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Laplace transform

From Wikipedia, the free encyclopedia
Integral transform useful in probability theory, physics, and engineering

Inmathematics, theLaplace transform, named afterPierre-Simon Laplace (/ləˈplɑːs/), is anintegral transform that converts afunction of arealvariable (usually⁠t{\displaystyle t}⁠, in thetime domain) to a function of acomplex variables{\displaystyle s} (in the complex-valuedfrequency domain, also known ass-domain ors-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){\displaystyle x(t)} and⁠X(s){\displaystyle X(s)}⁠.

The transform is useful for convertingdifferentiation andintegration in the time domain into much easiermultiplication anddivision in the Laplace domain (analogous to howlogarithms are useful for simplifying multiplication and division into addition and subtraction). This gives the transform many applications inscience andengineering, mostly as a tool for solving lineardifferential equations[1] anddynamical systems by simplifyingordinary differential equations andintegral equations intoalgebraic polynomial equations, and by simplifyingconvolution intomultiplication.[2][3]

For example, through the Laplace transform, the equation of thesimple harmonic oscillator (Hooke's law)x″(t)+kx(t)=0{\displaystyle x''(t)+kx(t)=0} is converted into the algebraic equations2X(s)−sx(0)−x′(0)+kX(s)=0,{\displaystyle s^{2}X(s)-sx(0)-x'(0)+kX(s)=0,} which incorporates theinitial conditionsx(0){\displaystyle x(0)} and⁠x′(0){\displaystyle x'(0)}⁠, and can be solved for the unknown functionX(s).{\displaystyle X(s).} 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 givenbelow.

The Laplace transform is defined (for suitable functions⁠f{\displaystyle f}⁠) by theintegralL{f}(s)=∫0∞f(t)e−stdt,{\displaystyle {\mathcal {L}}\{f\}(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt,}wheres is acomplex number.

The Laplace transform is related to many other transforms. It is essentially the same as theMellin transform and is closely related to theFourier transform. Unlike for the Fourier transform, the Laplace transform of a function is often ananalytic function, meaning that it has a convergentpower series, the coefficients of which represent themoments of the original function. Moreover, the techniques ofcomplex analysis, especiallycontour integrals, can be used for simplifying calculations.

History

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Pierre-Simon, marquis de Laplace

The Laplace transform is named aftermathematician andastronomerPierre-Simon, Marquis de Laplace, who used a similar transform in his work onprobability theory.[4][5] Laplace wrote extensively about the use ofgenerating functions (1814), and the integral form of the Laplace transform evolved naturally as a result.[6]

Laplace's use of generating functions was similar to what is now known as thez-transform, and he gave little attention to thecontinuous variable case which was discussed byNiels Henrik Abel.[7]

From 1744,Leonhard Euler investigated integrals of the formz=∫X(x)eaxdx and z=∫X(x)xAdx{\displaystyle 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.[8]Joseph-Louis Lagrange was an admirer of Euler and, in his work on integratingprobability density functions, investigated expressions of the form∫X(x)e−axaxdx,{\displaystyle \int X(x)e^{-ax}a^{x}\,dx,}which resembles a Laplace transform.[9][10]

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.[11] 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∫xsφ(x)dx,{\displaystyle \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.[12]

Laplace also recognised thatJoseph Fourier's method ofFourier series for solving thediffusion equation could 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.[13] In 1821,Cauchy developed anoperational calculus for the Laplace transform that could be used to studylinear differential equations in much the same way the transform is now used in basic engineering. This method was popularized, and perhaps rediscovered, byOliver Heaviside around the turn of the century.[14]

Bernhard Riemann used 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 his method is still used to relate themodular transformation law of theJacobi theta function, which is simple to prove viaPoisson summation, to the functional equation.[15]

Hjalmar Mellin was among the first to study the Laplace transform, rigorously in theKarl Weierstrass school of analysis, and apply it to the study ofdifferential equations andspecial functions, at the turn of the 20th century.[16] At around the same time, Heaviside was busy with his operational calculus.Thomas Joannes Stieltjes considered a generalization of the Laplace transform connected to hiswork on moments. Other contributors in this time period includedMathias Lerch,[17]Oliver Heaviside, andThomas Bromwich.[18]

In 1929,Vannevar Bush andNorbert Wiener publishedOperational 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 andNorbert Wiener published 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 inGodfrey Harold Hardy andJohn Edensor Littlewood's study oftauberian theorems, and this application was later expounded on byWidder (1941), who developed other aspects of the theory such as a new method for inversion.Edward Charles Titchmarsh wrote 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,[19] replacing the earlier Heavisideoperational calculus. The advantages of the Laplace transform had been emphasized byGustav Doetsch[20].

Formal definition

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ℜ(e−st){\displaystyle \Re (e^{-st})} for various complex frequencies in thes-domain(s=σ+iω),{\displaystyle (s=\sigma +i\omega ),} which can be expressed ase−σtcos⁡(ωt).{\displaystyle e^{-\sigma t}\cos(\omega t).} Theσ=0{\displaystyle \sigma =0} axis contains pure cosines. Positiveσ{\displaystyle \sigma } containsdamped cosines. Negativeσ{\displaystyle \sigma } containsexponentially growing cosines.

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

F(s)=∫0∞f(t)e−stdt,{\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt,}   (Eq. 1)

wheres is acomplex frequency-domain parameters=σ+iω{\displaystyle s=\sigma +i\omega }with real numbersσ andω.

An alternate notation for the Laplace transform isL{f}{\displaystyle {\mathcal {L}}\{f\}} instead ofF.[3] ThusF(s)=L{f}(s){\displaystyle F(s)={\mathcal {L}}\{f\}(s)} infunctional notation. This is often written, especially in engineering settings, as⁠F(s)=L{f(t)}{\displaystyle F(s)={\mathcal {L}}\{f(t)\}}⁠, with the understanding that thedummy variablet{\displaystyle t} does not appear in the function⁠F(s){\displaystyle F(s)}⁠.

The meaning of the integral depends on types of functions of interest. A necessary condition for existence of the integral is thatf must belocally integrable on[0, ∞). For locally integrable functions that decay at infinity or are ofexponential type (⁠|f(t)|≤AeB|t|{\displaystyle \vert f(t)\vert \leq Ae^{B\vert t\vert }}⁠), the integral can be understood to be a (proper)Lebesgue integral. However, for many applications it is necessary to regard it as aconditionally convergentimproper integral at∞. 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 measureμ by the Lebesgue integral[21]L{μ}(s)=∫[0,∞)e−stdμ(t).{\displaystyle {\mathcal {L}}\{\mu \}(s)=\int _{[0,\infty )}e^{-st}\,d\mu (t).}

An important special case is whereμ is 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 writesL{f}(s)=∫0−∞f(t)e−stdt,{\displaystyle {\mathcal {L}}\{f\}(s)=\int _{0^{-}}^{\infty }f(t)e^{-st}\,dt,}where the lower limit of0− is shorthand notation forlimε→0+∫−ε∞.{\displaystyle \lim _{\varepsilon \to 0^{+}}\int _{-\varepsilon }^{\infty }.}

This limit emphasizes that anypoint mass located at0 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 theLaplace–Stieltjes transform.

Bilateral Laplace transform

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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 includes being multiplied by theHeaviside step function.

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

F(s)=∫−∞∞e−stf(t)dt.{\displaystyle F(s)=\int _{-\infty }^{\infty }e^{-st}f(t)\,dt.}   (Eq. 2)

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

Inverse Laplace transform

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Main article:Inverse Laplace transform

Two integrable functions have the same Laplace transform only if they differ on a set ofLebesgue 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 aone-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 spaceL∞(0, ∞), or more generallytempered distributions on(0, ∞). The Laplace transform is also defined and injective for suitable spaces of tempered distributions.

In these cases, the image of the Laplace transform lives in a space ofanalytic functions in theregion of convergence. Theinverse Laplace transform is given by the following complex integral, which is known by various names (theBromwich integral, theFourier–Mellin integral, andMellin's inverse formula):

f(t)=L−1{F}(t)=12πilimT→∞∫γ−iTγ+iTestF(s)ds,{\displaystyle 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)

whereγ is 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

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Inpure andapplied probability, the Laplace transform is defined as anexpected value. IfX is arandom variable with probability density functionf, then the Laplace transform off is given by the expectationL{f}(s)=E⁡[e−sX],{\displaystyle {\mathcal {L}}\{f\}(s)=\operatorname {E} \left[e^{-sX}\right],}whereE⁡[r]{\displaystyle \operatorname {E} [r]} is theexpectation ofrandom variable⁠r{\displaystyle r}⁠.

Byconvention, this is referred to as the Laplace transform of the random variableX itself. Here, replacings by−t gives themoment generating function ofX. The Laplace transform has applications throughout probability theory, includingfirst passage times ofstochastic processes such asMarkov chains, andrenewal theory.

Of particular use is the ability to recover thecumulative distribution function of a continuous random variableX by means of the Laplace transform as follows:[22]FX(x)=L−1{1sE⁡[e−sX]}(x)=L−1{1sL{f}(s)}(x).{\displaystyle F_{X}(x)={\mathcal {L}}^{-1}\left\{{\frac {1}{s}}\operatorname {E} \left[e^{-sX}\right]\right\}(x)={\mathcal {L}}^{-1}\left\{{\frac {1}{s}}{\mathcal {L}}\{f\}(s)\right\}(x).}

Algebraic construction

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The Laplace transform can be alternatively defined in a purely algebraic manner by applying afield of fractions construction to the convolutionring of functions on the positive half-line. The resultingspace 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).[23]

Region of convergence

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

Iff is a locally integrable function (or more generally a Borel measure locally of bounded variation), then the Laplace transformF(s) off converges provided that the limitlimR→∞∫0Rf(t)e−stdt{\displaystyle \lim _{R\to \infty }\int _{0}^{R}f(t)e^{-st}\,dt}exists.

The Laplace transformconverges absolutely if the integral∫0∞|f(t)e−st|dt{\displaystyle \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) >a orRe(s) ≥a, wherea is anextended real constant with−∞ ≤a ≤ ∞ (a consequence of thedominated convergence theorem). The constanta is known as the abscissa of absolute convergence, and depends on the growth behavior off(t).[24] Analogously, the two-sided transform converges absolutely in a strip of the forma < Re(s) <b, and possibly including the linesRe(s) =a orRe(s) =b.[25] The subset of values ofs 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 ofFubini's theorem andMorera'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 alls withRe(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 off can be expressed byintegrating by parts as the integralF(s)=(s−s0)∫0∞e−(s−s0)tβ(t)dt,β(u)=∫0ue−s0tf(t)dt.{\displaystyle 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. In its most general form, the Laplace transform gives a one-to-one correspondence between the holomorphic functions which, for someσ∈R{\displaystyle \sigma \in \mathbb {R} }, are defined on{s∈C | Re(s)>σ}{\displaystyle \{s\in \mathbb {C} \ |\ \mathrm {Re} (s)>\sigma \}} and are bounded there in absolute value by a polynomial, and thedistributions on the real line supported on⁠[0,∞){\displaystyle [0,\infty )}⁠ which becometempered distributions after multiplied bye−σt{\displaystyle e^{-\sigma t}} for some⁠σ{\displaystyle \sigma }⁠.[26]

There are severalPaley–Wiener theorems concerning 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) system isstable 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 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

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The Laplace transform's key property is that it convertsdifferentiation andintegration in the time domain into multiplication and division bys in the Laplace domain. Thus, the Laplace variables is also known as anoperator variable in the Laplace domain: either thederivative operator or (fors−1) theintegration operator.

Given the functionsf(t) andg(t), and their respective Laplace transformsF(s) andG(s),f(t)=L−1{F(s)},g(t)=L−1{G(s)},{\displaystyle {\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:[27]

Properties of the unilateral Laplace transform
PropertyTime domains domainComment
Linearityaf(t)+bg(t) {\displaystyle af(t)+bg(t)\ }aF(s)+bG(s) {\displaystyle aF(s)+bG(s)\ }Can be proved using basic rules of integration.
Frequency-domain derivativetf(t) {\displaystyle tf(t)\ }−F′(s) {\displaystyle -F'(s)\ }F′ is the first derivative ofF with respect tos.
Frequency-domain general derivativetnf(t) {\displaystyle t^{n}f(t)\ }(−1)nF(n)(s) {\displaystyle (-1)^{n}F^{(n)}(s)\ }More general form,nth derivative ofF(s).
Derivativef′(t) {\displaystyle f'(t)\ }sF(s)−f(0−) {\displaystyle sF(s)-f(0^{-})\ }f is 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 derivativef″(t) {\displaystyle f''(t)\ }s2F(s)−sf(0−)−f′(0−) {\textstyle s^{2}F(s)-sf(0^{-})-f'(0^{-})\ }f is assumed twice differentiable and the second derivative to be of exponential type. Follows by applying the Differentiation property tof′(t).
General derivativef(n)(t) {\displaystyle f^{(n)}(t)\ }snF(s)−∑k=1nsn−kf(k−1)(0−) {\displaystyle s^{n}F(s)-\sum _{k=1}^{n}s^{n-k}f^{(k-1)}(0^{-})\ }f is assumed to ben-times differentiable, withnth derivative of exponential type. Follows bymathematical induction.
Frequency-domainintegration1tf(t) {\displaystyle {\frac {1}{t}}f(t)\ }∫s∞F(σ)dσ {\displaystyle \int _{s}^{\infty }F(\sigma )\,d\sigma \ }This is deduced using the nature of frequency differentiation and conditional convergence.
Time-domain integration∫0tf(τ)dτ=(u∗f)(t){\displaystyle \int _{0}^{t}f(\tau )\,d\tau =(u*f)(t)}1sF(s){\displaystyle {1 \over s}F(s)}u(t) is the Heaviside step function and(u ∗f)(t) is theconvolution ofu(t) andf(t).
Frequency shiftingeatf(t){\displaystyle e^{at}f(t)}F(s−a){\displaystyle F(s-a)}
Time shiftingf(t−a)u(t−a){\displaystyle f(t-a)u(t-a)}

f(t)u(t−a) {\displaystyle f(t)u(t-a)\ }

e−asF(s) {\displaystyle e^{-as}F(s)\ }

e−asL{f(t+a)}{\displaystyle e^{-as}{\mathcal {L}}\{f(t+a)\}}

a > 0,u(t) is the Heaviside step function
Time scalingf(at){\displaystyle f(at)}1aF(sa){\displaystyle {\frac {1}{a}}F\left({s \over a}\right)}a > 0
Multiplicationf(t)g(t){\displaystyle f(t)g(t)}12πilimT→∞∫c−iTc+iTF(σ)G(s−σ)dσ {\displaystyle {\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) =c that lies entirely within the region of convergence ofF.[28]
Convolution(f∗g)(t)=∫0tf(τ)g(t−τ)dτ{\displaystyle (f*g)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau }F(s)⋅G(s) {\displaystyle F(s)\cdot G(s)\ }
Circular convolution(f∗g)(t)=∫0Tf(τ)g(t−τ)dτ{\displaystyle (f*g)(t)=\int _{0}^{T}f(\tau )g(t-\tau )\,d\tau }F(s)⋅G(s) {\displaystyle F(s)\cdot G(s)\ }For periodic functions with periodT.
Complex conjugationf∗(t){\displaystyle f^{*}(t)}F∗(s∗){\displaystyle F^{*}(s^{*})}
Periodic functionf(t){\displaystyle f(t)}11−e−Ts∫0Te−stf(t)dt{\displaystyle {1 \over 1-e^{-Ts}}\int _{0}^{T}e^{-st}f(t)\,dt}f(t) is a periodic function of periodT so thatf(t) =f(t +T), for allt ≥ 0. This is the result of the time shifting property and thegeometric series.
Periodic summationfP(t)=∑n=0∞f(t−Tn){\displaystyle f_{P}(t)=\sum _{n=0}^{\infty }f(t-Tn)}

fP(t)=∑n=0∞(−1)nf(t−Tn){\displaystyle f_{P}(t)=\sum _{n=0}^{\infty }(-1)^{n}f(t-Tn)}

FP(s)=11−e−TsF(s){\displaystyle F_{P}(s)={\frac {1}{1-e^{-Ts}}}F(s)}

FP(s)=11+e−TsF(s){\displaystyle F_{P}(s)={\frac {1}{1+e^{-Ts}}}F(s)}

Initial value theorem
f(0+)=lims→∞sF(s).{\displaystyle f(0^{+})=\lim _{s\to \infty }{sF(s)}.}
Final value theorem
⁠f(∞)=lims→0sF(s){\displaystyle f(\infty )=\lim _{s\to 0}{sF(s)}}⁠, if allpoles ofsF(s){\displaystyle 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 fraction decompositions (or other difficult algebra). IfF(s) has a pole in the right-hand plane or poles on the imaginary axis (e.g., iff(t)=et{\displaystyle f(t)=e^{t}} or⁠f(t)=sin⁡(t){\displaystyle f(t)=\sin(t)}⁠), then the behaviour of this formula is undefined.

Relation to power series

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The Laplace transform can be viewed as acontinuous analogue of apower series.[29] Ifa(n) is a discrete function of a positive integern, then the power series associated toa(n) is the series∑n=0∞a(n)xn{\displaystyle \sum _{n=0}^{\infty }a(n)x^{n}}wherex is a real variable (seeZ-transform). Replacing summation overn with integration overt, a continuous version of the power series becomes∫0∞f(t)xtdt{\displaystyle \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 fromx toe gives∫0∞f(t)(eln⁡x)tdt{\displaystyle \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 = lnx gives just the Laplace transform:∫0∞f(t)e−stdt{\displaystyle \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 parametern is replaced by the continuous parametert, andx is replaced bye−s.

Analogously to a power series, if⁠a(n)=O(ρ−n){\displaystyle a(n)=O(\rho ^{-n})}⁠, then the power series converges to an analytic function in⁠|x|<ρ{\displaystyle \vert x\vert <\rho }⁠, if⁠f(t)=O(e−σt){\displaystyle f(t)=O(e^{-\sigma t})}⁠, the Laplace transform converges to an analytic function for⁠ℜ(s)>σ{\displaystyle \Re (s)>\sigma }⁠.[30]

Relation to moments

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Main article:Moment-generating function

The quantitiesμn=∫0∞tnf(t)dt{\displaystyle \mu _{n}=\int _{0}^{\infty }t^{n}f(t)\,dt}are themoments of the functionf. If the firstn moments off converge absolutely, then by repeateddifferentiation under the integral,(−1)n(Lf)(n)(0)=μn.{\displaystyle (-1)^{n}({\mathcal {L}}f)^{(n)}(0)=\mu _{n}.}This is of special significance in probability theory, where the moments of a random variableX are given by the expectation values⁠μn=E⁡[Xn]{\displaystyle \mu _{n}=\operatorname {E} [X^{n}]}⁠. Then, the relation holdsμn=(−1)ndndsnE⁡[e−sX](0).{\displaystyle \mu _{n}=(-1)^{n}{\frac {d^{n}}{ds^{n}}}\operatorname {E} \left[e^{-sX}\right](0).}

Transform of a function's derivative

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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:L{f(t)}=∫0−∞e−stf(t)dt=[f(t)e−st−s]0−∞−∫0−∞e−st−sf′(t)dt(by parts)=[−f(0−)−s]+1sL{f′(t)},{\displaystyle {\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}}}yieldingL{f′(t)}=s⋅L{f(t)}−f(0−),{\displaystyle {\mathcal {L}}\{f'(t)\}=s\cdot {\mathcal {L}}\{f(t)\}-f(0^{-}),}and in the bilateral case,L{f′(t)}=s∫−∞∞e−stf(t)dt=s⋅L{f(t)}.{\displaystyle {\mathcal {L}}\{f'(t)\}=s\int _{-\infty }^{\infty }e^{-st}f(t)\,dt=s\cdot {\mathcal {L}}\{f(t)\}.}

The general resultL{f(n)(t)}=sn⋅L{f(t)}−sn−1f(0−)−⋯−f(n−1)(0−),{\displaystyle {\mathcal {L}}\left\{f^{(n)}(t)\right\}=s^{n}\cdot {\mathcal {L}}\{f(t)\}-s^{n-1}f(0^{-})-\cdots -f^{(n-1)}(0^{-}),}wheref(n){\displaystyle f^{(n)}} denotes thenth derivative off, can then be established with an inductive argument.

Evaluating integrals over the positive real axis

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A useful property of the Laplace transform is the following:∫0∞f(x)g(x)dx=∫0∞(Lf)(s)⋅(L−1g)(s)ds{\displaystyle \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 off,g{\displaystyle f,g} in a right neighbourhood of0{\displaystyle 0} and on the decay rate off,g{\displaystyle f,g} in a left neighbourhood of⁠∞{\displaystyle \infty }⁠. The above formula is a variation of integration by parts, with the operatorsddx{\displaystyle {\frac {d}{dx}}} and∫dx{\displaystyle \int \,dx} being replaced byL{\displaystyle {\mathcal {L}}} and⁠L−1{\displaystyle {\mathcal {L}}^{-1}}⁠. Let us prove the equivalent formulation:∫0∞(Lf)(x)g(x)dx=∫0∞f(s)(Lg)(s)ds.{\displaystyle \int _{0}^{\infty }({\mathcal {L}}f)(x)g(x)\,dx=\int _{0}^{\infty }f(s)({\mathcal {L}}g)(s)\,ds.}

By plugging in(Lf)(x)=∫0∞f(s)e−sxds{\displaystyle ({\mathcal {L}}f)(x)=\int _{0}^{\infty }f(s)e^{-sx}\,ds} the left-hand side turns into:∫0∞∫0∞f(s)g(x)e−sxdsdx,{\displaystyle \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,∫0∞sin⁡xxdx=∫0∞L(1)(x)sin⁡xdx=∫0∞1⋅L(sin)(x)dx=∫0∞dxx2+1=π2.{\displaystyle \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

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Laplace–Stieltjes transform

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The (unilateral) Laplace–Stieltjes transform of a functiong : ℝ → ℝ is defined by theLebesgue–Stieltjes integral{L∗g}(s)=∫0∞e−stdg(t) .{\displaystyle \{{\mathcal {L}}^{*}g\}(s)=\int _{0}^{\infty }e^{-st}\,d\,g(t)~.}

The functiong is assumed to be ofbounded variation. Ifg is theantiderivative off:g(x)=∫0xf(t)dt{\displaystyle g(x)=\int _{0}^{x}f(t)\,d\,t}

then the Laplace–Stieltjes transform ofg and the Laplace transform off coincide. In general, the Laplace–Stieltjes transform is the Laplace transform of theStieltjes measure associated 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.[31]

Fourier transform

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Further information:Fourier transform § Laplace transform

Letf{\displaystyle f} be a complex-valued Lebesgue integrable function supported on⁠[0,∞){\displaystyle [0,\infty )}⁠, and letF(s)=Lf(s){\displaystyle F(s)={\mathcal {L}}f(s)} be its Laplace transform. Then, within the region of convergence, we haveF(σ+iτ)=∫0∞f(t)e−σte−iτtdt,{\displaystyle F(\sigma +i\tau )=\int _{0}^{\infty }f(t)e^{-\sigma t}e^{-i\tau t}\,dt,}which is the Fourier transform of the function⁠f(t)e−σt{\displaystyle f(t)e^{-\sigma t}}⁠.[32]

Indeed, theFourier 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 areal variable (frequency⁠τ{\displaystyle \tau }⁠), the Laplace transform of a function is a complex function of acomplex variable (damping factorσ{\displaystyle \sigma } and frequency⁠τ{\displaystyle \tau }⁠). The Laplace transform is usually restricted to transformation of functions oft witht ≥ 0. A consequence of this restriction is that the Laplace transform of a function is aholomorphic function of the variables. Unlike the Fourier transform, the Laplace transform of adistribution is generally awell-behaved function. Techniques of complex variables can also be used to directly study Laplace transforms. As a holomorphic function, the Laplace transform has apower series representation. This power series expresses a function as a linear superposition ofmoments of the function. This perspective has applications in probability theory.

Formally, the Fourier transform is equivalent to evaluating the bilateral Laplace transform with imaginary arguments =iω[33][34] when the condition explained below is fulfilled,f^(ω)=F{f(t)}=L{f(t)}|s=iω=F(s)|s=iω=∫−∞∞e−iωtf(t)dt .{\displaystyle {\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 (⁠f^3(ω){\displaystyle {\hat {f}}_{3}(\omega )}⁠ inFourier transform § Other conventions) requires a factor of⁠1/2π⁠ on the inverse Fourier transform. This relationship between the Laplace and Fourier transforms is often used to determine thefrequency spectrum of asignal or dynamical system.

The above relation is valid as statedif and only if the 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 =iω0 is a pole ofF(s), substitutings =iω inF(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 formlimσ→0+F(σ+iω)=f^(ω){\displaystyle \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 limit of 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

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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 transformG(s)=M{g(θ)}=∫0∞θsg(θ)dθθ{\displaystyle G(s)={\mathcal {M}}\{g(\theta )\}=\int _{0}^{\infty }\theta ^{s}g(\theta )\,{\frac {d\theta }{\theta }}}we setθ =e−t we get a two-sided Laplace transform.

Z-transform

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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 ofz=defesT,{\displaystyle z{\stackrel {\mathrm {def} }{{}={}}}e^{sT},}whereT = 1/fs is thesampling interval (in units of time e.g., seconds) andfs is thesampling rate (insamples per second orhertz).

LetΔT(t) =def ∑n=0∞δ(t−nT){\displaystyle \Delta _{T}(t)\ {\stackrel {\mathrm {def} }{=}}\ \sum _{n=0}^{\infty }\delta (t-nT)}be a sampling impulse train (also called aDirac comb) andxq(t)=defx(t)ΔT(t)=x(t)∑n=0∞δ(t−nT)=∑n=0∞x(nT)δ(t−nT)=∑n=0∞x[n]δ(t−nT){\displaystyle {\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]=defx(nT) .{\displaystyle x[n]{\stackrel {\mathrm {def} }{{}={}}}x(nT)~.}

The Laplace transform of the sampled signalxq(t) isXq(s)=∫0−∞xq(t)e−stdt=∫0−∞∑n=0∞x[n]δ(t−nT)e−stdt=∑n=0∞x[n]∫0−∞δ(t−nT)e−stdt=∑n=0∞x[n]e−nsT .{\displaystyle {\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)=∑n=0∞x[n]z−n{\displaystyle X(z)=\sum _{n=0}^{\infty }x[n]z^{-n}}with the substitution ofz →esT.

Comparing the last two equations, we find the relationship between the unilateral Z-transform and the Laplace transform of the sampled signal,Xq(s)=X(z)|z=esT.{\displaystyle 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

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The integral form of theBorel transformF(s)=∫0∞f(z)e−szdz{\displaystyle F(s)=\int _{0}^{\infty }f(z)e^{-sz}\,dz}is a special case of the Laplace transform forf anentire function of exponential type, meaning that|f(z)|≤AeB|z|{\displaystyle |f(z)|\leq Ae^{B|z|}}for some constantsA andB. 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

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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

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Main article:List of Laplace transforms

The following table provides Laplace transforms for many common functions of a single variable.[35][36] For definitions and explanations, see theExplanatory Notes at the end of the table.

Because the Laplace transform is a linear operator,

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-negative reals, which is why all of the time domain functions in the table below are multiples of theHeaviside step function,u(t).

The entries of the table that involve a time delayτ are required to becausal (meaning thatτ > 0). A causal system is a system where theimpulse responseh(t) is zero for all timet prior tot = 0. In general, the region of convergence for causal systems is not the same as that ofanticausal systems.

Selected Laplace transforms
FunctionTime domain
f(t)=L−1{F(s)}{\displaystyle f(t)={\mathcal {L}}^{-1}\{F(s)\}}		Laplaces-domain
F(s)=L{f(t)}{\displaystyle F(s)={\mathcal {L}}\{f(t)\}}		Region of convergenceReference
unit impulseδ(t) {\displaystyle \delta (t)\ }1{\displaystyle 1}allsinspection
delayed impulseδ(t−τ) {\displaystyle \delta (t-\tau )\ }e−τs {\displaystyle e^{-\tau s}\ }allstime shift of
unit impulse
unit stepu(t) {\displaystyle u(t)\ }1s{\displaystyle {1 \over s}}Re⁡(s)>0{\displaystyle \operatorname {Re} (s)>0}integrate unit impulse
delayed unit stepu(t−τ) {\displaystyle u(t-\tau )\ }1se−τs{\displaystyle {\frac {1}{s}}e^{-\tau s}}Re⁡(s)>0{\displaystyle \operatorname {Re} (s)>0}time shift of
unit step
product of delayed function and delayed stepf(t−τ)u(t−τ){\displaystyle f(t-\tau )u(t-\tau )}e−sτL{f(t)}{\displaystyle e^{-s\tau }{\mathcal {L}}\{f(t)\}}u-substitution,u=t−τ{\displaystyle u=t-\tau }
rectangular impulseu(t)−u(t−τ){\displaystyle u(t)-u(t-\tau )}1s(1−e−τs){\displaystyle {\frac {1}{s}}(1-e^{-\tau s})}Re⁡(s)>0{\displaystyle \operatorname {Re} (s)>0}
rampt⋅u(t) {\displaystyle t\cdot u(t)\ }1s2{\displaystyle {\frac {1}{s^{2}}}}Re⁡(s)>0{\displaystyle \operatorname {Re} (s)>0}integrate unit
impulse twice
nth power
(for integern)		tn⋅u(t){\displaystyle t^{n}\cdot u(t)}n!sn+1{\displaystyle {n! \over s^{n+1}}}Re⁡(s)>0{\displaystyle \operatorname {Re} (s)>0}
(n > −1)		integrate unit
stepn times
qth power
(for complexq)		tq⋅u(t){\displaystyle t^{q}\cdot u(t)}Γ⁡(q+1)sq+1{\displaystyle {\operatorname {\Gamma } (q+1) \over s^{q+1}}}Re⁡(s)>0{\displaystyle \operatorname {Re} (s)>0}
Re⁡(q)>−1{\displaystyle \operatorname {Re} (q)>-1}		[37][38]
nth roottn⋅u(t){\displaystyle {\sqrt[{n}]{t}}\cdot u(t)}1s1n+1Γ⁡(1n+1){\displaystyle {1 \over s^{{\frac {1}{n}}+1}}\operatorname {\Gamma } \left({\frac {1}{n}}+1\right)}Re⁡(s)>0{\displaystyle \operatorname {Re} (s)>0}Setq = 1/n above.
nth power with frequency shifttne−αt⋅u(t){\displaystyle t^{n}e^{-\alpha t}\cdot u(t)}n!(s+α)n+1{\displaystyle {\frac {n!}{(s+\alpha )^{n+1}}}}Re⁡(s)>−α{\displaystyle \operatorname {Re} (s)>-\alpha }Integrate unit step,
apply frequency shift
delayednth power
with frequency shift		(t−τ)ne−α(t−τ)⋅u(t−τ){\displaystyle (t-\tau )^{n}e^{-\alpha (t-\tau )}\cdot u(t-\tau )}n!⋅e−τs(s+α)n+1{\displaystyle {\frac {n!\cdot e^{-\tau s}}{(s+\alpha )^{n+1}}}}Re⁡(s)>−α{\displaystyle \operatorname {Re} (s)>-\alpha }integrate unit step,
apply frequency shift,
apply time shift
exponential decaye−αt⋅u(t){\displaystyle e^{-\alpha t}\cdot u(t)}1s+α{\displaystyle {1 \over s+\alpha }}Re⁡(s)>−α{\displaystyle \operatorname {Re} (s)>-\alpha }Frequency shift of
unit step
two-sided exponential decay
(only for bilateral transform)		e−α|t| {\displaystyle e^{-\alpha |t|}\ }2αα2−s2{\displaystyle {2\alpha \over \alpha ^{2}-s^{2}}}−α<Re⁡(s)<α{\displaystyle -\alpha <\operatorname {Re} (s)<\alpha }Frequency shift of
unit step
exponential approach(1−e−αt)⋅u(t) {\displaystyle (1-e^{-\alpha t})\cdot u(t)\ }αs(s+α){\displaystyle {\frac {\alpha }{s(s+\alpha )}}}Re⁡(s)>0{\displaystyle \operatorname {Re} (s)>0}unit step minus
exponential decay
sinesin⁡(ωt)⋅u(t) {\displaystyle \sin(\omega t)\cdot u(t)\ }ωs2+ω2{\displaystyle {\omega \over s^{2}+\omega ^{2}}}Re⁡(s)>0{\displaystyle \operatorname {Re} (s)>0}[39]
cosinecos⁡(ωt)⋅u(t) {\displaystyle \cos(\omega t)\cdot u(t)\ }ss2+ω2{\displaystyle {s \over s^{2}+\omega ^{2}}}Re⁡(s)>0{\displaystyle \operatorname {Re} (s)>0}[39]
hyperbolic sinesinh⁡(αt)⋅u(t) {\displaystyle \sinh(\alpha t)\cdot u(t)\ }αs2−α2{\displaystyle {\alpha \over s^{2}-\alpha ^{2}}}Re⁡(s)>|α|{\displaystyle \operatorname {Re} (s)>\left|\alpha \right|}[40]
hyperbolic cosinecosh⁡(αt)⋅u(t) {\displaystyle \cosh(\alpha t)\cdot u(t)\ }ss2−α2{\displaystyle {s \over s^{2}-\alpha ^{2}}}Re⁡(s)>|α|{\displaystyle \operatorname {Re} (s)>\left|\alpha \right|}[40]
exponentially decaying
sine wave		e−αtsin⁡(ωt)⋅u(t) {\displaystyle e^{-\alpha t}\sin(\omega t)\cdot u(t)\ }ω(s+α)2+ω2{\displaystyle {\omega \over (s+\alpha )^{2}+\omega ^{2}}}Re⁡(s)>−α{\displaystyle \operatorname {Re} (s)>-\alpha }[39]
exponentially decaying
cosine wave		e−αtcos⁡(ωt)⋅u(t) {\displaystyle e^{-\alpha t}\cos(\omega t)\cdot u(t)\ }s+α(s+α)2+ω2{\displaystyle {s+\alpha \over (s+\alpha )^{2}+\omega ^{2}}}Re⁡(s)>−α{\displaystyle \operatorname {Re} (s)>-\alpha }[39]
natural logarithmln⁡(t)⋅u(t){\displaystyle \ln(t)\cdot u(t)}−1s[ln⁡(s)+γ]{\displaystyle -{1 \over s}\left[\ln(s)+\gamma \right]}Re⁡(s)>0{\displaystyle \operatorname {Re} (s)>0}[40]
Bessel function
of the first kind,
of ordern		Jn(ωt)⋅u(t){\displaystyle J_{n}(\omega t)\cdot u(t)}(s2+ω2−s)nωns2+ω2{\displaystyle {\frac {\left({\sqrt {s^{2}+\omega ^{2}}}-s\right)^{\!n}}{\omega ^{n}{\sqrt {s^{2}+\omega ^{2}}}}}}Re⁡(s)>0{\displaystyle \operatorname {Re} (s)>0}
(n > −1)		[41]
Error functionerf⁡(t)⋅u(t){\displaystyle \operatorname {erf} (t)\cdot u(t)}1ses2/4(1−erf⁡s2){\displaystyle {\frac {1}{s}}e^{s^{2}/4}\!\left(1-\operatorname {erf} {\frac {s}{2}}\right)}Re⁡(s)>0{\displaystyle \operatorname {Re} (s)>0}[41]
Explanatory notes:
  • t, a real number, typically representstime, although it can representany independent dimension.
  • s is thecomplex frequency domain parameter, andRe(s) is itsreal part.
  • α,β,τ, andω arereal numbers.
  • n is aninteger.

s-domain equivalent circuits and impedances

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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 tophasor impedances.

Here is a summary of equivalents:

s-domain equivalent circuits
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

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The Laplace transform is used frequently inengineering andphysics; the output of alinear 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, 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.[42]

The Laplace transform can also be used to solve differential equations and is used extensively inmechanical engineering andelectrical 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 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

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Let⁠L{f(t)}=F(s){\displaystyle {\mathcal {L}}\left\{f(t)\right\}=F(s)}⁠. Then (see the table above)∂sL{f(t)t}=∂s∫0∞f(t)te−stdt=−∫0∞f(t)e−stdt=−F(s){\displaystyle \partial _{s}{\mathcal {L}}\left\{{\frac {f(t)}{t}}\right\}=\partial _{s}\int _{0}^{\infty }{\frac {f(t)}{t}}e^{-st}\,dt=-\int _{0}^{\infty }f(t)e^{-st}dt=-F(s)}

From which one gets:L{f(t)t}=∫s∞F(p)dp.{\displaystyle {\mathcal {L}}\left\{{\frac {f(t)}{t}}\right\}=\int _{s}^{\infty }F(p)\,dp.}

In the limit⁠s→0{\displaystyle s\rightarrow 0}⁠, one gets∫0∞f(t)tdt=∫0∞F(p)dp,{\displaystyle \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∫0∞cos⁡(at)−cos⁡(bt)tdt=∫0∞(pp2+a2−pp2+b2)dp=[12ln⁡p2+a2p2+b2]0∞=12ln⁡b2a2=ln⁡|ba|.{\displaystyle {\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}}}

Complex impedance of a capacitor

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In the theory ofelectrical circuits, the current flow in acapacitor is proportional to the capacitance and rate of change in the electrical potential (with equations as for theSI unit system). Symbolically, this is expressed by the differential equationi=Cdvdt,{\displaystyle i=C{dv \over dt},}whereC is the capacitance of the capacitor,i =i(t) is theelectric current through the capacitor as a function of time, andv =v(t) is thevoltage across the terminals of the capacitor, also as a function of time.

Taking the Laplace transform of this equation, we obtainI(s)=C(sV(s)−V0),{\displaystyle I(s)=C(sV(s)-V_{0}),}whereI(s)=L{i(t)},V(s)=L{v(t)},{\displaystyle {\begin{aligned}I(s)&={\mathcal {L}}\{i(t)\},\\V(s)&={\mathcal {L}}\{v(t)\},\end{aligned}}}andV0=v(0).{\displaystyle V_{0}=v(0).}

Solving forV(s) we haveV(s)=I(s)sC+V0s.{\displaystyle V(s)={I(s) \over sC}+{V_{0} \over s}.}

The definition of the complex impedanceZ (inohms) is the ratio of the complex voltageV divided by the complex currentI while holding the initial stateV0 at zero:Z(s)=V(s)I(s)|V0=0.{\displaystyle Z(s)=\left.{V(s) \over I(s)}\right|_{V_{0}=0}.}

Using this definition and the previous equation, we find:Z(s)=1sC,{\displaystyle 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

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Consider a linear time-invariant system withtransfer functionH(s)=1(s+α)(s+β).{\displaystyle H(s)={\frac {1}{(s+\alpha )(s+\beta )}}.}

Theimpulse response is simply the inverse Laplace transform of this transfer function:h(t)=L−1{H(s)}.{\displaystyle 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,1(s+α)(s+β)=Ps+α+Rs+β.{\displaystyle {\frac {1}{(s+\alpha )(s+\beta )}}={P \over s+\alpha }+{R \over s+\beta }.}

The unknown constantsP andR are theresidues located at the corresponding poles of the transfer function. Each residue represents the relative contribution of thatsingularity to 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 +α to get1s+β=P+R(s+α)s+β.{\displaystyle {\frac {1}{s+\beta }}=P+{R(s+\alpha ) \over s+\beta }.}

Then by lettings = −α, the contribution fromR vanishes and all that is left isP=1s+β|s=−α=1β−α.{\displaystyle P=\left.{1 \over s+\beta }\right|_{s=-\alpha }={1 \over \beta -\alpha }.}

Similarly, the residueR is given byR=1s+α|s=−β=1α−β.{\displaystyle R=\left.{1 \over s+\alpha }\right|_{s=-\beta }={1 \over \alpha -\beta }.}

Note thatR=−1β−α=−P{\displaystyle R={-1 \over \beta -\alpha }=-P}and so the substitution ofR andP into the expanded expression forH(s) givesH(s)=(1β−α)⋅(1s+α−1s+β).{\displaystyle 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 #3 in theTable of Laplace Transforms, above), we can take the inverse Laplace transform ofH(s) to obtainh(t)=L−1{H(s)}=1β−α(e−αt−e−βt),{\displaystyle 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 property as if the system is a series of filters with transfer functions1/(s +α) and1/(s +β). That is, the inverse ofH(s)=1(s+α)(s+β)=1s+α⋅1s+β{\displaystyle H(s)={\frac {1}{(s+\alpha )(s+\beta )}}={\frac {1}{s+\alpha }}\cdot {\frac {1}{s+\beta }}}isL−1{1s+α}∗L−1{1s+β}=e−αt∗e−βt=∫0te−αxe−β(t−x)dx=e−αt−e−βtβ−α.{\displaystyle {\mathcal {L}}^{-1}\!\left\{{\frac {1}{s+\alpha }}\right\}*{\mathcal {L}}^{-1}\!\left\{{\frac {1}{s+\beta }}\right\}=e^{-\alpha t}*e^{-\beta t}=\int _{0}^{t}e^{-\alpha x}e^{-\beta (t-x)}\,dx={\frac {e^{-\alpha t}-e^{-\beta t}}{\beta -\alpha }}.}

Phase delay

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Time functionLaplace transform
sin⁡(ωt+φ){\displaystyle \sin {(\omega t+\varphi )}}ssin⁡(φ)+ωcos⁡(φ)s2+ω2{\displaystyle {\frac {s\sin(\varphi )+\omega \cos(\varphi )}{s^{2}+\omega ^{2}}}}
cos⁡(ωt+φ){\displaystyle \cos {(\omega t+\varphi )}}scos⁡(φ)−ωsin⁡(φ)s2+ω2.{\displaystyle {\frac {s\cos(\varphi )-\omega \sin(\varphi )}{s^{2}+\omega ^{2}}}.}

Starting with the Laplace transform,X(s)=ssin⁡(φ)+ωcos⁡(φ)s2+ω2{\displaystyle X(s)={\frac {s\sin(\varphi )+\omega \cos(\varphi )}{s^{2}+\omega ^{2}}}}we find the inverse by first rearranging terms in the fraction:X(s)=ssin⁡(φ)s2+ω2+ωcos⁡(φ)s2+ω2=sin⁡(φ)(ss2+ω2)+cos⁡(φ)(ωs2+ω2).{\displaystyle {\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:x(t)=sin⁡(φ)L−1{ss2+ω2}+cos⁡(φ)L−1{ωs2+ω2}=sin⁡(φ)cos⁡(ωt)+cos⁡(φ)sin⁡(ωt).{\displaystyle {\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 sum of the arguments, yielding:x(t)=sin⁡(ωt+φ).{\displaystyle x(t)=\sin(\omega t+\varphi ).}

We can apply similar logic to find thatL−1{scos⁡φ−ωsin⁡φs2+ω2}=cos⁡(ωt+φ).{\displaystyle {\mathcal {L}}^{-1}\left\{{\frac {s\cos \varphi -\omega \sin \varphi }{s^{2}+\omega ^{2}}}\right\}=\cos {(\omega t+\varphi )}.}

Statistical mechanics

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Instatistical mechanics, the Laplace transform of the density of statesg(E){\displaystyle g(E)} defines thepartition function.[43] That is, the canonical partition functionZ(β){\displaystyle Z(\beta )} is given byZ(β)=∫0∞e−βEg(E)dE{\displaystyle Z(\beta )=\int _{0}^{\infty }e^{-\beta E}g(E)\,dE}and the inverse is given byg(E)=12πi∫β0−i∞β0+i∞eβEZ(β)dβ{\displaystyle 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

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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 distribution of matter of anastronomical source ofradiofrequencythermal radiation too distant toresolve as more than a point, given itsflux densityspectrum, rather than relating thetime 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 possiblemodel of the distribution of matter in it (density as a function of distance from the center) consistent with the spectrum.[44] 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

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Consider arandom walk, with steps{+1,−1}{\displaystyle \{+1,-1\}} occurring with probabilities⁠p,q=1−p{\displaystyle p,q=1-p}⁠.[45] Suppose also that the time step is aPoisson process, with parameter⁠λ{\displaystyle \lambda }⁠. Then the probability of the walk being at the lattice pointn{\displaystyle n} at timet{\displaystyle t} isPn(t)=∫0tλe−λ(t−s)(pPn−1(s)+qPn+1(s))ds(+e−λtwhen n=0).{\displaystyle 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 asystem of linear equations forπn(s)=L(Pn)(s),{\displaystyle \pi _{n}(s)={\mathcal {L}}(P_{n})(s),}namely:πn(s)=λλ+s(pπn−1(s)+qπn+1(s))(+1λ+swhen n=0){\displaystyle \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

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The Laplace transform of the measureμ{\displaystyle \mu } on[0,∞){\displaystyle [0,\infty )} is given byLμ(s)=∫0∞e−stdμ(t).{\displaystyle {\mathcal {L}}\mu (s)=\int _{0}^{\infty }e^{-st}d\mu (t).}It is intuitively clear that, for small⁠s>0{\displaystyle s>0}⁠, the exponentially decaying integrand will become more sensitive to the concentration of the measureμ{\displaystyle \mu } on larger subsets of the domain. To make this more precise, introduce the distribution function:M(t)=μ([0,t)).{\displaystyle M(t)=\mu ([0,t)).}Formally, we expect a limit of the following kind:lims→0+Lμ(s)=limt→∞M(t).{\displaystyle \lim _{s\to 0^{+}}{\mathcal {L}}\mu (s)=\lim _{t\to \infty }M(t).}Tauberian theorems are theorems relating the asymptotics of the Laplace transform, as⁠s→0+{\displaystyle s\to 0^{+}}⁠, to those of the distribution ofμ{\displaystyle \mu } as⁠t→∞{\displaystyle t\to \infty }⁠. They are thus of importance in asymptotic formulae ofprobability andstatistics, where often the spectral side has asymptotics that are simpler to infer.[45]

Two Tauberian theorems of note are theHardy–Littlewood Tauberian theorem andWiener's Tauberian theorem. The Wiener theorem generalizes theIkehara Tauberian theorem, which is the following statement:

LetA(x) be a non-negative,monotonic nondecreasing function ofx, defined for0 ≤x < ∞. Suppose thatf(s)=∫0∞A(x)e−xsdx{\displaystyle f(s)=\int _{0}^{\infty }A(x)e^{-xs}\,dx}converges for ℜ(s) > 1 to the functionf(s) and that, for some non-negative numberc,f(s)−cs−1{\displaystyle f(s)-{\frac {c}{s-1}}}has an extension as acontinuous function forℜ(s) ≥ 1.Then thelimit asx goes to infinity ofe−xA(x) is equal to c.

This statement can be applied in particular to thelogarithmic derivative ofRiemann zeta function, and thus provides an extremely short way to prove theprime number theorem.[46]

See also

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Notes

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  1. ^Lynn, Paul A. (1986), "The Laplace Transform and thez-transform",Electronic Signals and Systems, London: Macmillan Education UK, pp. 225–272,doi:10.1007/978-1-349-18461-3_6,ISBN 978-0-333-39164-8,Laplace Transform and the z-transform are closely related to the Fourier Transform. Laplace Transform is somewhat more general in scope than the Fourier Transform, and is widely used by engineers for describing continuous circuits and systems, including automatic control systems.
  2. ^"Differential Equations – Laplace Transforms",Pauls Online Math Notes, retrieved2020-08-08
  3. ^abWeisstein, Eric W.,"Laplace Transform",Wolfram MathWorld, retrieved2020-08-08
  4. ^Deakin, Michael A. B. (1981),"The Development of the Laplace Transform, 1737-1937: I. Euler to Spitzer, 1737-1880",Archive for History of Exact Sciences,25 (4):343–390,ISSN 0003-9519
  5. ^"Des Fonctions génératrices" [On generating functions],Théorie analytique des Probabilités [Analytical Probability Theory] (in French) (2nd ed.), Paris, 1814, chap.I sect.2-20
  6. ^Jaynes, E. T. (Edwin T.) (2003),Probability theory : the logic of science, Bretthorst, G. Larry, Cambridge, UK: Cambridge University Press,ISBN 0511065892,OCLC 57254076
  7. ^Abel, Niels H. (1820), "Sur les fonctions génératrices et leurs déterminantes",Œuvres Complètes (in French), vol. II (published 1839), pp. 77–881881 edition
  8. ^Euler 1744,Euler 1753,Euler 1769
  9. ^Lagrange 1773
  10. ^Grattan-Guinness 1997, p. 260
  11. ^Grattan-Guinness 1997, p. 261
  12. ^Grattan-Guinness 1997, pp. 261–262
  13. ^Grattan-Guinness 1997, pp. 262–266
  14. ^Heaviside, Oliver (January 2008),"The solution of definite integrals by differential transformation",Electromagnetic Theory, vol. III, London, section 526,ISBN 9781605206189{{citation}}: CS1 maint: location missing publisher (link)
  15. ^Edwards, H. M. (1974),Riemann's Zeta Function, New York: Academic Press,ISBN 0-12-232750-0,Zbl 0315.10035
  16. ^Gardner, Murray F.; Barnes, John L. (1942),Transients in Linear Systems studied by the Laplace Transform, New York: Wiley, Appendix C
  17. ^Lerch, Mathias (1903), "Sur un point de la théorie des fonctions génératrices d'Abel" [Proof of the inversion formula],Acta Mathematica (in French),27:339–351,doi:10.1007/BF02421315,hdl:10338.dmlcz/501554
  18. ^Bromwich, Thomas J. (1916),"Normal coordinates in dynamical systems",Proceedings of the London Mathematical Society,15:401–448,doi:10.1112/plms/s2-15.1.401
  19. ^An influential book was:Gardner, Murray F.; Barnes, John L. (1942),Transients in Linear Systems studied by the Laplace Transform, New York: Wiley
  20. ^Doetsch, Gustav (1937),Theorie und Anwendung der Laplacesche Transformation [Theory and Application of the Laplace Transform] (in German), Berlin: Springer translation 1943
  21. ^Feller 1971, §XIII.1.
  22. ^The cumulative distribution function is the integral of the probability density function.
  23. ^Mikusiński, Jan (14 July 2014),Operational Calculus, Elsevier,ISBN 9781483278933
  24. ^Widder 1941, Chapter II, §1
  25. ^Widder 1941, Chapter VI, §2
  26. ^Beffa, Federico (2024),"Laplace Transform of Distributions",Weakly Nonlinear Systems, Cham: Springer Nature Switzerland, pp. 75–85,doi:10.1007/978-3-031-40681-2_5,ISBN 978-3-031-40680-5, retrieved2026-01-14{{citation}}: CS1 maint: work parameter with ISBN (link)
  27. ^Korn & Korn 1967, pp. 226–227
  28. ^Bracewell 2000, Table 14.1, p. 385
  29. ^Archived atGhostarchive and theWayback Machine:Mattuck, Arthur (7 November 2008),"Where the Laplace Transform comes from",YouTube
  30. ^Widder 1941, p. 38.
  31. ^Feller 1971, p. 432
  32. ^Laurent Schwartz (1966),Mathematics for the physical sciences, Addison-Wesley, p 224.
  33. ^Titchmarsh, E. (1986) [1948],Introduction to the theory of Fourier integrals (2nd ed.),Clarendon Press, p. 6,ISBN 978-0-8284-0324-5
  34. ^Takacs 1953, p. 93
  35. ^Riley, K. F.; Hobson, M. P.; Bence, S. J. (2010),Mathematical methods for physics and engineering (3rd ed.), Cambridge University Press, p. 455,ISBN 978-0-521-86153-3
  36. ^Distefano, J. J.; Stubberud, A. R.; Williams, I. J. (1995),Feedback systems and control, Schaum's outlines (2nd ed.), McGraw-Hill, p. 78,ISBN 978-0-07-017052-0
  37. ^Lipschutz, S.; Spiegel, M. R.; Liu, J. (2009),Mathematical Handbook of Formulas and Tables, Schaum's Outline Series (3rd ed.), McGraw-Hill, p. 183,ISBN 978-0-07-154855-7 – provides the case for realq.
  38. ^http://mathworld.wolfram.com/LaplaceTransform.html – Wolfram Mathword provides case for complexq
  39. ^abcdBracewell 1978, p. 227.
  40. ^abcWilliams 1973, p. 88.
  41. ^abWilliams 1973, p. 89.
  42. ^Korn & Korn 1967, §8.1
  43. ^RK Pathria; Paul Beal (1996),Statistical mechanics (2nd ed.), Butterworth-Heinemann, p. 56,ISBN 9780750624695
  44. ^Salem, M.; Seaton, M. J. (1974), "I. Continuum spectra and brightness contours",Monthly Notices of the Royal Astronomical Society,167:493–510,Bibcode:1974MNRAS.167..493S,doi:10.1093/mnras/167.3.493, and
    Salem, M. (1974), "II. Three-dimensional models",Monthly Notices of the Royal Astronomical Society,167:511–516,Bibcode:1974MNRAS.167..511S,doi:10.1093/mnras/167.3.511
  45. ^abFeller,Introduction to Probability Theory, volume II,pp=479-483
  46. ^S. Ikehara (1931), "An extension of Landau's theorem in the analytic theory of numbers",Journal of Mathematics and Physics,10 (1–4):1–12,doi:10.1002/sapm19311011,Zbl 0001.12902

References

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Modern

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  • Bracewell, Ronald N. (1978),The Fourier Transform and its Applications (2nd ed.), McGraw-Hill Kogakusha,ISBN 978-0-07-007013-4
  • Bracewell, R. N. (2000),The Fourier Transform and Its Applications (3rd ed.), Boston: McGraw-Hill,ISBN 978-0-07-116043-8
  • Feller, William (1971),An introduction to probability theory and its applications. Vol. II., Second edition, New York:John Wiley & Sons,MR 0270403
  • Korn, G. A.;Korn, T. M. (1967),Mathematical Handbook for Scientists and Engineers (2nd ed.), McGraw-Hill Companies,ISBN 978-0-07-035370-1
  • Widder, David Vernon (1941),The Laplace Transform, Princeton Mathematical Series, v. 6,Princeton University Press,MR 0005923
  • Williams, J. (1973),Laplace Transforms, Problem Solvers, George Allen & Unwin,ISBN 978-0-04-512021-5
  • Takacs, J. (1953), "Fourier amplitudok meghatarozasa operatorszamitassal",Magyar Hiradastechnika (in Hungarian),IV (7–8):93–96

Historical

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  • 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,ISBN 978-3764314743{{citation}}: CS1 maint: work parameter with ISBN (link), 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,ISBN 978-0-691-01185-1
  • Lagrange, J. L. (1773),Mémoire sur l'utilité de la méthode, Œuvres de Lagrange, vol. 2, pp. 171–234

Further reading

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External links

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