Laplace transform

File:Laplace, Pierre-Simon, marquis de.jpg

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

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.{{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]

From 1744, Leonhard Euler investigated integrals of the form

$$z\; =\; \backslash int\; X(x)\; e^\{ax\}\backslash ,\; dx\; \backslash quad\backslash text\{\; and\; \}\backslash quad\; z\; =\; \backslash int\; X(x)\; x^A\; \backslash ,\; dx$$

as solutions of differential equations, introducing in particular the gamma function.{{harvnb|Euler|1744}}, {{harvnb|Euler|1753}}, {{harvnb|Euler|1769}} Joseph-Louis Lagrange was an admirer of Euler and, in his work on integrating probability density functions, investigated expressions of the form

$$\backslash int\; X(x)\; e^\{-\; a\; x\; \}\; a^x\backslash ,\; dx,$$

which resembles a Laplace transform.{{harvnb|Lagrange|1773}}{{harvnb|Grattan-Guinness| 1997|p=260}}

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.{{harvnb|Grattan-Guinness|1997|p=261}} 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

$$\backslash int\; x^s\; \backslash varphi\; (x)\backslash ,\; dx,$$

akin to a Mellin transform, to transform the whole of a difference equation, in order to look for solutions of the transformed equation. He then went on to apply the Laplace transform in the same way and started to derive some of its properties, beginning to appreciate its potential power.{{harvnb|Grattan-Guinness|1997|pp=261–262}}

Laplace also recognised that Joseph Fourier's method of Fourier series for solving the diffusion equation could only apply to a limited region of space, because those solutions were periodic. In 1809, Laplace applied his transform to find solutions that diffused indefinitely in space.{{harvnb|Grattan-Guinness|1997|pp=262–266}} 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.{{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 }}

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 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.{{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 At around the same time, Heaviside was busy with his operational calculus. Thomas Joannes Stieltjes considered a generalization of the Laplace transform connected to his work on moments. Other contributors in this time period included Mathias Lerch,{{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 }} Oliver Heaviside, and Thomas Bromwich.{{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 }}

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 theorems, and this application was later expounded on by Widder (1941), who developed other aspects of the theory such as a new method for inversion. Edward Charles 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,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}} replacing the earlier Heaviside operational calculus. The advantages of the Laplace transform had been emphasized by Gustav Doetsch,{{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 to whom the name Laplace transform is apparently due.

File:Complex frequency s-domain negative.jpg. Negative $\backslash sigma$ contains exponentially growing cosines.]]

The Laplace transform of a function {{math|f(t)}}, defined for all real numbers {{math|t ≥ 0}}, is the function {{math|F(s)}}, which is a unilateral transform defined by

{{Equation box 1

|indent = :

|equation = $F(s)\; =\; \backslash int\_0^\backslash infty\; f(t)e^\{-st\}\; \backslash ,\; dt,$

|ref = Eq. 1

|border

|border colour = #0073CF

|background colour=#F5FFFA}}

where s is a complex frequency-domain parameter

s = \sigma + i \omega

with real numbers {{mvar|σ}} and {{mvar|ω}}.

An alternate notation for the Laplace transform is {{anchor|ℒ}}$\backslash mathcal\{L\}\backslash \{f\backslash \}$ instead of {{math|F}}, often written as $$

F(s) = \mathcal{L}\{f(t)\} in an 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 ($|f(t)|\; \backslash le\; Ae^\{B|t|\}$), the integral can be understood to be a (proper) Lebesgue integral. However, for many applications it is necessary to regard it as a conditionally convergent improper integral at {{math|∞}}. Still more generally, the integral can be understood in a weak sense, and this is dealt with below.

One can define the Laplace transform of a finite Borel measure {{mvar|μ}} by the Lebesgue integral{{harvnb|Feller|1971|loc=§XIII.1}}.

\mathcal{L}\{\mu\}(s) = \int_{[0,\infty)} e^{-st}\, d\mu(t).

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

\mathcal{L}\{f\}(s) = \int_{0^-}^\infty f(t)e^{-st} \, dt,

where the lower limit of {{math|0⁻}} is shorthand notation for

\lim_{\varepsilon \to 0^+}\int_{-\varepsilon}^\infty.

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.

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

|ref = Eq. 2

|border

|border colour = #0073CF

|background colour=#F5FFFA}}

An alternate notation for the bilateral Laplace transform is $\backslash mathcal\{B\}\backslash \{f\backslash \}$, instead of {{mvar|F}}.

{{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 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|L^∞(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 functions in the region of convergence. The inverse Laplace transform is given by the following complex integral, which is known by various names (the Bromwich integral, the Fourier–Mellin integral, and Mellin's inverse formula):

{{Equation box 1

|indent = :

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

|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.

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.

In 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

\mathcal{L}\{f\}(s) = \operatorname{E}\left[e^{-sX}\right],

where $\backslash operatorname\{E\}[r]$ is the expectation of random variable $r$.

By 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 times of stochastic processes such as Markov chains, and renewal theory.

Of particular use is the ability to recover the cumulative distribution function of a continuous random variable {{mvar|X}} by means of the Laplace transform as follows:The cumulative distribution function is the integral of the probability density function.

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).

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

{{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

$$\backslash lim\_\{R\backslash to\backslash infty\}\backslash int\_0^R\; f(t)e^\{-st\}\backslash ,dt$$

exists.

The Laplace transform converges absolutely if the integral

$$\backslash int\_0^\backslash infty\; \backslash left|f(t)e^\{-st\}\backslash right|\backslash ,dt$$

exists as a proper Lebesgue integral. The Laplace transform is usually understood as conditionally convergent, meaning that it converges in the former but not in the latter sense.

The set of values for which {{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 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)}}.{{harvnb|Widder|1941|loc=Chapter II, §1}} 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}}.{{harvnb|Widder|1941|loc=Chapter VI, §2}} 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 = s₀}}, then it automatically converges for all {{math|s}} with {{math|Re(s) > Re(s₀)}}. 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(s₀)}}, the Laplace transform of {{math|f}} can be expressed by integrating by parts as the integral

$$F(s)\; =\; (s-s\_0)\backslash int\_0^\backslash infty\; e^\{-(s-s\_0)t\}\backslash beta(t)\backslash ,dt,\; \backslash quad\; \backslash beta(u)\; =\; \backslash int\_0^u\; e^\{-s\_0t\}f(t)\backslash ,dt.$$

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 theorems 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 (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.

The Laplace transform's key property is that it converts differentiation and 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⁻¹)}} 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)}},

$$\backslash begin\{align\}$$

f(t) &= \mathcal{L}^{-1}\{F(s)\},\\

g(t) &= \mathcal{L}^{-1}\{G(s)\},

\end{align}

the following table is a list of properties of unilateral Laplace transform:{{harvnb|Korn|Korn|1967|pp=226–227}}

; Initial value theorem

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

; Final value theorem

:$f(\backslash infty)=\backslash lim\_\{s\backslash to\; 0\}\{sF(s)\}$, if all poles of $sF(s)$ are in the left half-plane.

:The final value theorem is useful because it gives the long-term behaviour without having to perform partial fraction decompositions (or other difficult algebra). If {{math|F(s)}} has a pole in the right-hand plane or poles on the imaginary axis (e.g., if $f(t)\; =\; e^t$ or $f(t)\; =\; \backslash sin(t)$), then the behaviour of this formula is undefined.

The Laplace transform can be viewed as a continuous analogue of a power series.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}} 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

$$\backslash sum\_\{n=0\}^\{\backslash infty\}\; a(n)\; x^n$$

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

$$\backslash int\_\{0\}^\{\backslash infty\}\; f(t)\; x^t\backslash ,\; dt$$

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

$$\backslash int\_\{0\}^\{\backslash infty\}\; f(t)\; \backslash left(e^\{\backslash ln\{x\}\}\backslash right)^t\backslash ,\; dt$$

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=−s = ln x}} gives just the Laplace transform:

$$\backslash int\_\{0\}^\{\backslash infty\}\; f(t)\; e^\{-st\}\backslash ,\; dt$$

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^−s}}.

{{main article|Moment-generating function}}

The quantities

$$\backslash mu\_n\; =\; \backslash int\_0^\backslash infty\; t^nf(t)\backslash ,\; dt$$

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,

$$(-1)^n(\backslash mathcal\; L\; f)^\{(n)\}(0)\; =\; \backslash mu\_n\; .$$

This is of special significance in probability theory, where the moments of a random variable {{math|X}} are given by the expectation values $\backslash mu\_n=\backslash operatorname\{E\}[X^n]$. Then, the relation holds

$$\backslash mu\_n\; =\; (-1)^n\backslash frac\{d^n\}\{ds^n\}\backslash operatorname\{E\}\backslash left[e^\{-sX\}\backslash right](0).$$

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:

$$\backslash begin\{align\}$$

\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}

yielding

$$\backslash mathcal\{L\}\; \backslash \{\; f\text{'}(t)\; \backslash \}\; =\; s\backslash cdot\backslash mathcal\{L\}\; \backslash \{\; f(t)\; \backslash \}-f(0^-),$$

and in the bilateral case,

$$\backslash mathcal\{L\}\; \backslash \{\; f\text{'}(t)\; \backslash \}\; =\; s\; \backslash int\_\{-\backslash infty\}^\backslash infty\; e^\{-st\}\; f(t)\backslash ,dt\; =\; s\; \backslash cdot\; \backslash mathcal\{L\}\; \backslash \{\; f(t)\; \backslash \}.$$

The general result

$$\backslash mathcal\{L\}\; \backslash left\backslash \{\; f^\{(n)\}(t)\; \backslash right\backslash \}\; =\; s^n\; \backslash cdot\; \backslash mathcal\{L\}\; \backslash \{\; f(t)\; \backslash \}\; -\; s^\{n\; -\; 1\}\; f(0^-)\; -\; \backslash cdots\; -\; f^\{(n\; -\; 1)\}(0^-),$$

where $f^\{(n)\}$ denotes the {{math|n}}th derivative of {{math|f}}, can then be established with an inductive argument.

A useful property of the Laplace transform is the following:

$$\backslash int\_0^\backslash infty\; f(x)g(x)\backslash ,dx\; =\; \backslash int\_0^\backslash infty(\backslash mathcal\{L\}\; f)(s)\backslash cdot(\backslash mathcal\{L\}^\{-1\}g)(s)\backslash ,ds$$

under suitable assumptions on the behaviour of $f,g$ in a right neighbourhood of $0$ and on the decay rate of $f,g$ in a left neighbourhood of $\backslash infty$. The above formula is a variation of integration by parts, with the operators

$\backslash frac\{d\}\{dx\}$ and $\backslash int\; \backslash ,dx$ being replaced by $\backslash mathcal\{L\}$ and $\backslash mathcal\{L\}^\{-1\}$. Let us prove the equivalent formulation:

$$\backslash int\_0^\backslash infty(\backslash mathcal\{L\}\; f)(x)g(x)\backslash ,dx\; =\; \backslash int\_0^\backslash infty\; f(s)(\backslash mathcal\{L\}g)(s)\backslash ,ds.$$

By plugging in $(\backslash mathcal\{L\}f)(x)=\backslash int\_0^\backslash infty\; f(s)e^\{-sx\}\backslash ,ds$ the left-hand side turns into:

$$\backslash int\_0^\backslash infty\backslash int\_0^\backslash infty\; f(s)g(x)\; e^\{-sx\}\backslash ,ds\backslash ,dx,$$

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,

$$\backslash int\_0^\backslash infty\backslash frac\{\backslash sin\; x\}\{x\}dx\; =\; \backslash int\_0^\backslash infty\; \backslash mathcal\{L\}(1)(x)\backslash sin\; x\; dx\; =\; \backslash int\_0^\backslash infty\; 1\; \backslash cdot\; \backslash mathcal\{L\}(\backslash sin)(x)dx\; =\; \backslash int\_0^\backslash infty\; \backslash frac\{dx\}\{x^2\; +\; 1\}\; =\; \backslash frac\{\backslash pi\}\{2\}.$$

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

$$\backslash \{\; \backslash mathcal\{L\}^*g\; \backslash \}(s)\; =\; \backslash int\_0^\backslash infty\; e^\{-st\}\; \backslash ,\; d\backslash ,g(t)\; ~.$$

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

$$g(x)\; =\; \backslash int\_0^x\; f(t)\backslash ,d\backslash ,t$$

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.{{harvnb|Feller|1971|p=432}}

{{further|Fourier transform#Laplace transform}}

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

:$F(\backslash sigma\; +\; i\backslash tau)\; =\; \backslash int\_0^\backslash infty\; f(t)e^\{-\backslash sigma\; t\}e^\{-i\backslash tau\; t\}\backslash ,dt,$

which is the Fourier transform of the function $f(t)e^\{-\backslash sigma\; t\}$.{{cite book|author=Laurent Schwartz|title=Mathematics for the physical sciences|year=1966|publisher=Addison-Wesley}}, p 224.

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 is generally a well-behaved function. Techniques of complex variables can also be used to directly study Laplace transforms. As a holomorphic function, the Laplace transform has a power series representation. This power series expresses a function as a linear superposition of moments of the function. This perspective has applications in probability theory.

Formally, the Fourier transform is equivalent to evaluating the bilateral Laplace transform with imaginary argument {{math|1=s = iω}}{{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

}}{{harvnb|Takacs|1953|p=93}} when the condition explained below is fulfilled,

$$\backslash begin\{align\}$$

\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}

This convention of the Fourier transform ($\backslash hat\; f\_3(\backslash omega)$ in {{Section link|Fourier transform|Other_conventions}}) requires a factor of {{math|{{sfrac|1|2π}}}} on the inverse Fourier transform. This relationship between the Laplace and Fourier transforms is often used to determine the frequency spectrum of a signal or dynamical system.

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

For example, the function {{math|1=f(t) = cos(ω₀t)}} has a Laplace transform {{math|1=F(s) = s/(s² + ω₀²)}} whose ROC is {{math|Re(s) > 0}}. As {{math|1=s = iω₀}} 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 ± ω₀)}}.

However, a relation of the form

$$\backslash lim\_\{\backslash sigma\backslash to\; 0^+\}\; F(\backslash sigma+i\backslash omega)\; =\; \backslash hat\{f\}(\backslash omega)$$

holds under much weaker conditions. For instance, this holds for the above example provided that the limit is understood as a weak limit of measures (see vague topology). General conditions relating the limit of the Laplace transform of a function on the boundary to the Fourier transform take the form of Paley–Wiener theorems.

{{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

$$G(s)\; =\; \backslash mathcal\{M\}\backslash \{g(\backslash theta)\backslash \}\; =\; \backslash int\_0^\backslash infty\; \backslash theta^s\; g(\backslash theta)\; \backslash ,\; \backslash frac\{d\backslash theta\}\; \backslash theta$$

we set {{math|1=θ = e^−t}} we get a two-sided Laplace 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

$$z\; \backslash stackrel\{\backslash mathrm\{def\}\; \}\{\; \{\}=\{\}\; \}\; e^\{sT\}\; ,$$

where {{math|1=T = 1/f_s}} is the sampling interval (in units of time e.g., seconds) and {{math|f_s}} is the sampling rate (in samples per second or hertz).

Let

$$\backslash Delta\_T(t)\; \backslash \; \backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\backslash \; \backslash sum\_\{n=0\}^\{\backslash infty\}\; \backslash delta(t\; -\; n\; T)$$

be a sampling impulse train (also called a Dirac comb) and

$$\backslash begin\{align\}$$

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}

be the sampled representation of the continuous-time {{math|x(t)}}

$$x[n]\; \backslash stackrel\{\backslash mathrm\{def\}\; \}\{\; \{\}=\{\}\; \}\; x(nT)\; ~.$$

The Laplace transform of the sampled signal {{math|x_q(t) }} is

$$\backslash begin\{align\}$$

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}

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

$$X(z)\; =\; \backslash sum\_\{n=0\}^\{\backslash infty\}\; x[n]\; z^\{-n\}$$

with the substitution of {{math|z → e^sT}}.

Comparing the last two equations, we find the relationship between the unilateral Z-transform and the Laplace transform of the sampled signal,

$$X\_q(s)\; =\; X(z)\; \backslash Big|\_\{z=e^\{sT\}\}.$$

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

The integral form of the Borel transform

$$F(s)\; =\; \backslash int\_0^\backslash infty\; f(z)e^\{-sz\}\backslash ,\; dz$$

is a special case of the Laplace transform for {{math|f}} an entire function of exponential type, meaning that

$$|f(z)|\backslash le\; Ae^\{B|z|\}$$

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.

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.

{{main article|List of Laplace transforms}}

The following table provides Laplace transforms for many common functions of a single variable.{{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}}{{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}} 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.$$\backslash mathcal\{L\}\backslash \{f(t)\; +\; g(t)\backslash \}\; =\; \backslash mathcal\{L\}\backslash \{f(t)\backslash \}\; +\; \backslash mathcal\{L\}\backslash \{\; g(t)\backslash \}$$
  • The Laplace transform of a multiple of a function is that multiple times the Laplace transformation of that function.$$\backslash mathcal\{L\}\backslash \{a\; f(t)\backslash \}\; =\; a\; \backslash mathcal\{L\}\backslash \{\; f(t)\backslash \}$$

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

    The unilateral Laplace transform takes as input a function whose time domain is the non-negative reals, which is why all of the time domain functions in the table below are multiples of the Heaviside step function, {{math|u(t)}}.

    The entries of the table that involve a time delay {{math|τ}} are required to be 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 systems.

    class="wikitable" style="text-align: center;" |+ Selected Laplace transforms
    scope="col" | Function ! scope="col" | Time domain $f(t)\; =\; \backslash mathcal\{L\}^\{-1\}\backslash \{F(s)\backslash \}$ ! scope="col" | Laplace {{math|s}}-domain $F(s)\; =\; \backslash mathcal\{L\}\backslash \{f(t)\backslash \}$ ! scope="col" | Region of convergence ! scope="col" | Reference
    scope="row" | unit impulse | $\backslash delta(t)\; \backslash$ | $1$ | all {{math|s}} | inspection
    scope="row" | delayed impulse | $\backslash delta(t\; -\; \backslash tau)\; \backslash$ | $e^\{-\backslash tau\; s\}\; \backslash$ | all {{math|s}} | time shift of unit impulse
    scope="row"| unit step | $u(t)\; \backslash$ | $\{\; 1\; \backslash over\; s\; \}$ | $\backslash operatorname\{Re\}(s)\; >\; 0$ | integrate unit impulse
    scope="row" | delayed unit step | $u(t\; -\; \backslash tau)\; \backslash$ | $\backslash frac\; 1\; s\; e^\{-\backslash tau\; s\}$ | $\backslash operatorname\{Re\}(s)\; >\; 0$ | time shift of unit step
    scope="row" | product of delayed function and delayed step | $f(t-\backslash tau)u(t-\backslash tau)$ | $e^\{-s\backslash tau\}\backslash mathcal\{L\}\backslash \{f(t)\backslash \}$ | | u-substitution, $u=t-\backslash tau$
    rectangular impulse | $u\; (t)\; -\; u(t\; -\; \backslash tau)$ | $\backslash frac\{1\}\{s\}(1\; -\; e^\{-\backslash tau\; s\})$ | $\backslash operatorname\{Re\}(s)\; >\; 0$ |
    scope="row" | ramp | $t\; \backslash cdot\; u(t)\backslash$ | $\backslash frac\; 1\; \{s^2\}$ | $\backslash operatorname\{Re\}(s)\; >\; 0$ | integrate unit impulse twice
    scope="row" | {{math|n}}th power (for integer {{math|n}}) | $t^n\; \backslash cdot\; u(t)$ | $\{\; n!\; \backslash over\; s^\{n\; +\; 1\}\; \}$ | $\backslash operatorname\{Re\}(s)\; >\; 0$ ({{math|n > −1}}) | integrate unit step {{math|n}} times
    scope="row" | {{math|q}}th power (for complex {{math|q}}) | $t^q\; \backslash cdot\; u(t)$ | $\{\; \backslash operatorname\{\backslash Gamma\}(q\; +\; 1)\; \backslash over\; s^\{q\; +\; 1\}\; \}$ | $\backslash operatorname\{Re\}(s)\; >\; 0$ $\backslash operatorname\{Re\}(q)\; >\; -1$ | {{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}}.http://mathworld.wolfram.com/LaplaceTransform.html – Wolfram Mathword provides case for complex {{math|q}}
    scope="row" | {{math|n}}th root | $\backslash sqrt[n]\{t\}\; \backslash cdot\; u(t)$ | $\{\; 1\; \backslash over\; s^\{\backslash frac\; 1\; n\; +\; 1\}\; \}\; \backslash operatorname\{\backslash Gamma\}\backslash left(\backslash frac\; 1\; n\; +\; 1\backslash right)$ | $\backslash operatorname\{Re\}(s)\; >\; 0$ | Set {{math|q {{=}} 1/n}} above.
    scope="row" | {{math|n}}th power with frequency shift | $t^\{n\}\; e^\{-\backslash alpha\; t\}\; \backslash cdot\; u(t)$ | $\backslash frac\{n!\}\{(s+\backslash alpha)^\{n+1\}\}$ | $\backslash operatorname\{Re\}(s)\; >\; -\backslash alpha$ | Integrate unit step, apply frequency shift
    scope="row" | delayed {{math|n}}th power with frequency shift | $(t-\backslash tau)^n\; e^\{-\backslash alpha\; (t-\backslash tau)\}\; \backslash cdot\; u(t-\backslash tau)$ | $\backslash frac\{n!\; \backslash cdot\; e^\{-\backslash tau\; s\}\}\{(s+\backslash alpha)^\{n+1\}\}$ | $\backslash operatorname\{Re\}(s)\; >\; -\backslash alpha$ | integrate unit step, apply frequency shift, apply time shift
    scope="row" | exponential decay | $e^\{-\backslash alpha\; t\}\; \backslash cdot\; u(t)$ | $\{\; 1\; \backslash over\; s+\backslash alpha\; \}$ | $\backslash operatorname\{Re\}(s)\; >\; -\backslash alpha$ | Frequency shift of unit step
    scope="row" | two-sided exponential decay (only for bilateral transform) | $e^\{-\backslash alpha|t$

    \

    | $\{\; 2\backslash alpha\; \backslash over\; \backslash alpha^2\; -\; s^2\; \}$

    |

    | Frequency shift of
    unit step

    |-

    ! scope="row" | exponential approach

    | $(1-e^\{-\backslash alpha\; t\})\; \backslash cdot\; u(t)\; \backslash$

    | $\backslash frac\{\backslash alpha\}\{s(s+\backslash alpha)\}$

    | $\backslash operatorname\{Re\}(s)\; >\; 0$

    | unit step minus
    exponential decay

    |-

    ! scope="row" | sine

    | $\backslash sin(\backslash omega\; t)\; \backslash cdot\; u(t)\; \backslash$

    | $\{\; \backslash omega\; \backslash over\; s^2\; +\; \backslash omega^2\; \}$

    | $\backslash operatorname\{Re\}(s)\; >\; 0$

    | {{sfn|Bracewell|1978|p=227}}

    |-

    ! scope="row" | cosine

    | $\backslash cos(\backslash omega\; t)\; \backslash cdot\; u(t)\; \backslash$

    | $\{\; s\; \backslash over\; s^2\; +\; \backslash omega^2\; \}$

    | $\backslash operatorname\{Re\}(s)\; >\; 0$

    | {{sfn|Bracewell|1978|p=227}}

    |-

    ! scope="row" | hyperbolic sine

    | $\backslash sinh(\backslash alpha\; t)\; \backslash cdot\; u(t)\; \backslash$

    | $\{\; \backslash alpha\; \backslash over\; s^2\; -\; \backslash alpha^2\; \}$

    | $\backslash operatorname\{Re\}(s)\; >\; \backslash left|\; \backslash alpha\; \backslash right|$

    | {{sfn|Williams|1973|p=88}}

    |-

    ! scope="row" | hyperbolic cosine

    | $\backslash cosh(\backslash alpha\; t)\; \backslash cdot\; u(t)\; \backslash$

    | $\{\; s\; \backslash over\; s^2\; -\; \backslash alpha^2\; \}$

    | $\backslash operatorname\{Re\}(s)\; >\; \backslash left|\; \backslash alpha\; \backslash right|$

    | {{sfn|Williams|1973|p=88}}

    |-

    ! scope="row" | exponentially decaying
    sine wave

    | $e^\{-\backslash alpha\; t\}\; \backslash sin(\backslash omega\; t)\; \backslash cdot\; u(t)\; \backslash$

    | $\{\; \backslash omega\; \backslash over\; (s+\backslash alpha)^2\; +\; \backslash omega^2\; \}$

    | $\backslash operatorname\{Re\}(s)\; >\; -\; \backslash alpha$

    | {{sfn|Bracewell|1978|p=227}}

    |-

    ! scope="row" | exponentially decaying
    cosine wave

    | $e^\{-\backslash alpha\; t\}\; \backslash cos(\backslash omega\; t)\; \backslash cdot\; u(t)\; \backslash$

    | $\{\; s+\backslash alpha\; \backslash over\; (s+\backslash alpha)^2\; +\; \backslash omega^2\; \}$

    | $\backslash operatorname\{Re\}(s)\; >\; -\; \backslash alpha$

    | {{sfn|Bracewell|1978|p=227}}

    |-

    ! scope="row" | natural logarithm

    | $\backslash ln(t)\; \backslash cdot\; u(t)$

    | $-\{1\; \backslash over\; s\}\; \backslash left[\; \backslash ln(s)+\backslash gamma\; \backslash right]$

    | $\backslash operatorname\{Re\}(s)\; >\; 0$

    | {{sfn|Williams|1973|p=88}}

    |-

    ! scope="row" | Bessel function
    of the first kind,
    of order {{math|n}}

    | $J\_n(\backslash omega\; t)\; \backslash cdot\; u(t)$

    | $\backslash frac\{\; \backslash left(\backslash sqrt\{s^2+\; \backslash omega^2\}-s\backslash right)^\{\backslash !n\}\}\{\backslash omega^n\; \backslash sqrt\{s^2\; +\; \backslash omega^2\}\}$

    | $\backslash operatorname\{Re\}(s)\; >\; 0$
    ({{math|n > −1}})

    | {{sfn|Williams|1973|p=89}}

    |-

    ! scope="row" | Error function

    | $\backslash operatorname\{erf\}(t)\; \backslash cdot\; u(t)$

    | $\backslash frac\{1\}\{s\}\; e^\{s^2\; /\; 4\}\; \backslash !\backslash left(1\; -\; \backslash operatorname\{erf\}\; \backslash frac\{s\}\{2\}\; \backslash right)$

    | $\backslash operatorname\{Re\}(s)\; >\; 0$

    | {{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–Mascheroni constant.

    {{col-break}}

    • {{math|t}}, a real number, typically represents time, although it can represent any independent dimension.
    • {{math|s}} is the 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}}

    |}

The Laplace transform is often used in circuit analysis, and simple conversions to the {{math|s}}-domain of circuit elements can be made. Circuit elements can be transformed into impedances, very similar to phasor impedances.

Here is a summary of equivalents:

: File:S-Domain circuit equivalents.svg

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.

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.{{harvnb|Korn|Korn|1967|loc=§8.1}}

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.

Let $\backslash mathcal\{L\}\backslash left\backslash \{f(t)\backslash right\backslash \}\; =\; F(s)$. Then (see the table above)

$$\backslash partial\_s\backslash mathcal\{L\}\; \backslash left\backslash \{\backslash frac\{f(t)\}\; t\; \backslash right\backslash \}\; =\; \backslash partial\_s\backslash int\_0^\backslash infty\; \backslash frac\{f(t)\}\{t\}e^\{-st\}\backslash ,\; dt\; =\; -\backslash int\_0^\backslash infty\; f(t)e^\{-st\}dt\; =\; -\; F(s)$$

From which one gets:

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

In the limit $s\; \backslash rightarrow\; 0$, one gets

$$\backslash int\_0^\backslash infty\; \backslash frac\{f(t)\}\; t\; \backslash ,\; dt\; =\; \backslash int\_0^\backslash infty\; F(p)\backslash ,\; dp,$$

provided that the interchange of limits can be justified. This is often possible as a consequence of the final value theorem. Even when the interchange cannot be justified the calculation can be suggestive. For example, with {{math|a ≠ 0 ≠ b}}, proceeding formally one has

\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}

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

Another example is Dirichlet integral.

In the theory of electrical circuits, the current flow in a capacitor is proportional to the capacitance and rate of change in the electrical potential (with equations as for the SI unit system). Symbolically, this is expressed by the differential equation

$$i\; =\; C\; \{\; dv\; \backslash over\; dt\}\; ,$$

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 voltage across the terminals of the capacitor, also as a function of time.

Taking the Laplace transform of this equation, we obtain

$$I(s)\; =\; C(s\; V(s)\; -\; V\_0),$$

where

$$\backslash begin\{align\}$$

I(s) &= \mathcal{L} \{ i(t) \},\\

V(s) &= \mathcal{L} \{ v(t) \},

\end{align}

and

$$V\_0\; =\; v(0).$$

Solving for {{math|V(s)}} we have

$$V(s)\; =\; \{\; I(s)\; \backslash over\; sC\; \}\; +\; \{\; V\_0\; \backslash over\; s\; \}.$$

The definition of the complex impedance {{math|Z}} (in ohms) is the ratio of the complex voltage {{math|V}} divided by the complex current {{math|I}} while holding the initial state {{math|V₀}} at zero:

$$Z(s)\; =\; \backslash left.\; \{\; V(s)\; \backslash over\; I(s)\; \}\; \backslash right|\_\{V\_0\; =\; 0\}.$$

Using this definition and the previous equation, we find:

$$Z(s)\; =\; \backslash frac\{1\}\{sC\},$$

which is the correct expression for the complex impedance of a capacitor. In addition, the Laplace transform has large applications in control theory.

Consider a linear time-invariant system with transfer function

$$H(s)\; =\; \backslash frac\{1\}\{(s\; +\; \backslash alpha)(s\; +\; \backslash beta)\}.$$

The impulse response is simply the inverse Laplace transform of this transfer function:

$$h(t)\; =\; \backslash mathcal\{L\}^\{-1\}\backslash \{H(s)\backslash \}.$$

;Partial fraction expansion

To evaluate this inverse transform, we begin by expanding {{math|H(s)}} using the method of partial fraction expansion,

$$\backslash frac\{1\}\{(s\; +\; \backslash alpha)(s\; +\; \backslash beta)\}\; =\; \{\; P\; \backslash over\; s\; +\; \backslash alpha\; \}\; +\; \{\; R\; \backslash over\; s+\backslash beta\; \}.$$

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

By the residue theorem, the inverse Laplace transform depends only upon the poles and their residues. To find the residue {{math|P}}, we multiply both sides of the equation by {{math|s + α}} to get

$$\backslash frac\{1\}\{s\; +\; \backslash beta\}\; =\; P\; +\; \{\; R\; (s\; +\; \backslash alpha)\; \backslash over\; s\; +\; \backslash beta\; \}.$$

Then by letting {{math|1=s = −α}}, the contribution from {{math|R}} vanishes and all that is left is

$$P\; =\; \backslash left.\{1\; \backslash over\; s+\backslash beta\}\backslash right|\_\{s=-\backslash alpha\}\; =\; \{1\; \backslash over\; \backslash beta\; -\; \backslash alpha\}.$$

Similarly, the residue {{math|R}} is given by

$$R\; =\; \backslash left.\{1\; \backslash over\; s\; +\; \backslash alpha\}\backslash right|\_\{s=-\backslash beta\}\; =\; \{1\; \backslash over\; \backslash alpha\; -\; \backslash beta\}.$$

Note that

$$R\; =\; \{-1\; \backslash over\; \backslash beta\; -\; \backslash alpha\}\; =\; -\; P$$

and so the substitution of {{math|R}} and {{math|P}} into the expanded expression for {{math|H(s)}} gives

$$H(s)\; =\; \backslash left(\backslash frac\{1\}\{\backslash beta\; -\; \backslash alpha\}\; \backslash right)\; \backslash cdot\; \backslash left(\; \{\; 1\; \backslash over\; s\; +\; \backslash alpha\; \}\; -\; \{\; 1\; \backslash over\; s\; +\; \backslash beta\; \}\; \backslash right).$$

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

$$h(t)\; =\; \backslash mathcal\{L\}^\{-1\}\backslash \{H(s)\backslash \}\; =\; \backslash frac\{1\}\{\backslash beta\; -\; \backslash alpha\}\backslash left(e^\{-\backslash alpha\; t\}\; -\; e^\{-\backslash beta\; t\}\backslash right),$$

which is the impulse response of the system.

;Convolution

The same result can be achieved using the convolution property as if the system is a series of filters with transfer functions {{math|1/(s + α)}} and {{math|1/(s + β)}}. That is, the inverse of

$$H(s)\; =\; \backslash frac\{1\}\{(s\; +\; \backslash alpha)(s\; +\; \backslash beta)\}\; =\; \backslash frac\{1\}\{s+\backslash alpha\}\; \backslash cdot\; \backslash frac\{1\}\{s\; +\; \backslash beta\}$$

is

$$\backslash mathcal\{L\}^\{-1\}\backslash !\; \backslash left\backslash \{\; \backslash frac\{1\}\{s\; +\; \backslash alpha\}\; \backslash right\backslash \}\; *\; \backslash mathcal\{L\}^\{-1\}\backslash !\; \backslash left\backslash \{\backslash frac\{1\}\{s\; +\; \backslash beta\}\; \backslash right\backslash \}\; =\; e^\{-\backslash alpha\; t\}\; *\; e^\{-\backslash beta\; t\}\; =\; \backslash int\_0^t\; e^\{-\backslash alpha\; x\}e^\{-\backslash beta\; (t\; -\; x)\}\backslash ,\; dx\; =\; \backslash frac\{e^\{-\backslash alpha\; t\}-e^\{-\backslash beta\; t\}\}\{\backslash beta\; -\; \backslash alpha\}.$$

Starting with the Laplace transform,

$$X(s)\; =\; \backslash frac\{s\backslash sin(\backslash varphi)\; +\; \backslash omega\; \backslash cos(\backslash varphi)\}\{s^2\; +\; \backslash omega^2\}$$

we find the inverse by first rearranging terms in the fraction:

$$\backslash begin\{align\}$$

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}

We are now able to take the inverse Laplace transform of our terms:

$$\backslash begin\{align\}$$

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}

This is just the sine of the sum of the arguments, yielding:

$$x(t)\; =\; \backslash sin\; (\backslash omega\; t\; +\; \backslash varphi).$$

We can apply similar logic to find that

$$\backslash mathcal\{L\}^\{-1\}\; \backslash left\backslash \{\; \backslash frac\{s\backslash cos\backslash varphi\; -\; \backslash omega\; \backslash sin\backslash varphi\}\{s^2\; +\; \backslash omega^2\}\; \backslash right\backslash \}\; =\; \backslash cos\{(\backslash omega\; t\; +\; \backslash varphi)\}.$$

In statistical mechanics, the Laplace transform of the density of states $g(E)$ defines the partition function.{{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 }} That is, the canonical partition function $Z(\backslash beta)$ is given by

$$Z(\backslash beta)\; =\; \backslash int\_0^\backslash infty\; e^\{-\backslash beta\; E\}g(E)\backslash ,dE$$

and the inverse is given by

$$g(E)\; =\; \backslash frac\{1\}\{2\backslash pi\; i\}\; \backslash int\_\{\backslash beta\_0-i\backslash infty\}^\{\backslash beta\_0+i\backslash infty\}\; e^\{\backslash beta\; E\}Z(\backslash beta)\; \backslash ,\; d\backslash beta$$

The wide and general applicability of the Laplace transform and its inverse is illustrated by an application in astronomy which provides some information on the spatial distribution of matter of an astronomical source of radiofrequency thermal radiation too distant to resolve as more than a point, given its flux density spectrum, rather than relating the time domain with the spectrum (frequency domain).

Assuming certain properties of the object, e.g. spherical shape and constant temperature, calculations based on carrying out an inverse Laplace transformation on the spectrum of the object can produce the only possible model of the distribution of matter in it (density as a function of distance from the center) consistent with the spectrum.{{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
{{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}} When independent information on the structure of an object is available, the inverse Laplace transform method has been found to be in good agreement.

Consider a random walk, with steps $\backslash \{+1,-1\backslash \}$ occurring with probabilities $p,q=1-p$.{{cite book|author=Feller|title=Introduction to Probability Theory, volume II,pp=479-483}} Suppose also that the time step is an Poisson process, with parameter $\backslash lambda$. Then the probability of the walk being at the lattice point $n$ at time $t$ is

:$P\_n(t)\; =\; \backslash int\_0^t\backslash lambda\; e^\{-\backslash lambda(t-s)\}(pP\_\{n-1\}(s)\; +\; qP\_\{n+1\}(s))\backslash ,ds\backslash quad\; (+e^\{-\backslash lambda\; t\}\backslash quad\backslash text\{when\}\backslash \; n=0).$

This leads to a system of integral equations (or equivalently a system of differential equations). However, because it is a system of convolution equations, the Laplace transform converts it into a system of linear equations for

:$\backslash pi\_n(s)\; =\; \backslash mathcal\; L(P\_n)(s),$

namely:

:$\backslash pi\_n(s)\; =\; \backslash frac\{\backslash lambda\}\{\backslash lambda+s\}(p\backslash pi\_\{n-1\}(s)\; +\; q\backslash pi\_\{n+1\}(s))\backslash quad\; (+\backslash frac1\{\backslash lambda\; +\; s\}\backslash quad\; \backslash text\{when\}\backslash \; n=0)$

which may now be solved by standard methods.

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

:$\backslash mathcal\; L\backslash mu(s)\; =\; \backslash int\_0^\backslash infty\; e^\{-st\}d\backslash mu(t).$

It is intuitively clear that, for small $s>0$, the exponentially decaying integrand will become more sensitive to the concentration of the measure $\backslash mu$ on larger subsets of the domain. To make this more precise, introduce the distribution function:

:$M(t)\; =\; \backslash mu([0,t)).$

Formally, we expect a limit of the following kind:

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

Tauberian theorems are theorems relating the asymptotics of the Laplace transform, as $s\backslash to\; 0^+$, to those of the distribution of $\backslash mu$ as $t\backslash to\backslash infty$. They are thus of importance in asymptotic formulae of probability and statistics, where often the spectral side has asymptotics that are simpler to infer.{{cite book|author=Feller|title=Introduction to Probability Theory, volume II,pp=479-483}}

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 nondecreasing function of x, defined for 0 ≤ x < ∞. Suppose that

:$f(s)=\backslash int\_0^\backslash infty\; A(x)\; e^\{-xs\}\backslash ,dx$

converges for ℜ(s) > 1 to the function ƒ(s) and that, for some non-negative number c,

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

has an extension as a continuous function for ℜ(s) ≥ 1.

Then the limit as x goes to infinity of e^−x A(x) is equal to 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.{{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}}

{{Portal|Mathematics}}

{{div col}}

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

{{div col end}}

{{Reflist|30em}}

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• {{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}}
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• {{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}}
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• {{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}}
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• {{Citation | last1=Widder | first1=David Vernon | title=What is the Laplace transform? | mr=0013447 | year=1945 | journal=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. 985–1006. https://doi.org/10.1007/s11075-022-01368-x .

{{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.

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Category:Differential equations

Category:Fourier analysis

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