$(1-e^{-\ Alpha t})\cdot u(t)\$	${\frac {\ Alpha }{s(s+\ Alpha )}}$	$\operatorname {Re} (s)>0$	unit step minus exponential decay
sine	$\sin(\ Omega t)\cdot u(t)\$	${\ Omega \over s^{2}+\ Omega ^{2}}$	$\operatorname {Re} (s)>0$
cosine	$\cos(\ Omega t)\cdot u(t)\$	${s \over s^{2}+\ Omega ^{2}}$	$\operatorname {Re} (s)>0$
hyperbolic sine	$\sinh(\ Alpha t)\cdot u(t)\$	${\ Alpha \over s^{2}-\ Alpha ^{2}}$	$\operatorname {Re} (s)>\left|\ Alpha \right|$
hyperbolic cosine	$\cosh(\ Alpha t)\cdot u(t)\$	${s \over s^{2}-\ Alpha ^{2}}$	$\operatorname {Re} (s)>\left|\ Alpha \right|$
exponentially decaying sine wave	$e^{-\ Alpha t}\sin(\ Omega t)\cdot u(t)\$	${\ Omega \over (s+\ Alpha )^{2}+\ Omega ^{2}}$	$\operatorname {Re} (s)>-\ Alpha$
exponentially decaying cosine wave	$e^{-\ Alpha t}\cos(\ Omega t)\cdot u(t)\$	${s+\ Alpha \over (s+\ Alpha )^{2}+\ Omega ^{2}}$	$\operatorname {Re} (s)>-\ Alpha$
natural logarithm	$\ln(t)\cdot u(t)$	$-{1 \over s}\left[\ln(s)+\gamma \right]$	$\operatorname {Re} (s)>0$
Bessel function of the first kind, of ordern	$J_{n}(\ Omega t)\cdot u(t)$	${\frac {\left({\sqrt {s^{2}+\ Omega ^{2}}}-s\right)^{\!n}}{\ Omega ^{n}{\sqrt {s^{2}+\ Omega ^{2}}}}}$	$\operatorname {Re} (s)>0$ (n> −1)
Error function	$\operatorname {erf} (t)\cdot u(t)$	${\frac {1}{s}}e^{(1/4)s^{2}}\!\left(1-\operatorname {erf} {\frac {s}{2}}\right)$	$\operatorname {Re} (s)>0$
Explanatory notes: u(t)represents theHeaviside step function. δrepresents theDirac delta function. Γ(z)represents thegamma function. γis theEuler–Mascheroni constant. t,a real number, typically representstime,although it can representanyindependent dimension. sis thecomplexfrequency domain parameter, andRe(s)is itsreal part. α,β,τ,andωarereal numbers. nis aninteger.

s-domain equivalent circuits and impedances

The Laplace transform is often used incircuit analysis,and simple conversions to thes-domain of circuit elements can be made. Circuit elements can be transformed intoimpedances,very similar tophasorimpedances.

Here is a summary of equivalents:

s-domain equivalent circuits

Note that the resistor is exactly the same in the time domain and thes-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 thes-domain account for that.

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

Examples and applications

The Laplace transform is used frequently inengineeringandphysics;the output of alinear time-invariant systemcan 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, seecontrol 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 asystem,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 inmechanical engineeringandelectrical 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 engineerOliver Heavisidefirst 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${\mathcal {L}}\left\{f(t)\right\}=F(s)$.Then (see the table above)

$$\partial _{s}{\mathcal {L}}\left\{{\frac {f(t)}{t}}\right\}=\partial _{s}\int _{0}^{\infty }{\frac {f(t)}{t}}e^{-st}\,dt=-\int _{0}^{\infty }f(t)e^{-st}dt=-F(s)$$

From which one gets:

$${\mathcal {L}}\left\{{\frac {f(t)}{t}}\right\}=\int _{s}^{\infty }F(p)\,dp.$$

In the limit$s\rightarrow 0$,one gets $$\int _{0}^{\infty }{\frac {f(t)}{t}}\,dt=\int _{0}^{\infty }F(p)\,dp,$$ provided that the interchange of limits can be justified. This is often possible as a consequence of thefinal value theorem.Even when the interchange cannot be justified the calculation can be suggestive. For example, witha≠ 0 ≠b,proceeding formally one has $${\begin{aligned}\int _{0}^{\infty }{\frac {\cos(at)-\cos(bt)}{t}}\,dt&=\int _{0}^{\infty }\left({\frac {p}{p^{2}+a^{2}}}-{\frac {p}{p^{2}+b^{2}}}\right)\,dp\\[6pt]&=\left[{\frac {1}{2}}\ln {\frac {p^{2}+a^{2}}{p^{2}+b^{2}}}\right]_{0}^{\infty }={\frac {1}{2}}\ln {\frac {b^{2}}{a^{2}}}=\ln \left|{\frac {b}{a}}\right|.\end{aligned}}$$

The validity of this identity can be proved by other means. It is an example of aFrullani integral.

Another example isDirichlet integral.

Complex impedance of a capacitor

In the theory ofelectrical circuits,the current flow in acapacitoris proportional to the capacitance and rate of change in the electrical potential (with equations as for theSIunit system). Symbolically, this is expressed by the differential equation $$i=C{dv \over dt},$$ whereCis the capacitance of the capacitor,i=i(t)is theelectric currentthrough the capacitor as a function of time, andv=v(t)is thevoltageacross 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 $${\begin{aligned}I(s)&={\mathcal {L}}\{i(t)\},\\V(s)&={\mathcal {L}}\{v(t)\},\end{aligned}}$$ and $$V_{0}=v(0).$$

Solving forV(s)we have $$V(s)={I(s) \over sC}+{V_{0} \over s}.$$

The definition of the complex impedanceZ(inohms) is the ratio of the complex voltageVdivided by the complex currentIwhile holding the initial stateV₀at zero: $$Z(s)=\left.{V(s) \over I(s)}\right|_{V_{0}=0}.$$

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 withtransfer function $$H(s)={\frac {1}{(s+\ Alpha )(s+\beta )}}.$$

Theimpulse responseis simply the inverse Laplace transform of this transfer function: $$h(t)={\mathcal {L}}^{-1}\{H(s)\}.$$

Partial fraction expansion

To evaluate this inverse transform, we begin by expandingH(s)using the method of partial fraction expansion, $${\frac {1}{(s+\ Alpha )(s+\beta )}}={P \over s+\ Alpha }+{R \over s+\beta }.$$

The unknown constantsPandRare theresidueslocated at the corresponding poles of the transfer function. Each residue represents the relative contribution of thatsingularityto the transfer function's overall shape.

By theresidue theorem,the inverse Laplace transform depends only upon the poles and their residues. To find the residueP,we multiply both sides of the equation bys+αto get $${\frac {1}{s+\beta }}=P+{R(s+\ Alpha ) \over s+\beta }.$$

Then by lettings= −α,the contribution fromRvanishes and all that is left is $$P=\left.{1 \over s+\beta }\right|_{s=-\ Alpha }={1 \over \beta -\ Alpha }.$$

Similarly, the residueRis given by $$R=\left.{1 \over s+\ Alpha }\right|_{s=-\beta }={1 \over \ Alpha -\beta }.$$

Note that $$R={-1 \over \beta -\ Alpha }=-P$$ and so the substitution ofRandPinto the expanded expression forH(s)gives $$H(s)=\left({\frac {1}{\beta -\ Alpha }}\right)\cdot \left({1 \over s+\ Alpha }-{1 \over s+\beta }\right).$$

Finally, using the linearity property and the known transform for exponential decay (seeItem#3in theTable of Laplace Transforms,above), we can take the inverse Laplace transform ofH(s)to obtain $$h(t)={\mathcal {L}}^{-1}\{H(s)\}={\frac {1}{\beta -\ Alpha }}\left(e^{-\ Alpha t}-e^{-\beta t}\right),$$ which is the impulse response of the system.

Convolution

The same result can be achieved using theconvolution propertyas if the system is a series of filters with transfer functions1/(s+α)and1/(s+β).That is, the inverse of $$H(s)={\frac {1}{(s+\ Alpha )(s+\beta )}}={\frac {1}{s+\ Alpha }}\cdot {\frac {1}{s+\beta }}$$ is $${\mathcal {L}}^{-1}\!\left\{{\frac {1}{s+\ Alpha }}\right\}*{\mathcal {L}}^{-1}\!\left\{{\frac {1}{s+\beta }}\right\}=e^{-\ Alpha t}*e^{-\beta t}=\int _{0}^{t}e^{-\ Alpha x}e^{-\beta (t-x)}\,dx={\frac {e^{-\ Alpha t}-e^{-\beta t}}{\beta -\ Alpha }}.$$

Phase delay

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

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

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

This is just thesine of the sumof the arguments, yielding: $$x(t)=\sin(\ Omega t+\varphi ).$$

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

Statistical mechanics

Instatistical mechanics,the Laplace transform of the density of states$g(E)$defines thepartition function.That is, the canonical partition function$Z(\beta )$is given by $$Z(\beta )=\int _{0}^{\infty }e^{-\beta E}g(E)\,dE$$ and the inverse is given by $$g(E)={\frac {1}{2\pi i}}\int _{\beta _{0}-i\infty }^{\beta _{0}+i\infty }e^{\beta E}Z(\beta )\,d\beta$$

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 thespatial distributionof matter of anastronomicalsource ofradiofrequencythermal radiationtoo distant toresolveas more than a point, given itsflux densityspectrum,rather than relating thetimedomain 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 possiblemodelof 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 arandom walk,with steps$\{+1,-1\}$occurring with probabilities$p,q=1-p$.Suppose also that the time step is anPoisson process,with parameter$\lambda$.Then the probability of the walk being at the lattice point$n$at time$t$is

$P_{n}(t)=\int _{0}^{t}\lambda e^{-\lambda (t-s)}(pP_{n-1}(s)+qP_{n+1}(s))\,ds\quad (+e^{-\lambda t}\quad {\text{when}}\ n=0).$

This leads to a system ofintegral 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

$\pi _{n}(s)={\mathcal {L}}(P_{n})(s),$

namely:

$\pi _{n}(s)={\frac {\lambda }{\lambda +s}}(p\pi _{n-1}(s)+q\pi _{n+1}(s))\quad (+{\frac {1}{\lambda +s}}\quad {\text{when}}\ n=0)$

which may now be solved by standard methods.

Tauberian theory

The Laplace transform of the measure$\mu$on$[0,\infty )$is given by

${\mathcal {L}}\mu (s)=\int _{0}^{\infty }e^{-st}d\mu (t).$

It is intuitively clear that, for small$s>0$,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:

$\lim _{s\to 0^{+}}{\mathcal {L}}\mu (s)=\lim _{t\to \infty }M(t).$

Tauberian theoremsare theorems relating the asymptotics of the Laplace transform, as$s\to 0^{+}$,to those of the distribution of$\mu$as$t\to \infty$.They are thus of importance in asymptotic formulae ofprobabilityandstatistics,where often the spectral side has asymptotics that are simpler to infer.

Two tauberian theorems of note are theHardy–Littlewood tauberian theoremand theWiener tauberian theorem.The Wiener theorem generalizes theIkehara tauberian theorem,which is the following statement:

LetA(x) be a non-negative,monotonicnondecreasing function ofx,defined for 0 ≤x< ∞. Suppose that

$f(s)=\int _{0}^{\infty }A(x)e^{-xs}\,dx$

converges for ℜ(s) > 1 to the functionƒ(s) and that, for some non-negative numberc,

$f(s)-{\frac {c}{s-1}}$

has an extension as acontinuous functionfor ℜ(s) ≥ 1. Then thelimitasxgoes to infinity ofe^−x A(x) is equal to c.

This statement can be applied in particular to thelogarithmic derivativeofRiemann zeta function,and thus provides an extremely short way to prove theprime number theorem.

See also

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

References

Modern

• Bracewell, Ronald N. (1978),The Fourier Transform and its Applications(2nd ed.), McGraw-Hill Kogakusha,ISBN978-0-07-007013-4
• Bracewell, R. N. (2000),The Fourier Transform and Its Applications(3rd ed.), Boston: McGraw-Hill,ISBN978-0-07-116043-8
• Feller, William(1971),An introduction to probability theory and its applications. Vol. II.,Second edition, New York:John Wiley & Sons,MR0270403
• Korn, G. A.;Korn, T. M.(1967),Mathematical Handbook for Scientists and Engineers(2nd ed.), McGraw-Hill Companies,ISBN978-0-07-035370-1
• Widder, David Vernon (1941),The Laplace Transform,Princeton Mathematical Series, v. 6,Princeton University Press,MR0005923
• Williams, J. (1973),Laplace Transforms,Problem Solvers, George Allen & Unwin,ISBN978-0-04-512021-5
• Takacs, J. (1953), "Fourier amplitudok meghatarozasa operatorszamitassal",Magyar Hiradastechnika(in Hungarian),IV(7–8): 93–96

Historical

• Euler, L.(1744), "De constructione aequationum" [The Construction of Equations],Opera Omnia,1st series (in Latin),22:150–161
• Euler, L.(1753), "Methodus aequationes differentiales" [A Method for Solving Differential Equations],Opera Omnia,1st series (in Latin),22:181–213
• Euler, L.(1992) [1769], "Institutiones calculi integralis, Volume 2" [Institutions of Integral Calculus],Opera Omnia,1st series (in Latin),12,Basel: Birkhäuser,ISBN978-3764314743,Chapters 3–5
• Euler, Leonhard(1769),Institutiones calculi integralis[Institutions of Integral Calculus] (in Latin), vol. II, Paris: Petropoli, ch. 3–5, pp. 57–153
• Grattan-Guinness, I(1997), "Laplace's integral solutions to partial differential equations", in Gillispie, C. C. (ed.),Pierre Simon Laplace 1749–1827: A Life in Exact Science,Princeton: Princeton University Press,ISBN978-0-691-01185-1
• Lagrange, J. L.(1773),Mémoire sur l'utilité de la méthode,Œuvres de Lagrange, vol. 2, pp. 171–234

Further reading

• Arendt, Wolfgang; Batty, Charles J.K.; Hieber, Matthias; Neubrander, Frank (2002),Vector-Valued Laplace Transforms and Cauchy Problems,Birkhäuser Basel,ISBN978-3-7643-6549-3.
• Davies, Brian (2002),Integral transforms and their applications(Third ed.), New York: Springer,ISBN978-0-387-95314-4
• 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,S2CID117913073
• 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,S2CID123071842
• Doetsch, Gustav(1974),Introduction to the Theory and Application of the Laplace Transformation,Springer,ISBN978-0-387-06407-9
• Mathews, Jon; Walker, Robert L. (1970),Mathematical methods of physics(2nd ed.), New York: W. A. Benjamin,ISBN0-8053-7002-1
• Polyanin, A. D.; Manzhirov, A. V. (1998),Handbook of Integral Equations,Boca Raton: CRC Press,ISBN978-0-8493-2876-3
• Schwartz, Laurent(1952), "Transformation de Laplace des distributions",Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.](in French),1952:196–206,MR0052555
• Schwartz, Laurent(2008) [1966],Mathematics for the Physical Sciences,Dover Books on Mathematics, New York: Dover Publications, pp. 215–241,ISBN978-0-486-46662-0- See Chapter VI. The Laplace transform.
• Siebert, William McC. (1986),Circuits, Signals, and Systems,Cambridge, Massachusetts: MIT Press,ISBN978-0-262-19229-3
• Widder, David Vernon (1945), "What is the Laplace transform?",The American Mathematical Monthly,52(8): 419–425,doi:10.2307/2305640,ISSN0002-9890,JSTOR2305640,MR0013447
• 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/s11075-022-01368-x.

External links

• "Laplace transform",Encyclopedia of Mathematics,EMS Press,2001 [1994]
• Online Computationof the transform or inverse transform, wims.unice.fr
• Tables of Integral Transformsat EqWorld: The World of Mathematical Equations.
• Weisstein, Eric W."Laplace Transform".MathWorld.
• Good explanations of the initial and final value theoremsArchived2009-01-08 at theWayback Machine
• Laplace Transformsat MathPages
• Computational Knowledge Engineallows to easily calculate Laplace Transforms and its inverse Transform.
• Laplace Calculatorto calculate Laplace Transforms online easily.
• Code to visualize Laplace Transformsand many example videos.