Editing Laplace transform

{{short description|Integral transform useful in probability theory, physics, and engineering}}
In [[mathematics]], the '''Laplace transform''', named after [[Pierre-Simon Laplace]] ({{IPAc-en|l|ə|ˈ|p|l|ɑ:|s}}), is an [[integral transform]] that converts a [[Function (mathematics)|function]] of a [[Real number|real]] [[Variable (mathematics)|variable]] (usually <math>t</math>, in the ''[[time domain]]'') to a function of a [[Complex number|complex]] variable <math>s</math> (in the complex-valued [[frequency domain]], also known as '''''s''-domain''', or '''''s''-plane''').

The transform is useful for converting [[derivative|differentiation]] and [[integral|integration]] in the time domain into much easier [[multiplication]] and [[Division (mathematics)|division]] in the Laplace domain (analogous to how [[logarithm]]s are useful for simplifying multiplication and division into addition and subtraction). This gives the transform many applications in [[science]] and [[engineering]], mostly as a tool for solving linear [[differential equation]]s<ref name="Lynn 1986 pp. 225–272">{{cite book | last=Lynn | first=Paul A. | title=Electronic Signals and Systems | chapter=The Laplace Transform and the ''z''-transform | publisher=Macmillan Education UK | publication-place=London | year=1986 | isbn=978-0-333-39164-8 | doi=10.1007/978-1-349-18461-3_6 | pages=225–272|quote=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.}}</ref> and [[dynamical system]]s by simplifying [[ordinary differential equation]]s and [[integral equation]]s into [[algebraic equation|algebraic polynomial equation]]s, and by simplifying [[convolution]] into [[multiplication]].<ref>{{Cite web |title=Differential Equations – Laplace Transforms |url=https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx |access-date=2020-08-08 |website=Pauls Online Math Notes}}</ref><ref name=":1">{{Cite web |last=Weisstein |first=Eric W. |title=Laplace Transform |url=https://mathworld.wolfram.com/LaplaceTransform.html |access-date=2020-08-08 |website=Wolfram MathWorld |language=en}}</ref> Once solved, the inverse Laplace transform reverts to the original domain.

The Laplace transform is defined (for suitable functions <math>f</math>) by the [[integral]]
<math display="block">
 \mathcal{L}\{f\}(s) = \int_0^\infty f(t) e^{-st} \, dt,
</math>
where ''s'' is a [[complex number]]. It is related to many other transforms, most notably the [[Fourier transform]] and the [[Mellin transform]].  [[Formal calculation|Formally]], the Laplace transform is converted into a Fourier transform by the substitution <math>s = i\omega</math> where <math>\omega</math> 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 an [[analytic function]], and so has a convergent [[power series]], the coefficients of which give the decomposition of a function into its [[moment (mathematics)|moments]]. Also unlike the Fourier transform, when regarded in this way as an analytic function, the techniques of [[complex analysis]], and especially [[contour integral]]s, can be used for calculations.

== History ==
[[File:Laplace, Pierre-Simon, marquis de.jpg|thumb|Pierre-Simon, marquis de Laplace]]
The Laplace transform is named after [[mathematician]] and [[astronomer]] [[Pierre-Simon Laplace|Pierre-Simon, Marquis de Laplace]], who used a similar transform in his work on [[probability theory]].<ref>{{citation |url=https://archive.org/details/thorieanalytiqu01laplgoog |title=Théorie analytique des Probabilités |location=Paris |date=1814 |edition=2nd |at=chap.I sect.2-20 |chapter=Des Fonctions génératrices |trans-title=Analytical Probability Theory |trans-chapter=On generating functions |language=fr}}</ref> Laplace wrote extensively about the use of [[generating function]]s  (1814), and the integral form of the Laplace transform evolved naturally as a result.<ref>{{Cite book|title=Probability theory : the logic of science|last=Jaynes, E. T. (Edwin T.)|date=2003|publisher=Cambridge University Press|others=Bretthorst, G. Larry|isbn=0511065892|location=Cambridge, UK|oclc=57254076}}</ref>

Laplace's use of generating functions was similar to what is now known as the [[z-transform]], and he gave little attention to the [[continuous variable]] case which was discussed by [[Niels Henrik Abel]].<ref>{{citation |first=Niels H. |last=Abel|author-link=Niels Henrik Abel |chapter=Sur les fonctions génératrices et leurs déterminantes |date=1820 |title=Œuvres Complètes |language=fr |publication-date=1839 |volume=II |pages=77–88}} [https://books.google.com/books?id=6FtDAQAAMAAJ&pg=RA2-PA67 1881 edition]</ref> 

From 1744, [[Leonhard Euler]] investigated integrals of the form
<math display=block> z = \int X(x) e^{ax}\, dx \quad\text{ and }\quad z = \int X(x) x^A \, dx</math>
as solutions of differential equations, introducing in particular the [[gamma function]].<ref>{{harvnb|Euler|1744}}, {{harvnb|Euler|1753}}, {{harvnb|Euler|1769}}</ref> [[Joseph-Louis Lagrange]] was an admirer of Euler and, in his work on integrating [[probability density function]]s, investigated expressions of the form
<math display=block> \int X(x) e^{- a x } a^x\, dx,</math>
which resembles a Laplace transform.<ref>{{harvnb|Lagrange|1773}}</ref><ref>{{harvnb|Grattan-Guinness| 1997|p=260}}</ref>

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.<ref>{{harvnb|Grattan-Guinness|1997|p=261}}</ref> However, in 1785, Laplace took the critical step forward when, rather than simply looking for a solution in the form of an integral, he started to apply the transforms in the sense that was later to become popular. He used an integral of the form
<math display=block> \int x^s \varphi (x)\, dx,</math>
akin to a [[Mellin transform]], to transform the whole of a [[difference equation]], in order to look for solutions of the transformed equation. He then went on to apply the Laplace transform in the same way and started to derive some of its properties, beginning to appreciate its potential power.<ref>{{harvnb|Grattan-Guinness|1997|pp=261–262}}</ref>

Laplace also recognised that [[Joseph Fourier]]'s method of [[Fourier series]] for solving the [[diffusion equation]] could only apply to a limited region of space, because those solutions were [[Periodic function|periodic]]. In 1809, Laplace applied his transform to find solutions that diffused indefinitely in space.<ref>{{harvnb|Grattan-Guinness|1997|pp=262&ndash;266}}</ref>  In 1821, [[Cauchy]] developed an [[operational calculus]] for the Laplace transform that could be used to study linear differential equations in much the same way the transform is now used in basic engineering.  This method was popularized, and perhaps rediscovered, by [[Oliver Heaviside]] around the turn of the century.<ref>{{citation |first=Oliver |last=Heaviside |author-link=Oliver Heaviside |chapter=The solution of definite integrals by differential transformation |title=Electromagnetic Theory |location=London |at=section 526 |volume=III |chapter-url=https://books.google.com/books?id=y9auR0L6ZRcC&pg=PA234|isbn=9781605206189 |date=January 2008 }}</ref>

[[Bernhard Riemann]] used the Laplace transform in his 1859 paper ''[[On the number of primes less than a given magnitude]]'', in which he also developed the inversion theorem.  Riemann used the Laplace transform to develop the functional equation of the [[Riemann zeta function]], and this method{{clarify|reason=what method?|date=April 2025}} is still used to relate the [[modular form|modular transformation law]] of the [[Jacobi theta function]] to the functional equation{{clarify|reason=which functional equation?|date=April 2025}} .  

[[Hjalmar Mellin]] was among the first to study the Laplace transform, rigorously in the [[Karl Weierstrass]] school of analysis, and apply it to the study of [[differential equations]] and [[special functions]], at the turn of the 20th century.<ref>{{citation |first1=Murray F. |last1=Gardner |first2=John L. |last2=Barnes |title=Transients in Linear Systems studied by the Laplace Transform |date=1942 |location=New York |publisher=Wiley}}, Appendix C</ref>  At around the same time, Heaviside was busy with his operational calculus.  [[Thomas Joannes Stieltjes]] considered a generalization of the Laplace transform connected to his [[Stieltjes moment problem|work on moments]].  Other contributors in this time period included [[Mathias Lerch]],<ref>{{citation |first=Mathias |last=Lerch |author-link=Mathias Lerch |title=Sur un point de la théorie des fonctions génératrices d'Abel |journal=[[Acta Mathematica]] |volume=27 |date=1903 |pages=339–351 |doi=10.1007/BF02421315 |trans-title=Proof of the inversion formula |language=fr|doi-access=free |hdl=10338.dmlcz/501554 |hdl-access=free }}</ref> [[Oliver Heaviside]], and [[Thomas John I'Anson Bromwich|Thomas Bromwich]].<ref>{{citation |first=Thomas J. |last=Bromwich |author-link=Thomas John I'Anson Bromwich |title=Normal coordinates in dynamical systems |journal=[[Proceedings of the London Mathematical Society]] |volume=15 |pages=401–448 |date=1916 |doi=10.1112/plms/s2-15.1.401|url=https://zenodo.org/record/2319588 }}</ref>

In 1929, [[Vannevar Bush]] and [[Norbert Wiener]] published ''Operational Circuit Analysis'' as a text for engineering analysis of electrical circuits, applying both Fourier transforms and operational calculus, and in which they included one of the first predecessors of the modern table of Laplace transforms.
In 1934, [[Raymond Paley]] and [[Norbert Wiener]] published the important work ''Fourier transforms in the complex domain'', about what is now called the Laplace transform (see below).   Also during the 30s, the Laplace transform was instrumental in [[G H Hardy]] and [[John Edensor Littlewood]]'s study of [[tauberian theorem]]s, 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 Titchmarsh]] wrote the influential ''Introduction to the theory of the Fourier integral'' (1937).

The current widespread use of the transform (mainly in engineering) came about during and soon after [[World War II]],<ref>An influential book was: {{citation |first1=Murray F. |last1=Gardner |first2=John L. |last2=Barnes |title=Transients in Linear Systems studied by the Laplace Transform |date=1942 |location=New York |publisher=Wiley}}</ref> replacing the earlier Heaviside [[operational calculus]]. The advantages of the Laplace transform had been emphasized by [[Gustav Doetsch]],<ref>{{citation |first=Gustav |last=Doetsch |title=Theorie und Anwendung der Laplacesche Transformation |location=Berlin |date=1937 |publisher=Springer |language=de |trans-title=Theory and Application of the Laplace Transform}} translation 1943</ref> to whom the name Laplace transform is apparently due.

== Formal definition ==
[[File:Complex frequency s-domain negative.jpg|class=skin-invert-image|thumb|<math>\Re(e^{-st})</math> for various complex frequencies in the ''s''-domain <math>(s = \sigma + i \omega),</math> which can be expressed as <math>e^{-\sigma t} \cos(\omega t).</math> The <math>\sigma = 0</math> axis contains pure cosines. Positive <math>\sigma</math> contains [[Damped sinusoid|damped cosines]]. Negative <math>\sigma</math> contains [[Exponential growth|exponentially growing]] cosines.]]

The Laplace transform of a [[function (mathematics)|function]] {{math|''f''(''t'')}}, defined for all [[real number]]s {{math|''t'' ≥ 0}}, is the function {{math|''F''(''s'')}}, which is a unilateral transform defined by
{{Equation box 1
|indent = :
|equation = <math>F(s) = \int_0^\infty f(t)e^{-st} \, dt,</math>
|ref = Eq. 1
|border
|border colour = #0073CF
|background colour=#F5FFFA}}
where ''s'' is a [[Complex number|complex]] frequency-domain parameter
<math display=block>
 s = \sigma + i \omega
</math>
with real numbers {{mvar|σ}} and {{mvar|ω}}.

An alternate notation for the Laplace transform is {{anchor|ℒ}}<math>\mathcal{L}\{f\}</math> instead of {{math|''F''}},<ref name=":1" /> often written as <math>
F(s) = \mathcal{L}\{f(t)\}</math> in an [[Function_(mathematics)#Functional_notation|abuse of notation]].

The meaning of the integral depends on types of functions of interest. A necessary condition for existence of the integral is that {{mvar|f}} must be [[locally integrable]] on {{closed-open|0, ∞}}. For locally integrable functions that decay at infinity or are of [[exponential type]] (<math>|f(t)| \le Ae^{B|t|}</math>), the integral can be understood to be a (proper) [[Lebesgue integral]]. However, for many applications it is necessary to regard it as a [[conditionally convergent]] [[improper integral]] at {{math|∞}}. Still more generally, the integral can be understood in a [[distribution (mathematics)|weak sense]], and this is dealt with below.

One can define the Laplace transform of a finite [[Borel measure]] {{mvar|μ}} by the Lebesgue integral<ref>{{harvnb|Feller|1971|loc=§XIII.1}}.</ref>
<math display=block>
 \mathcal{L}\{\mu\}(s) = \int_{[0,\infty)} e^{-st}\, d\mu(t).
</math>

An important special case is where {{mvar|μ}} is a [[probability measure]], for example, the [[Dirac delta function]]. In [[operational calculus]], the Laplace transform of a measure is often treated as though the measure came from a probability density function {{mvar|f}}. In that case, to avoid potential confusion, one often writes
<math display=block>
 \mathcal{L}\{f\}(s) = \int_{0^-}^\infty f(t)e^{-st} \, dt,
</math>
where the lower limit of {{math|0<sup>−</sup>}} is shorthand notation for
<math display=block>
 \lim_{\varepsilon \to 0^+}\int_{-\varepsilon}^\infty.
</math>

This limit emphasizes that any point mass located at {{math|0}} is entirely captured by the Laplace transform. Although with the Lebesgue integral, it is not necessary to take such a limit, it does appear more naturally in connection with the [[Laplace–Stieltjes transform]].

=== Bilateral Laplace transform ===
{{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 the ''bilateral Laplace transform'', or [[two-sided Laplace transform]], by extending the limits of integration to be the entire real axis. If that is done, the common unilateral transform simply becomes a special case of the bilateral transform, where the definition of the function being transformed is multiplied by the [[Heaviside step function]].

The bilateral Laplace transform {{math|''F''(''s'')}} is defined as follows:
{{Equation box 1
|indent = :
|equation = <math>F(s) = \int_{-\infty}^\infty e^{-st} f(t)\, dt.</math>
|ref = Eq. 2
|border
|border colour = #0073CF
|background colour=#F5FFFA}}
An alternate notation for the bilateral Laplace transform is <math>\mathcal{B}\{f\}</math>, instead of {{mvar|F}}.

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

Typical function spaces in which this is true include the spaces of bounded continuous functions, the space {{math|[[Lp space|''L''<sup>∞</sup>(0, ∞)]]}}, or more generally [[tempered distributions]] on {{open-open|0, ∞}}. The Laplace transform is also defined and injective for suitable spaces of tempered distributions.

In these cases, the image of the Laplace transform lives in a space of [[analytic function]]s in the [[#Region of convergence|region of convergence]].  The [[inverse Laplace transform]] is given by the following complex integral, which is known by various names (the '''Bromwich integral''', the '''Fourier–Mellin integral''', and '''Mellin's inverse formula'''):
{{Equation box 1
|indent = :
|equation = <math>f(t) = \mathcal{L}^{-1}\{F\}(t) = \frac{1}{2 \pi i} \lim_{T\to\infty} \int_{\gamma - i T}^{\gamma + i T} e^{st} F(s)\, ds,</math>
|ref = Eq. 3
|border
|border colour = #0073CF
|background colour=#F5FFFA}}
where {{mvar|γ}} is a real number so that the contour path of integration is in the region of convergence of {{math|''F''(''s'')}}. In most applications, the contour can be closed, allowing the use of the [[residue theorem]]. An alternative formula for the inverse Laplace transform is given by [[Post's inversion formula]]. The limit here is interpreted in the [[Weak topology#Weak-* topology|weak-* topology]].

In practice, it is typically more convenient to decompose a Laplace transform into known transforms of functions obtained from a table and construct the inverse by inspection.

=== Probability theory ===
In [[probability theory|pure]] and [[applied probability]], the Laplace transform is defined as an [[expected value]]. If {{mvar|X}} is a [[random variable]] with probability density function {{mvar|f}}, then the Laplace transform of {{mvar|f}} is given by the expectation
<math display=block>
 \mathcal{L}\{f\}(s) = \operatorname{E}\left[e^{-sX}\right],
</math>
where <math>\operatorname{E}[r]</math> is the [[Expected value|expectation]] of [[random variable]] <math>r</math>.

By [[Abuse of notation|convention]], this is referred to as the Laplace transform of the random variable {{mvar|X}} itself. Here, replacing {{mvar|s}} by {{math|−''t''}} gives the [[moment generating function]] of {{mvar|X}}. The Laplace transform has applications throughout probability theory, including [[first passage time]]s of [[stochastic process]]es such as [[Markov chain]]s, and [[renewal theory]].

Of particular use is the ability to recover the [[cumulative distribution function]] of a continuous random variable {{mvar|X}} by means of the Laplace transform as follows:<ref>The cumulative distribution function is the integral of the probability density function.</ref>
<math display=block>
 F_X(x) = \mathcal{L}^{-1}\left\{\frac{1}{s} \operatorname{E}\left[e^{-sX}\right]\right\}(x) = \mathcal{L}^{-1}\left\{\frac{1}{s} \mathcal{L}\{f\}(s)\right\}(x).
</math>

=== Algebraic construction ===

The Laplace transform can be alternatively defined in a purely algebraic manner by applying a [[field of fractions]] construction to the convolution [[ring (abstract algebra)|ring]] of functions on the positive half-line. The resulting [[convolution quotient|space of abstract operators]] is exactly equivalent to Laplace space, but in this construction the forward and reverse transforms never need to be explicitly defined (avoiding the related difficulties with proving convergence).<ref>{{cite book | first=Jan | last=Mikusiński | url=https://books.google.com/books?id=e8LSBQAAQBAJ | title=Operational Calculus | date=14 July 2014 | publisher=Elsevier | isbn=9781483278933 }}</ref>

== Region of convergence ==
{{See also|Pole–zero plot#Continuous-time systems}}
If {{math|''f''}} is a locally integrable function (or more generally a Borel measure locally of bounded variation), then the Laplace transform {{math|''F''(''s'')}} of {{math|''f''}} converges provided that the limit
<math display=block>\lim_{R\to\infty}\int_0^R f(t)e^{-st}\,dt</math>
exists.

The Laplace transform [[Absolute convergence|converges absolutely]] if the integral
<math display=block>\int_0^\infty \left|f(t)e^{-st}\right|\,dt</math>
exists as a proper Lebesgue integral. The Laplace transform is usually understood as [[Conditional convergence|conditionally convergent]], meaning that it converges in the former but not in the latter sense.

The set of values for which {{math|''F''(''s'')}} converges absolutely is either of the form {{math|Re(''s'') > ''a''}} or {{math|Re(''s'') ≥ ''a''}}, where {{math|''a''}} is an [[extended real number|extended real constant]] with {{math|−∞ ≤ ''a'' ≤ ∞}} (a consequence of the [[dominated convergence theorem]]). The constant {{math|''a''}} is known as the abscissa of absolute convergence, and depends on the growth behavior of {{math|''f''(''t'')}}.<ref>{{harvnb|Widder|1941|loc=Chapter II, §1}}</ref> Analogously, the two-sided transform converges absolutely in a strip of the form  {{math|''a'' < Re(''s'') < ''b''}}, and possibly including the lines {{math|1=Re(''s'') = ''a''}} or {{math|1=Re(''s'') = ''b''}}.<ref>{{harvnb|Widder|1941|loc=Chapter VI, §2}}</ref> The subset of values of {{math|''s''}} for which the Laplace transform converges absolutely is called the region of absolute convergence, or the domain of absolute convergence.  In the two-sided case, it is sometimes called the strip of absolute convergence. The Laplace transform is analytic in the region of absolute convergence: this is a consequence of [[Fubini's theorem]] and [[Morera's theorem]].

Similarly, the set of values for which {{math|''F''(''s'')}} converges (conditionally or absolutely) is known as the region of conditional convergence, or simply the '''region of convergence''' (ROC). If the Laplace transform converges (conditionally) at {{math|1=''s'' = ''s''<sub>0</sub>}}, then it automatically converges for all {{math|''s''}} with {{math|Re(''s'') > Re(''s''<sub>0</sub>)}}. Therefore, the region of convergence is a half-plane of the form {{math|Re(''s'') > ''a''}}, possibly including some points of the boundary line {{math|1=Re(''s'') = ''a''}}.

In the region of convergence {{math|Re(''s'') > Re(''s''<sub>0</sub>)}}, the Laplace transform of {{math|''f''}} can be expressed by [[integration by parts|integrating by parts]] as the integral
<math display=block>F(s) = (s-s_0)\int_0^\infty e^{-(s-s_0)t}\beta(t)\,dt, \quad \beta(u) = \int_0^u e^{-s_0t}f(t)\,dt.</math>

That is, {{math|''F''(''s'')}} can effectively be expressed, in the region of convergence, as the absolutely convergent Laplace transform of some other function.  In particular, it is analytic.

There are several [[Paley–Wiener theorem]]s concerning the relationship between the decay properties of {{math|''f''}}, and the properties of the Laplace transform within the region of convergence.

In engineering applications, a function corresponding to a [[Linear time-invariant system|linear time-invariant (LTI) system]] is ''stable'' if every bounded input produces a bounded output. This is equivalent to the absolute convergence of the Laplace transform of the impulse response function in the region {{math|Re(''s'') ≥ 0}}. As a result, LTI systems are stable, provided that the poles of the Laplace transform of the impulse response function have negative real part.

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

== Properties and theorems ==
The Laplace transform's key property is that it converts [[derivative|differentiation]] and [[integral|integration]] in the time domain into multiplication and division by {{math|''s''}} in the Laplace domain. Thus, the Laplace variable {{math|''s''}} is also known as an ''operator variable'' in the Laplace domain: either the ''derivative operator'' or (for {{math|''s''<sup>−1</sup>)}} the ''integration operator''.

Given the functions {{math|''f''(''t'')}} and {{math|''g''(''t'')}}, and their respective Laplace transforms {{math|''F''(''s'')}} and {{math|''G''(''s'')}},
<math display=block>\begin{align}
f(t) &= \mathcal{L}^{-1}\{F(s)\},\\
g(t) &= \mathcal{L}^{-1}\{G(s)\},
\end{align}</math>

the following table is a list of properties of unilateral Laplace transform:<ref>{{harvnb|Korn|Korn|1967|pp=226&ndash;227}}</ref>

{| class="wikitable" id="291017_tableid"
|+ Properties of the unilateral Laplace transform
|-
 ! scope="col" | Property
 ! scope="col" | Time domain
 ! scope="col" | {{math|''s''}} domain
 ! scope="col" | Comment
|-
 ! scope="row" | [[Linearity]]
 | <math> a f(t) + b g(t) \ </math>
 | <math> a F(s) + b G(s) \ </math>
 | Can be proved using basic rules of integration.
|-
 ! scope="row" | Frequency-domain derivative
 | <math> t f(t) \ </math>
 | <math> -F'(s) \ </math>
 | {{math|''F''′}} is the first derivative of {{math|''F''}} with respect to {{math|''s''}}.
|-
 ! scope="row" | Frequency-domain general derivative
 | <math> t^{n} f(t) \ </math>
 | <math> (-1)^{n} F^{(n)}(s) \ </math>
 | More general form, {{math|''n''}}th derivative of {{math|''F''(''s'')}}.
|-
 ! scope="row" | [[Derivative]]
 | <math> f'(t) \ </math>
 | <math> s F(s) - f(0^{-}) \ </math>
 | {{math|''f''}} is assumed to be a [[differentiable function]], and its derivative is assumed to be of exponential type.  This can then be obtained by integration by parts
|-
 ! scope="row" | Second derivative
 | <math> f''(t) \ </math>
 | <math display="inline"> s^2 F(s) - s f(0^{-}) - f'(0^{-}) \ </math>
 | {{math|''f''}} is assumed twice differentiable and the second derivative to be of exponential type. Follows by applying the Differentiation property to {{math|''f''′(''t'')}}.
|-
 ! scope="row" | General derivative
 | <math> f^{(n)}(t)  \ </math>
 | <math> s^n F(s) - \sum_{k=1}^{n} s^{n-k} f^{(k-1)}(0^{-}) \ </math>
 | {{math|''f''}} is assumed to be {{math|''n''}}-times differentiable, with {{math|''n''}}th derivative of exponential type.  Follows by [[mathematical induction]].
|-
 ! scope="row" | Frequency-domain [[Integral|integration]]
 | <math> \frac{1}{t}f(t)  \ </math>
 | <math> \int_s^\infty F(\sigma)\, d\sigma \ </math>
 | This is deduced using the nature of frequency differentiation and conditional convergence.
|-
 ! scope="row" | Time-domain integration
 | <math> \int_0^t f(\tau)\, d\tau = (u * f)(t)</math>
 | <math> {1 \over s} F(s) </math>
 | {{math|''u''(''t'')}} is the Heaviside step function and {{math|(''u'' ∗ ''f'')(''t'')}} is the [[convolution]] of {{math|''u''(''t'')}} and {{math|''f''(''t'')}}.
|-
 ! scope="row" | Frequency shifting
 | <math> e^{at} f(t) </math>
 | <math> F(s - a) \ </math>
 |
|-
 ! scope="row" | Time shifting
 | <math> f(t - a) u(t - a) </math>
<math> f(t) u(t - a) \ </math>
 | <math> e^{-as} F(s) \ </math>
<math> e^{-as} \mathcal{L}\{f(t + a)\} </math>
 | {{math|''a'' > 0}}, {{math|''u''(''t'')}} is the Heaviside step function
|-
 ! scope="row" | Time scaling
 | <math>f(at)</math>
 | <math> \frac{1}{a} F \left ({s \over a} \right)</math>
 | {{math|''a'' > 0}}
|-
 ! scope="row" | [[Multiplication]]
 | <math>f(t)g(t)</math>
 | <math> \frac{1}{2\pi i}\lim_{T\to\infty}\int_{c - iT}^{c + iT}F(\sigma)G(s - \sigma)\,d\sigma \ </math>
 | The integration is done along the vertical line {{math|1=Re(''σ'') = ''c''}} that lies entirely within the region of convergence of {{math|''F''}}.<ref>{{harvnb|Bracewell|2000|loc=Table 14.1, p. 385}}</ref>
|-
 ! scope="row" | [[Convolution]]
 | <math> (f * g)(t) = \int_{0}^{t} f(\tau)g(t - \tau)\,d\tau</math>
 | <math> F(s) \cdot G(s) \ </math>
 | 
|-
 ! scope="row" | [[Circular convolution]]
 | <math> (f * g)(t) = \int_{0}^T f(\tau)g(t - \tau)\,d\tau</math>
 | <math> F(s) \cdot G(s) \ </math>
 | For periodic functions with period {{math|''T''}}.
|-
 ! scope="row" | [[Complex conjugation]]
 | <math> f^*(t) </math>
 | <math> F^*(s^*) </math>
 |
|-
 ! scope="row" | [[Periodic function]]
 | <math>f(t)</math>
 | <math>{1 \over 1 - e^{-Ts}} \int_0^T e^{-st} f(t)\,dt </math>
 | {{math|''f''(''t'')}} is a periodic function of period {{math|''T''}} so that {{math|1=''f''(''t'') = ''f''(''t'' + ''T'')}}, for all {{math|''t'' ≥ 0}}. This is the result of the time shifting property and the [[geometric series]].
|-
! scope="row" | [[Periodic summation]]
| <math>  f_P(t) = \sum_{n=0}^{\infty} f(t-Tn) </math>
<math> f_P(t) = \sum_{n=0}^{\infty} (-1)^n f(t-Tn) </math>
| <math>  F_P(s) = \frac{1}{1-e^{-Ts}} F(s) </math>
<math> F_P(s) = \frac{1}{1+e^{-Ts}} F(s) </math>
| 
|}

; [[Initial value theorem]]
:<math>f(0^+)=\lim_{s\to \infty}{sF(s)}.</math>
; [[Final value theorem]]
:<math>f(\infty)=\lim_{s\to 0}{sF(s)}</math>, if all [[Pole (complex analysis)|poles]] of <math>sF(s)</math> are in the left half-plane.
:The final value theorem is useful because it gives the long-term behaviour without having to perform [[partial fraction]] decompositions (or other difficult algebra). If {{math|''F''(''s'')}} has a pole in the right-hand plane or poles on the imaginary axis (e.g., if <math>f(t) = e^t</math> or <math>f(t) = \sin(t)</math>), then the behaviour of this formula is undefined.

=== Relation to power series ===
The Laplace transform can be viewed as a [[continuous function|continuous]] analogue of a [[power series]].<ref>Archived at [https://ghostarchive.org/varchive/youtube/20211211/zvbdoSeGAgI Ghostarchive]{{cbignore}} and the [https://web.archive.org/web/20141220033002/https://www.youtube.com/watch?v=zvbdoSeGAgI&gl=US&hl=en Wayback Machine]{{cbignore}}: {{cite web |last1=Mattuck |first1=Arthur |title=Where the Laplace Transform comes from |website=[[YouTube]] |date=7 November 2008 |url=https://www.youtube.com/watch?v=zvbdoSeGAgI}}{{cbignore}}</ref> If  {{math|''a''(''n'')}} is a discrete function of a positive integer {{math|''n''}}, then the power series associated to {{math|''a''(''n'')}} is the series
<math display=block>\sum_{n=0}^{\infty} a(n) x^n</math>
where {{math|''x''}} is a real variable (see ''[[Z-transform]]''). Replacing summation over {{math|''n''}} with integration over  {{math|''t''}}, a continuous version of the power series becomes
<math display=block>\int_{0}^{\infty} f(t) x^t\, dt</math>
where the discrete function {{math|''a''(''n'')}} is replaced by the continuous one {{math|''f''(''t'')}}.

Changing the base of the power from {{math|''x''}} to {{math|''e''}} gives
<math display=block>\int_{0}^{\infty} f(t) \left(e^{\ln{x}}\right)^t\, dt</math>

For this to converge for, say, all bounded functions {{math|''f''}}, it is necessary to require that {{math|ln ''x'' < 0}}. Making the substitution {{math|1=&minus;''s'' = ln ''x''}} gives just the Laplace transform:
<math display=block>\int_{0}^{\infty} f(t) e^{-st}\, dt</math>

In other words, the Laplace transform is a continuous analog of a power series, in which the discrete parameter {{math|''n''}} is replaced by the continuous parameter {{math|''t''}}, and {{math|''x''}} is replaced by {{math|''e''<sup>&minus;''s''</sup>}}.

=== Relation to moments ===
{{main article|Moment-generating function}}
The quantities
<math display=block>\mu_n = \int_0^\infty t^nf(t)\, dt</math>

are the ''moments'' of the function {{math|''f''}}.  If the first {{math|''n''}} moments of {{math|''f''}} converge absolutely, then by repeated [[differentiation under the integral]], 
<math display=block>(-1)^n(\mathcal L f)^{(n)}(0) = \mu_n .</math>
This is of special significance in probability theory, where the moments of a random variable {{math|''X''}} are given by the expectation values <math>\mu_n=\operatorname{E}[X^n]</math>.  Then, the relation holds
<math display=block>\mu_n = (-1)^n\frac{d^n}{ds^n}\operatorname{E}\left[e^{-sX}\right](0).</math>

=== Transform of a function's derivative ===
It is often convenient to use the differentiation property of the Laplace transform to find the transform of a function's derivative. This can be derived from the basic expression for a Laplace transform as follows:
<math display=block>\begin{align}
  \mathcal{L} \left\{f(t)\right\} &= \int_{0^-}^\infty e^{-st} f(t)\, dt \\[6pt]
                                  &= \left[\frac{f(t)e^{-st}}{-s} \right]_{0^-}^\infty -
                                       \int_{0^-}^\infty \frac{e^{-st}}{-s} f'(t) \, dt\quad \text{(by parts)} \\[6pt]
                                  &= \left[-\frac{f(0^-)}{-s}\right] + \frac 1 s \mathcal{L} \left\{f'(t)\right\},
\end{align}</math>
yielding
<math display=block>\mathcal{L} \{ f'(t) \} = s\cdot\mathcal{L} \{ f(t) \}-f(0^-), </math>
and in the bilateral case,
<math display=block> \mathcal{L} \{ f'(t) \} = s \int_{-\infty}^\infty e^{-st} f(t)\,dt  = s \cdot \mathcal{L} \{ f(t) \}. </math>

The general result
<math display=block>\mathcal{L} \left\{ f^{(n)}(t) \right\} = s^n \cdot \mathcal{L} \{ f(t) \} - s^{n - 1} f(0^-) - \cdots - f^{(n - 1)}(0^-),</math>
where <math>f^{(n)}</math> denotes the {{math|''n''}}th derivative of {{math|''f''}}, can then be established with an inductive argument.

=== Evaluating integrals over the positive real axis ===
A useful property of the Laplace transform is the following:
<math display=block>\int_0^\infty f(x)g(x)\,dx = \int_0^\infty(\mathcal{L} f)(s)\cdot(\mathcal{L}^{-1}g)(s)\,ds </math>
under suitable assumptions on the behaviour of <math>f,g</math> in a right neighbourhood of <math>0</math> and on the decay rate of <math>f,g</math> in a left neighbourhood of <math>\infty</math>. The above formula is a variation of integration by parts, with the operators 
<math>\frac{d}{dx}</math> and <math>\int \,dx</math> being replaced by <math>\mathcal{L}</math> and <math>\mathcal{L}^{-1}</math>. Let us prove the equivalent formulation:
<math display=block>\int_0^\infty(\mathcal{L} f)(x)g(x)\,dx = \int_0^\infty f(s)(\mathcal{L}g)(s)\,ds. </math>

By plugging in <math>(\mathcal{L}f)(x)=\int_0^\infty f(s)e^{-sx}\,ds</math> the left-hand side turns into:
<math display=block>\int_0^\infty\int_0^\infty f(s)g(x) e^{-sx}\,ds\,dx, </math>
but assuming Fubini's theorem holds, by reversing the order of integration we get the wanted 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,
<math display=block>\int_0^\infty\frac{\sin x}{x}dx = \int_0^\infty \mathcal{L}(1)(x)\sin x dx = \int_0^\infty 1 \cdot \mathcal{L}(\sin)(x)dx = \int_0^\infty \frac{dx}{x^2 + 1} = \frac{\pi}{2}. </math>

== Relationship to other transforms ==

=== Laplace–Stieltjes transform ===
The (unilateral) Laplace–Stieltjes transform of a function {{math|''g'' : ℝ → ℝ}} is defined by the [[Lebesgue–Stieltjes integral]]

<math display=block> \{ \mathcal{L}^*g \}(s) = \int_0^\infty e^{-st} \, d\,g(t) ~.</math>

The function {{math|''g''}} is assumed to be of [[bounded variation]].  If {{math|''g''}} is the [[antiderivative]] of {{math|''f''}}:

<math display=block> g(x) = \int_0^x f(t)\,d\,t </math>

then the Laplace–Stieltjes transform of {{mvar|g}} and the Laplace transform of {{mvar|f}} coincide. In general, the Laplace–Stieltjes transform is the Laplace transform of the [[Stieltjes measure]] associated to {{mvar|g}}. So in practice, the only distinction between the two transforms is that the Laplace transform is thought of as operating on the density function of the measure, whereas the Laplace–Stieltjes transform is thought of as operating on its [[cumulative distribution function]].<ref>{{harvnb|Feller|1971|p=432}}</ref>

=== Fourier transform ===
{{further|Fourier transform#Laplace transform}}

Let <math>f</math> be a complex-valued Lebesgue integrable function supported on <math>[0,\infty)</math>, and let <math>F(s) = \mathcal Lf(s)</math> be its Laplace transform.  Then, within the region of convergence, we have
:<math>F(\sigma + i\tau) = \int_0^\infty f(t)e^{-\sigma t}e^{-i\tau t}\,dt,</math>
which is the Fourier transform of the function <math>f(t)e^{-\sigma t}</math>.<ref>{{cite book|author=[[Laurent Schwartz]]|title=Mathematics for the physical sciences|year=1966|publisher=Addison-Wesley}}, p 224.</ref>

Indeed, the [[Fourier transform]] is a special case (under certain conditions) of the bilateral Laplace transform. The main difference is that the Fourier transform of a function is a complex function of a ''real'' variable (frequency), the Laplace transform of a function is a complex function of a ''complex'' variable. The Laplace transform is usually restricted to transformation of functions of {{math|''t''}} with {{math|''t'' ≥ 0}}. A consequence of this restriction is that the Laplace transform of a function is a [[holomorphic function]] of the variable {{math|''s''}}. Unlike the Fourier transform, the Laplace transform of a [[distribution (mathematics)|distribution]] is generally a [[well-behaved]] function. Techniques of complex variables can also be used to directly study Laplace transforms. As a holomorphic function, the Laplace transform has a [[power series]] representation. This power series expresses a function as a linear superposition of [[moment (mathematics)|moments]] of the function. This perspective has applications in probability theory.

Formally, the Fourier transform is equivalent to evaluating the bilateral Laplace transform with imaginary argument {{math|1=''s'' = ''iω''}}<ref>{{citation
 | last = Titchmarsh | first = E. | author-link = Edward Charles Titchmarsh
 | title = Introduction to the theory of Fourier integrals
 | isbn = 978-0-8284-0324-5
 | orig-year = 1948
 | year = 1986
 | edition = 2nd
 | publisher = [[Clarendon Press]]
 | page = 6
}}</ref><ref>{{harvnb|Takacs|1953|p=93}}</ref> when the condition explained below is fulfilled,

<math display="block">\begin{align}
  \hat{f}(\omega) &= \mathcal{F}\{f(t)\} \\[4pt]
                  &= \mathcal{L}\{f(t)\}|_{s = i \omega}  =  F(s)|_{s = i \omega} \\[4pt]
                  &= \int_{-\infty}^\infty e^{-i \omega t} f(t)\,dt~.
\end{align}</math>

This convention of the Fourier transform (<math>\hat f_3(\omega)</math> in {{Section link|Fourier transform|Other_conventions}}) requires a factor of {{math|{{sfrac|1|2''π''}}}} on the inverse Fourier transform. This relationship between the Laplace and Fourier transforms is often used to determine the [[frequency spectrum]] of a [[signal (information theory)|signal]] or dynamical system.

The above relation is valid as stated [[if and only if]] the region of convergence (ROC) of {{math|''F''(''s'')}} contains the imaginary axis, {{math|1=''σ'' = 0}}.

For example, the function {{math|1=''f''(''t'') = cos(''ω''<sub>0</sub>''t'')}} has a Laplace transform {{math|1=''F''(''s'') =  ''s''/(''s''<sup>2</sup> + ''ω''<sub>0</sub><sup>2</sup>)}} whose ROC is {{math|Re(''s'') > 0}}. As {{math|1=''s'' = ''iω''<sub>0</sub>}} is a pole of  {{math|''F''(''s'')}}, substituting {{math|1=''s'' = ''iω''}} in {{math|''F''(''s'')}} does not yield the Fourier transform of {{math|''f''(''t'')''u''(''t'')}}, which contains terms proportional to the [[Dirac delta functions]] {{math|''δ''(''ω'' ± ''ω''<sub>0</sub>)}}.

However, a relation of the form
<math display="block">\lim_{\sigma\to 0^+} F(\sigma+i\omega) = \hat{f}(\omega)</math>
holds under much weaker conditions. For instance, this holds for the above example provided that the limit is understood as a [[weak limit]] of measures (see [[vague topology]]). General conditions relating the limit of the Laplace transform of a function on the boundary to the Fourier transform take the form of [[Paley–Wiener theorem]]s.

=== Mellin transform ===
{{Main|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
<math display=block>G(s) = \mathcal{M}\{g(\theta)\} = \int_0^\infty \theta^s g(\theta) \, \frac{d\theta} \theta </math>
we set {{math|1=''θ'' = ''e''<sup>−''t''</sup>}} we get a two-sided Laplace transform.

=== Z-transform ===
{{further|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
<math display=block> z \stackrel{\mathrm{def} }{ {}={} } e^{sT} ,</math>
where {{math|1=''T'' = 1/''f<sub>s</sub>''}} is the [[sampling interval]] (in units of time e.g., seconds) and  {{math|''f<sub>s</sub>''}} is the [[sampling rate]] (in [[samples per second]] or [[hertz]]).

Let
<math display=block> \Delta_T(t) \ \stackrel{\mathrm{def}}{=}\  \sum_{n=0}^{\infty}  \delta(t - n T) </math>
be a sampling impulse train (also called a [[Dirac comb]]) and
<math display=block>\begin{align}
  x_q(t)   &\stackrel{\mathrm{def} }{ {}={} }  x(t) \Delta_T(t) = x(t) \sum_{n=0}^{\infty}  \delta(t - n T) \\
           &= \sum_{n=0}^{\infty} x(n T) \delta(t - n T) = \sum_{n=0}^{\infty} x[n] \delta(t - n T)
\end{align}</math>
be the sampled representation of the continuous-time {{math|''x''(''t'')}}
<math display=block> x[n] \stackrel{\mathrm{def} }{ {}={} }  x(nT) ~.</math>

The Laplace transform of the sampled signal {{math|''x''<sub>''q''</sub>(''t'') }} is
<math display=block>\begin{align}
  X_q(s) &= \int_{0^-}^\infty x_q(t) e^{-s t} \,dt \\
         &= \int_{0^-}^\infty \sum_{n=0}^\infty x[n] \delta(t - n T) e^{-s t} \, dt \\
         &= \sum_{n=0}^\infty x[n] \int_{0^-}^\infty \delta(t - n T) e^{-s t} \, dt \\
         &= \sum_{n=0}^\infty x[n] e^{-n s T}~.
\end{align}</math>

This is the precise definition of the unilateral Z-transform of the discrete function {{math|''x''[''n'']}}

<math display=block> X(z) = \sum_{n=0}^{\infty} x[n] z^{-n} </math>
with the substitution of {{math|''z'' → ''e''<sup>''sT''</sup>}}.

Comparing the last two equations, we find the relationship between the unilateral Z-transform and the Laplace transform of the sampled signal,
<math display=block>X_q(s) = X(z) \Big|_{z=e^{sT}}.</math>

The similarity between the Z- and Laplace transforms is expanded upon in the theory of [[time scale calculus]].

=== Borel transform ===
The integral form of the [[Borel summation|Borel transform]]
<math display=block>F(s) = \int_0^\infty f(z)e^{-sz}\, dz</math>
is a special case of the Laplace transform for {{math|''f''}} an [[entire function]] of exponential type, meaning that
<math display=block>|f(z)|\le Ae^{B|z|}</math>
for some constants {{math|''A''}} and {{math|''B''}}.  The generalized Borel transform allows a different weighting function to be used, rather than the exponential function, to transform functions not of exponential type. [[Nachbin's theorem]] gives necessary and sufficient conditions for the Borel transform to be well defined.

=== Fundamental relationships ===
Since an ordinary Laplace transform can be written as a special case of a 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.<ref>{{Citation |edition=3rd |page=455 |first1=K. F. |last1=Riley |first2=M. P. |last2=Hobson |first3=S. J. |last3=Bence |title=Mathematical methods for physics and engineering |publisher=Cambridge University Press |year=2010 |isbn=978-0-521-86153-3}}</ref><ref>{{Citation |first1=J. J. |last1=Distefano |first2=A. R. |last2=Stubberud |first3=I. J. |last3=Williams |page=78 |title=Feedback systems and control |edition=2nd |publisher=McGraw-Hill |series=Schaum's outlines |year=1995 |isbn=978-0-07-017052-0}}</ref> For definitions and explanations, see the ''Explanatory Notes'' at the end of the table.

Because the Laplace transform is a linear operator,
* The Laplace transform of a sum is the sum of Laplace transforms of each term.<!--
--><math display=block>\mathcal{L}\{f(t) + g(t)\}  = \mathcal{L}\{f(t)\} + \mathcal{L}\{ g(t)\}  </math>
* The Laplace transform of a multiple of a function is that multiple times the Laplace transformation of that function.<!--
--><math display=block>\mathcal{L}\{a f(t)\}  = a \mathcal{L}\{ f(t)\}</math>

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

The unilateral Laplace transform takes as input a function whose time domain is the [[non-negative]] reals, which is why all of the time domain functions in the table below are multiples of the Heaviside step function, {{math|''u''(''t'')}}.

The entries of the table that involve a time delay {{math|''τ''}} are required to be [[causal system|causal]] (meaning that {{math|''τ'' > 0}}).  A causal system is a system where the [[impulse response]] {{math|''h''(''t'')}} is zero for all time {{mvar|t}} prior to {{math|1=''t'' = 0}}. In general, the region of convergence for causal systems is not the same as that of [[anticausal system]]s.

{| class="wikitable" style="text-align: center;"
|+ Selected Laplace transforms
|-
 ! scope="col" | Function
 ! scope="col" | Time domain <br> <math>f(t) = \mathcal{L}^{-1}\{F(s)\}</math> 
 ! scope="col" | Laplace {{math|s}}-domain <br/> <math>F(s) = \mathcal{L}\{f(t)\}</math> 
 ! scope="col" | Region of convergence 
 ! scope="col" | Reference
|-
 ! scope="row" | unit impulse
 | <math> \delta(t) \ </math> 
 | <math> 1  </math> 
 | all {{math|''s''}}
 | inspection
|-
 ! scope="row" | delayed impulse 
 | <math> \delta(t - \tau) \ </math> 
 | <math> e^{-\tau s} \ </math> 
 | all {{math|''s''}} 
 | time shift of<br>unit impulse
|-
 ! scope="row"| unit step
 | <math> u(t) \ </math> 
 | <math> { 1 \over s } </math> 
 | <math> \operatorname{Re}(s) > 0 </math> 
 | integrate unit impulse
|-
 ! scope="row" | delayed unit step 
 | <math> u(t - \tau) \ </math> 
 | <math> \frac 1 s e^{-\tau s} </math> 
 | <math> \operatorname{Re}(s) > 0 </math> 
 | time shift of<br>unit step
|-
 ! scope="row" | product of delayed function and delayed step
 | <math> f(t-\tau)u(t-\tau) </math>
 | <math> e^{-s\tau}\mathcal{L}\{f(t)\}</math>
 |
 | u-substitution, <math>u=t-\tau</math>
|-
 !rectangular impulse
 | <math> u (t) - u(t - \tau) </math>
 | <math> \frac{1}{s}(1 - e^{-\tau s}) </math>
 | <math> \operatorname{Re}(s) > 0 </math> 
 |
|-
 ! scope="row" | [[ramp function|ramp]]
 | <math> t \cdot u(t)\ </math>
 | <math>\frac 1 {s^2}</math>
 | <math> \operatorname{Re}(s) > 0 </math> 
 | integrate unit<br>impulse twice
|-
 ! scope="row" | {{math|''n''}}th power <br/> (for integer {{math|''n''}}) 
 | <math> t^n \cdot u(t) </math>
 | <math> { n! \over s^{n + 1} } </math>
 | <math> \operatorname{Re}(s) > 0 </math> <br/> ({{math|''n'' > −1}})
 | integrate unit<br>step {{math|''n''}} times
|-
 ! scope="row" | {{math|''q''}}th power <br /> (for complex {{math|''q''}}) 
 | <math> t^q \cdot u(t) </math>
 | <math> { \operatorname{\Gamma}(q + 1) \over s^{q + 1} } </math>
 |  <math> \operatorname{Re}(s) > 0 </math>  <br/>  <math> \operatorname{Re}(q) > -1 </math> 
 | <ref>{{cite book |title=Mathematical Handbook of Formulas and Tables |edition=3rd |first1=S. |last1=Lipschutz |first2=M. R. |last2=Spiegel |first3=J. |last3=Liu |series=Schaum's Outline Series |publisher=McGraw-Hill |page=183 |year=2009 |isbn=978-0-07-154855-7}} – provides the case for real {{math|''q''}}.</ref><ref>http://mathworld.wolfram.com/LaplaceTransform.html – Wolfram Mathword provides case for complex {{math|''q''}}</ref>
|-
 ! scope="row" | {{math|''n''}}th root 
 | <math> \sqrt[n]{t} \cdot u(t) </math>
 | <math> { 1 \over s^{\frac 1 n + 1} } \operatorname{\Gamma}\left(\frac 1 n + 1\right) </math>
 | <math> \operatorname{Re}(s) > 0 </math> 
 | Set {{math|''q'' {{=}} 1/''n''}} above.
|-
 ! scope="row" | {{math|''n''}}th power with frequency shift 
 | <math>t^{n} e^{-\alpha t} \cdot u(t) </math>
 | <math>\frac{n!}{(s+\alpha)^{n+1}} </math>
 | <math> \operatorname{Re}(s) > -\alpha </math> 
 | Integrate unit step,<br/>apply frequency shift
|-
 ! scope="row" | delayed {{math|''n''}}th power <br /> with frequency shift 
 | <math>(t-\tau)^n e^{-\alpha (t-\tau)} \cdot u(t-\tau) </math>
 | <math> \frac{n! \cdot e^{-\tau s}}{(s+\alpha)^{n+1}} </math>
 |  <math> \operatorname{Re}(s) > -\alpha </math> 
 | integrate unit step,<br>apply frequency shift,<br>apply time shift
|-
 ! scope="row" | [[exponential decay]]
 | <math> e^{-\alpha t} \cdot u(t)   </math>
 | <math> { 1 \over s+\alpha } </math>
 | <math> \operatorname{Re}(s) > -\alpha </math> 
 | Frequency shift of<br>unit step
|-
 ! scope="row" | [[Two-sided Laplace transform|two-sided]] exponential decay <br>(only for bilateral transform)
 | <math> e^{-\alpha|t|}  \ </math>
 | <math> { 2\alpha \over \alpha^2 - s^2 } </math>
 | <math>  -\alpha < \operatorname{Re}(s) < \alpha </math> 
 | Frequency shift of<br>unit step
|-
 ! scope="row" | exponential approach 
 | <math>(1-e^{-\alpha t})  \cdot u(t)  \ </math>
 | <math>\frac{\alpha}{s(s+\alpha)} </math>
 | <math> \operatorname{Re}(s) > 0 </math> 
 | unit step minus<br/>exponential decay
|-
 ! scope="row" | [[sine]]
 | <math> \sin(\omega t) \cdot u(t) \ </math>
 | <math> { \omega \over s^2 + \omega^2  } </math>
 | <math> \operatorname{Re}(s) > 0 </math> 
 | {{sfn|Bracewell|1978|p=227}}
|-
 ! scope="row" | [[cosine]]
 | <math> \cos(\omega t) \cdot u(t) \ </math>
 | <math> { s \over s^2 + \omega^2  } </math>
 | <math> \operatorname{Re}(s) > 0 </math> 
 | {{sfn|Bracewell|1978|p=227}}
|-
 ! scope="row" | [[hyperbolic sine]]
 | <math> \sinh(\alpha t) \cdot u(t) \ </math>
 | <math> { \alpha \over s^2 - \alpha^2 } </math>
 | <math> \operatorname{Re}(s) > \left| \alpha \right| </math> 
 | {{sfn|Williams|1973|p=88}}
|-
 ! scope="row" | [[hyperbolic cosine]]
 | <math> \cosh(\alpha t) \cdot u(t) \ </math>
 | <math> { s \over s^2 - \alpha^2  } </math>
 |  <math> \operatorname{Re}(s) > \left| \alpha \right| </math> 
 | {{sfn|Williams|1973|p=88}}
|-
 ! scope="row" | exponentially decaying <br /> sine wave 
 | <math>e^{-\alpha t}  \sin(\omega t) \cdot u(t) \ </math>
 | <math> { \omega \over (s+\alpha)^2 + \omega^2  } </math>
 | <math> \operatorname{Re}(s) > - \alpha </math> 
 | {{sfn|Bracewell|1978|p=227}}
|-
 ! scope="row" | exponentially decaying <br /> cosine wave 
 | <math>e^{-\alpha t}  \cos(\omega t) \cdot u(t) \ </math>
 | <math> { s+\alpha \over (s+\alpha)^2 + \omega^2  } </math>
 | <math> \operatorname{Re}(s) > - \alpha </math> 
 | {{sfn|Bracewell|1978|p=227}}
|-
 ! scope="row" | [[natural logarithm]]
 | <math> \ln(t) \cdot u(t) </math>
 | <math> -{1 \over s} \left[ \ln(s)+\gamma \right] </math>
 |  <math> \operatorname{Re}(s) > 0 </math> 
 | {{sfn|Williams|1973|p=88}}
|-
 ! scope="row" | [[Bessel function]] <br> of the first kind, <br /> of order {{math|''n''}}
 | <math> J_n(\omega t) \cdot u(t)</math>
 | <math>\frac{ \left(\sqrt{s^2+ \omega^2}-s\right)^{\!n}}{\omega^n \sqrt{s^2 + \omega^2}}</math>
 | <math> \operatorname{Re}(s) > 0 </math> <br/> ({{math|''n'' > −1}}) 
 | {{sfn|Williams|1973|p=89}}
|-
 ! scope="row" | [[Error function]] 
 | <math> \operatorname{erf}(t) \cdot u(t) </math> 
 | <math> \frac{1}{s} e^{s^2 / 4} \!\left(1 - \operatorname{erf} \frac{s}{2} \right)</math> 
 | <math> \operatorname{Re}(s) > 0 </math>
 | {{sfn|Williams|1973|p=89}}
|-
 | colspan=5 style="text-align: left;" |'''Explanatory notes:'''
{{col-begin}}
{{col-break}}
* {{math|''u''(''t'')}} represents the [[Heaviside step function]].
* {{math|''δ''}}  represents the [[Dirac delta function]].
* {{math|Γ(''z'')}} represents the [[gamma function]].
* {{math|''γ''}} is the [[Euler&ndash;Mascheroni constant]].
{{col-break}}
*  {{math|''t''}}, a real number, typically represents ''time'', although it can represent ''any'' independent dimension.
*  {{math|''s''}} is the [[complex number|complex]] frequency domain parameter, and  {{math|Re(''s'')}} is its [[real part]].
*  {{math|''α'', ''β'', ''τ'', and ''ω''}} are [[real numbers]].
*  {{math|''n''}} is an [[integer]].
{{col-end}}
|}

== ''s''-domain equivalent circuits and impedances ==
The Laplace transform is often used in [[Network analysis (electrical circuits)|circuit analysis]], and simple conversions to the {{math|''s''}}-domain of circuit elements can be made. Circuit elements can be transformed into [[Electrical impedance|impedances]], very similar to [[Phasor (sine waves)|phasor]] impedances.

Here is a summary of equivalents:

: [[File:S-Domain circuit equivalents.svg|class=skin-invert-image|alt={{math|''s''}}-domain equivalent circuits|centre|frameless|400x400px|{{math|''s''}}-domain equivalent circuits]]

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

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

== Examples and applications ==
<!--A few worked examples are provided here to enable the reader to assess comprehension of the factual presentation.  Elaboration beyond the role of supporting factual comprehension belongs at [[v:|Wikiversity]] or [[b:|Wikibooks]].-->

The Laplace transform is used frequently in [[engineering]] and [[physics]]; the output of a [[linear time-invariant system]] can be calculated by convolving its unit impulse response with the input signal. Performing this calculation in Laplace space turns the convolution into a multiplication; the latter being easier to solve because of its algebraic form. For more information, see [[control theory]]. The Laplace transform is invertible on a large class of functions.  Given a simple mathematical or functional description of an input or output to a [[system]], the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.<ref>{{harvnb|Korn|Korn|1967|loc=§8.1}}</ref>

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

=== Evaluating improper integrals ===
Let <math>\mathcal{L}\left\{f(t)\right\} = F(s)</math>. Then (see the table above)

<math display="block">\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) </math>

From which one gets:

<math display=block>
 \mathcal{L} \left\{\frac{f(t)} t \right\} = \int_s^\infty F(p)\, dp.</math>

In the limit <math>s \rightarrow 0</math>, one gets
<math display=block>\int_0^\infty \frac{f(t)} t \, dt = \int_0^\infty F(p)\, dp,</math>
provided that the interchange of limits can be justified. This is often possible as a consequence of the [[Final value theorem#Final Value Theorem for improperly integrable functions (Abel's theorem for integrals)|final value theorem]]. Even when the interchange cannot be justified the calculation can be suggestive. For example, with {{math|''a'' ≠ 0 ≠ ''b''}}, proceeding formally one has
<math display=block>
\begin{align}
\int_0^\infty \frac{ \cos(at) - \cos(bt) }{t} \, dt 
&=\int_0^\infty \left(\frac p {p^2 + a^2} - \frac{p}{p^2 + b^2}\right)\, dp \\[6pt]
&=\left[ \frac{1}{2}  \ln\frac{p^2 + a^2}{p^2 + b^2} \right]_0^\infty = \frac{1}{2} \ln \frac{b^2}{a^2} = \ln \left| \frac {b}{a} \right|.
\end{align}
</math>

The validity of this identity can be proved by other means. It is an example of a [[Frullani integral]].

Another example is [[Dirichlet integral]].

=== Complex impedance of a capacitor ===
In the theory of [[electrical circuit]]s, the current flow in a [[capacitor]] is proportional to the capacitance and rate of change in the electrical potential (with equations as for the [[International System of Units|SI]] unit system). Symbolically, this is expressed by the differential equation
<math display=block>i = C { dv \over dt} ,</math>
where {{math|''C''}} is the capacitance of the capacitor, {{math|1=''i'' = ''i''(''t'')}} is the [[electric current]] through the capacitor as a function of time, and {{math|1=''v'' = ''v''(''t'')}} is the [[electrostatic potential|voltage]] across the terminals of the capacitor, also as a function of time.

Taking the Laplace transform of this equation, we obtain
<math display=block>I(s) = C(s V(s) - V_0),</math>
where
<math display=block>\begin{align}
  I(s) &= \mathcal{L} \{ i(t) \},\\
  V(s) &= \mathcal{L} \{ v(t) \},
\end{align}</math>
and
<math display=block>V_0 = v(0). </math>

Solving for {{math|''V''(''s'')}} we have
<math display=block>V(s) = { I(s) \over sC } + { V_0 \over s }.</math>

The definition of the complex impedance {{math|''Z''}} (in [[ohm]]s) is the ratio of the complex voltage {{math|''V''}} divided by the complex current {{math|''I''}} while holding the initial state {{math|''V''<sub>0</sub>}} at zero:
<math display=block>Z(s) = \left. { V(s) \over I(s) } \right|_{V_0 = 0}.</math>

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

=== Impulse response ===
Consider a linear time-invariant system with [[transfer function]]
<math display=block>H(s) = \frac{1}{(s + \alpha)(s + \beta)}.</math>

The [[impulse response]] is simply the inverse Laplace transform of this transfer function:
<math display=block>h(t) = \mathcal{L}^{-1}\{H(s)\}.</math>

;Partial fraction expansion
<!-- [[Partial fractions in Laplace transforms]] redirect here -->
To evaluate this inverse transform, we begin by expanding {{math|''H''(''s'')}} using the method of partial fraction expansion,
<math display=block>\frac{1}{(s + \alpha)(s + \beta)} = { P \over s + \alpha } + { R \over s+\beta }.</math>

The unknown constants {{math|''P''}} and {{math|''R''}} are the [[residue (complex analysis)|residues]] located at the corresponding poles of the transfer function. Each residue represents the relative contribution of that [[mathematical singularity|singularity]] to the transfer function's overall shape.

By the [[residue theorem]], the inverse Laplace transform depends only upon the poles and their residues. To find the residue {{math|''P''}}, we multiply both sides of the equation by {{math|''s'' + ''α''}} to get
<math display=block>\frac{1}{s + \beta} = P  + { R (s + \alpha) \over s + \beta }.</math>

Then by letting {{math|1=''s'' = −''α''}}, the contribution from {{math|''R''}} vanishes and all that is left is
<math display=block>P = \left.{1 \over s+\beta}\right|_{s=-\alpha} = {1 \over \beta - \alpha}.</math>

Similarly, the residue {{math|''R''}} is given by
<math display=block>R = \left.{1 \over s + \alpha}\right|_{s=-\beta} = {1 \over \alpha - \beta}.</math>

Note that
<math display=block>R = {-1 \over \beta - \alpha} = - P</math>
and so the substitution of {{math|''R''}} and {{math|''P''}} into the expanded expression for {{math|''H''(''s'')}} gives
<math display=block>H(s)  = \left(\frac{1}{\beta - \alpha} \right) \cdot \left( { 1 \over s + \alpha } - { 1  \over s + \beta }  \right).</math>

Finally, using the linearity property and the known transform for exponential decay (see ''Item'' #''3'' in the ''Table of Laplace Transforms'', above), we can take the inverse Laplace transform of {{math|''H''(''s'')}} to obtain
<math display=block>h(t) = \mathcal{L}^{-1}\{H(s)\} = \frac{1}{\beta - \alpha}\left(e^{-\alpha t} - e^{-\beta t}\right),</math>
which is the impulse response of the system.

;Convolution
The same result can be achieved using the [[Convolution theorem|convolution property]] as if the system is a series of filters with transfer functions {{math|1/(''s'' + ''α'')}} and {{math|1/(''s'' + ''β'')}}. That is, the inverse of
<math display=block>H(s) = \frac{1}{(s + \alpha)(s + \beta)} = \frac{1}{s+\alpha} \cdot \frac{1}{s + \beta}</math>
is
<math display=block> \mathcal{L}^{-1}\! \left\{ \frac{1}{s + \alpha} \right\} * \mathcal{L}^{-1}\! \left\{\frac{1}{s + \beta} \right\} = e^{-\alpha t} * e^{-\beta t} = \int_0^t e^{-\alpha x}e^{-\beta (t - x)}\, dx = \frac{e^{-\alpha t}-e^{-\beta t}}{\beta - \alpha}.</math>

=== Phase delay ===
{| class="wikitable"
|-
 ! scope="col" | Time function
 ! scope="col" | Laplace transform
|-
 | <math>\sin{(\omega t + \varphi)}</math>
 | <math>\frac{s\sin(\varphi) + \omega \cos(\varphi)}{s^2 + \omega^2}</math>
|-
 | <math>\cos{(\omega t + \varphi)}</math>
 | <math>\frac{s\cos(\varphi) - \omega \sin(\varphi)}{s^2 + \omega^2}.</math>
|}

Starting with the Laplace transform,
<math display=block>X(s) = \frac{s\sin(\varphi) + \omega \cos(\varphi)}{s^2 + \omega^2}</math>
we find the inverse by first rearranging terms in the fraction:
<math display=block>\begin{align}
  X(s) &= \frac{s \sin(\varphi)}{s^2 + \omega^2} + \frac{\omega \cos(\varphi)}{s^2 + \omega^2} \\
       &= \sin(\varphi) \left(\frac{s}{s^2 + \omega^2} \right) + \cos(\varphi) \left(\frac{\omega}{s^2 + \omega^2} \right).
\end{align}</math>

We are now able to take the inverse Laplace transform of our terms:
<math display=block>\begin{align}
  x(t) &= \sin(\varphi) \mathcal{L}^{-1}\left\{\frac{s}{s^2 + \omega^2} \right\} + \cos(\varphi) \mathcal{L}^{-1}\left\{\frac{\omega}{s^2 + \omega^2} \right\} \\
       &= \sin(\varphi)\cos(\omega t) + \cos(\varphi)\sin(\omega t).
\end{align}</math>

This is just the [[Trigonometric identity#Angle sum and difference identities|sine of the sum]] of the arguments, yielding:
<math display=block>x(t) = \sin (\omega t + \varphi).</math>

We can apply similar logic to find that
<math display=block>\mathcal{L}^{-1} \left\{ \frac{s\cos\varphi - \omega \sin\varphi}{s^2 + \omega^2} \right\} = \cos{(\omega t + \varphi)}.</math>

=== Statistical mechanics ===
In [[statistical mechanics]], the Laplace transform of the density of states <math>g(E)</math> defines the [[partition function (statistical mechanics)|partition function]].<ref>{{cite book|author1=RK Pathria|author2=Paul Beal|title=Statistical mechanics|url=https://archive.org/details/statisticalmecha00path_911|url-access=limited|edition=2nd|publisher=Butterworth-Heinemann|year=1996|page=[https://archive.org/details/statisticalmecha00path_911/page/n66 56]|isbn=9780750624695 }}</ref> That is, the canonical partition function <math>Z(\beta)</math> is given by
<math display=block> Z(\beta) = \int_0^\infty e^{-\beta E}g(E)\,dE</math>
and the inverse is given by
<math display=block> g(E) = \frac{1}{2\pi i} \int_{\beta_0-i\infty}^{\beta_0+i\infty} e^{\beta E}Z(\beta) \, d\beta</math>

===Spatial (not time) structure from astronomical spectrum===
The wide and general applicability of the Laplace transform and its inverse is illustrated by an application in astronomy which provides some information on the ''spatial distribution'' of matter of an [[Astronomy|astronomical]] source of [[radiofrequency]] [[thermal radiation]] too distant to [[Angular resolution|resolve]] as more than a point, given its [[flux density]] [[spectrum]], rather than relating the ''time'' domain with the spectrum (frequency domain).

Assuming certain properties of the object, e.g. spherical shape and constant temperature, calculations based on carrying out an inverse Laplace transformation on the spectrum of the object can produce the only possible [[Mathematical model|model]] of the distribution of matter in it (density as a function of distance from the center) consistent with the spectrum.<ref>{{citation |first1=M. |last1=Salem |first2=M. J. |last2=Seaton |year=1974 |title=I. Continuum spectra and brightness contours |journal=[[Monthly Notices of the Royal Astronomical Society]] |volume=167 |pages=493–510 |doi=10.1093/mnras/167.3.493|bibcode=1974MNRAS.167..493S |doi-access=free}}, and<br/>{{citation |first1=M. |last1=Salem |year=1974 |title=II. Three-dimensional models |journal=Monthly Notices of the Royal Astronomical Society |volume=167 |pages=511–516 |doi=10.1093/mnras/167.3.511|bibcode=1974MNRAS.167..511S |doi-access=free}}</ref> When independent information on the structure of an object is available, the inverse Laplace transform method has been found to be in good agreement.

===Birth and death processes===
Consider a [[random walk]], with steps <math>\{+1,-1\}</math> occurring with probabilities <math>p,q=1-p</math>.<ref>{{cite book|author=Feller|title=Introduction to Probability Theory, volume II,pp=479-483}}</ref> Suppose also that the time step is an [[Poisson process]], with parameter <math>\lambda</math>.  Then the probability of the walk being at the lattice point <math>n</math> at time <math>t</math> is
:<math>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).</math>
This leads to a system of [[integral equation]]s (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
:<math>\pi_n(s) = \mathcal L(P_n)(s),</math>
namely:
:<math>\pi_n(s) = \frac{\lambda}{\lambda+s}(p\pi_{n-1}(s) + q\pi_{n+1}(s))\quad (+\frac1{\lambda + s}\quad \text{when}\ n=0)</math>
which may now be solved by standard methods.

===Tauberian theory===
The Laplace transform of the measure <math>\mu</math> on <math>[0,\infty)</math> is given by
:<math>\mathcal L\mu(s) = \int_0^\infty e^{-st}d\mu(t).</math>
It is intuitively clear that, for small <math>s>0</math>, the exponentially decaying integrand will become more sensitive to the concentration of the measure <math>\mu</math> on larger subsets of the domain.  To make this more precise, introduce the distribution function:
:<math>M(t) = \mu([0,t)).</math>
Formally, we expect a limit of the following kind:
:<math>\lim_{s\to 0^+}\mathcal L\mu(s) = \lim_{t\to\infty} M(t).</math>
[[Tauberian theorem]]s are theorems relating the asymptotics of the Laplace transform, as <math>s\to 0^+</math>, to those of the distribution of <math>\mu</math> as <math>t\to\infty</math>.  They are thus of importance in asymptotic formulae of [[probability]] and [[statistics]], where often the spectral side has asymptotics that are simpler to infer.<ref>{{cite book|author=Feller|title=Introduction to Probability Theory, volume II,pp=479-483}}</ref>

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

Let ''A''(''x'') be a non-negative, [[monotonic function|monotonic]] nondecreasing function of ''x'', defined for 0&nbsp;≤&nbsp;''x''&nbsp;<&nbsp;∞. Suppose that

:<math>f(s)=\int_0^\infty A(x) e^{-xs}\,dx</math>

converges for ℜ(''s'')&nbsp;>&nbsp;1 to the function ''&fnof;''(''s'') and that, for some non-negative number ''c'',

:<math>f(s) - \frac{c}{s-1}</math>

has an extension as a [[continuous function]] for ℜ(''s'')&nbsp;≥&nbsp;1.
Then the [[Limit of a function|limit]] as ''x'' goes to infinity of ''e''<sup>&minus;''x''</sup>&thinsp;''A''(''x'') is equal to&nbsp;c.

This statement can be applied in particular to the [[logarithmic derivative]] of [[Riemann zeta function]], and thus provides an extremely short way to prove the [[prime number theorem]].<ref>{{citation| author=S. Ikehara | authorlink=Shikao Ikehara | title=An extension of Landau's theorem in the analytic theory of numbers | journal=Journal of Mathematics and Physics of the Massachusetts Institute of Technology | year=1931 | volume=10 | issue=1–4 | pages=1–12 | doi=10.1002/sapm19311011 |zbl=0001.12902}}</ref>

== See also ==
{{Portal|Mathematics}}
{{div col}}
* [[Analog signal processing]]
* [[Bernstein's theorem on monotone functions]]
* [[Continuous-repayment mortgage#Mortgage difference and differential equation|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]]
{{div col end}}

== Notes ==
{{Reflist|30em}}

== References ==

=== Modern ===
* {{Citation |last=Bracewell |first=Ronald N. |title=The Fourier Transform and its Applications |edition=2nd |year=1978 |publisher=McGraw-Hill Kogakusha |isbn=978-0-07-007013-4 }}<!-- This edition is used for pinpoint citations in the transform table. -->
* {{citation|first=R. N.|last=Bracewell|title=The Fourier Transform and Its Applications|edition=3rd|location=Boston|publisher=McGraw-Hill|year=2000|isbn=978-0-07-116043-8}}
* {{Citation | last1=Feller | first1=William | author1-link=William Feller | title=An introduction to probability theory and its applications. Vol. II. | publisher=[[John Wiley & Sons]] | location=New York | series=Second edition | mr=0270403  | year=1971}}
* {{citation |first1=G. A. |last1=Korn |first2=T. M. |last2=Korn | author2-link= Theresa M. Korn |title=Mathematical Handbook for Scientists and Engineers |publisher=McGraw-Hill Companies |edition=2nd |year=1967 |isbn=978-0-07-035370-1 }}
* {{Citation | last1=Widder | first1=David Vernon | title=The Laplace Transform | publisher=[[Princeton University Press]] | series=Princeton Mathematical Series, v. 6 | mr=0005923  | year=1941}}
* {{Citation | last=Williams |first=J. |title=Laplace Transforms |series=Problem Solvers |publisher=George Allen & Unwin |year=1973 |isbn= 978-0-04-512021-5 }}
* {{Citation | last=Takacs | first= J.|title=Fourier amplitudok meghatarozasa operatorszamitassal | year=1953 | journal=Magyar Hiradastechnika | volume=IV | issue=7–8|pages=93–96 |language=hu }}

=== Historical ===
<!-- Citations to Opera omnia [The Complete Works] are wrong. Opera omnia was published 1911 and after, so the citations should be |origyear=17xx |year=1992... Handling of Euler's volume number and Opera omnia volume is problematic -->
* {{citation |last=Euler |first=L. |author-link=Leonhard Euler |year=1744 |title=De constructione aequationum |trans-title=The Construction of Equations |language=la |journal=Opera Omnia |series=1st series |volume=22 |pages=150–161}}
* {{citation |last=Euler |first=L. |author-link=Leonhard Euler |year=1753 |title=Methodus aequationes differentiales |trans-title=A Method for Solving Differential Equations |language=la |journal=Opera Omnia |series=1st series |volume=22 |pages=181–213}}
* {{citation |last=Euler |first=L. |author-link=Leonhard Euler |orig-year=1769 |title=Institutiones calculi integralis, Volume 2 |trans-title=Institutions of Integral Calculus |language=la |journal=Opera Omnia |series=1st series |volume=12 |year=1992 |location=Basel |publisher=Birkhäuser |isbn=978-3764314743 <!-- isbn for the entire first series-->}}, Chapters 3–5
* {{citation |last=Euler |first=Leonhard |author-link=Leonhard Euler |year=1769 |title=Institutiones calculi integralis |trans-title=Institutions of Integral Calculus |language=la |volume=II <!--Secundum--> |at=ch. 3–5, pp. 57–153 |location=Paris |publisher=Petropoli |url=https://books.google.com/books?id=BFqWNwpfqo8C }}
* {{citation|last=Grattan-Guinness|first=I|author-link=Ivor Grattan-Guinness|year=1997|contribution=Laplace's integral solutions to partial differential equations|editor=Gillispie, C. C.|title=Pierre Simon Laplace 1749–1827: A Life in Exact Science|location=Princeton|publisher=Princeton University Press|isbn=978-0-691-01185-1}}
* {{citation|last=Lagrange|first=J. L.|author-link=Joseph Louis Lagrange|year=1773|title=Mémoire sur l'utilité de la méthode|series=Œuvres de Lagrange|volume=2|pages=171–234}}

==Further reading==
* {{citation|first1=Wolfgang|last1=Arendt|first2=Charles J.K.|last2=Batty|first3=Matthias|last3=Hieber|first4=Frank|last4=Neubrander|title=Vector-Valued Laplace Transforms and Cauchy Problems|publisher=Birkhäuser Basel|year=2002|isbn=978-3-7643-6549-3 |ref=none}}.
* {{citation|last=Davies|first=Brian|title=Integral transforms and their applications|edition=Third|publisher=Springer|location=New York|year=2002|isbn= 978-0-387-95314-4 |ref=none}}
* {{citation | last=Deakin|first= M. A. B. | year=1981 | title=The development of the Laplace transform | journal=Archive for History of Exact Sciences | volume=25 | pages=343–390 | doi=10.1007/BF01395660 | issue=4 |s2cid= 117913073 |ref=none}}
* {{citation | last=Deakin|first= M. A. B. | year=1982 | title=The development of the Laplace transform | journal=Archive for History of Exact Sciences | volume=26 | pages=351–381 | doi=10.1007/BF00418754 | issue=4 |s2cid= 123071842 |ref=none}}
* {{citation |last=Doetsch |first=Gustav |author-link=Gustav Doetsch |date=1974 |title=Introduction to the Theory and Application of the Laplace Transformation |publisher=Springer |isbn=978-0-387-06407-9 |ref=none}}
* Mathews, Jon; Walker, Robert L. (1970), ''Mathematical methods of physics'' (2nd ed.), New York: W. A. Benjamin, {{isbn|0-8053-7002-1}}
* {{citation|first1=A. D.|last1=Polyanin|first2=A. V.|last2=Manzhirov|title=Handbook of Integral Equations|publisher=CRC Press|location=Boca Raton|year=1998|isbn=978-0-8493-2876-3 |ref=none}}
* {{Citation | last1=Schwartz | first1=Laurent | author-link=Laurent Schwartz | title=Transformation de Laplace des distributions | mr=0052555  | year=1952 | journal=Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] | volume=1952 | pages=196–206 |language=fr |ref=none}}
* {{Citation |last=Schwartz |first=Laurent |author-link=Laurent Schwartz |year=2008 |orig-year=1966 |title=Mathematics for the Physical Sciences |publisher=Dover Publications |location=New York |series=Dover Books on Mathematics |pages=215–241 |isbn=978-0-486-46662-0 |url={{Google books|-_AuDQAAQBAJ|Mathematics for the Physical Sciences|page=215|plainurl=yes}} |ref=none}} - See Chapter VI. The Laplace transform.
* {{citation|first=William McC.|last=Siebert|title=Circuits, Signals, and Systems|publisher=MIT Press|location=Cambridge, Massachusetts|year=1986|isbn=978-0-262-19229-3 |ref=none}}
* {{Citation | last1=Widder | first1=David Vernon | title=What is the Laplace transform? | mr=0013447  | year=1945 | journal=[[American Mathematical Monthly|The American Mathematical Monthly]] | issn=0002-9890 |volume=52 |issue=8 | pages=419–425 | doi=10.2307/2305640 | jstor=2305640 |ref=none}}
* J.A.C.Weidman and Bengt Fornberg: "Fully numerical Laplace transform methods", Numerical Algorithms, vol.92 (2023), pp.&nbsp;985–1006. https://doi.org/10.1007/s11075-022-01368-x .

== External links ==
{{wikiquote}}
{{commons category|Laplace transformation}}
* {{springer|title=Laplace transform|id=p/l057540}}
* [http://wims.unice.fr/wims/wims.cgi?lang=en&+module=tool%2Fanalysis%2Ffourierlaplace Online Computation] of the transform or inverse transform, wims.unice.fr
* [http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm Tables of Integral Transforms] at EqWorld: The World of Mathematical Equations.
* {{MathWorld|title=Laplace Transform|urlname=LaplaceTransform}}
* [http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/ Good explanations of the initial and final value theorems] {{Webarchive|url=https://web.archive.org/web/20090108132440/http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/ |date=2009-01-08 }}
* [http://www.mathpages.com/home/kmath508/kmath508.htm Laplace Transforms] at MathPages
* [http://www.wolframalpha.com/input/?i=laplace+transform+example Computational Knowledge Engine] allows to easily calculate Laplace Transforms and its inverse Transform.
* [http://www.laplacetransformcalculator.com/easy-laplace-transform-calculator/ Laplace Calculator] to calculate Laplace Transforms online easily.
* [https://johnflux.com/2019/02/12/laplace-transform-visualized/ Code to visualize Laplace Transforms] and many example videos.

{{Authority control}}

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