Laplace transform

 ( Differential Equations ) 

In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a function of a Real number variable (usually $t$, in the time domain) to a function of a Complex number $s$ (in the complex-valued frequency domain, also known as s -domain , or s-plane). The functions are often denoted by $x(t)$ for the time-domain representation, and $X(s)$ for the frequency-domain.

The transform is useful for converting derivative and integral in the time domain into much easier multiplication and division in the Laplace domain (analogous to how are useful for simplifying multiplication and division into addition and subtraction). This gives the transform many applications in science and engineering, mostly as a tool for solving linear differential equations

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

The Laplace transform is defined (for suitable functions $f$) by the integral $$$$

\mathcal{L}\{f\}(s) = \int_0^\infty f(t) e^{-st} \, dt,
     

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

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. 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., , 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.

These types of integrals seem first to have attracted Laplace's attention in 1782, where he was following in the spirit of Euler in using the integrals themselves as solutions of equations. However, in 1785, Laplace took the critical step forward when, rather than simply looking for a solution in the form of an integral, he started to apply the transforms in the sense that was later to become popular. He used an integral of the form $$\backslash int\; x^s\; \backslash varphi\; (x)\backslash ,\; dx,$$ 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.

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

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

Hjalmar Mellin was among the first to study the Laplace transform, rigorously in the Karl Weierstrass school of analysis, and apply it to the study of differential equations and special functions, at the turn of the 20th century., 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, Oliver Heaviside, and Thomas Bromwich.

In 1929, Vannevar Bush and Norbert Wiener published Operational Circuit Analysis as a text for engineering analysis of electrical circuits, applying both Fourier transforms and operational calculus, and in which they included one of the first predecessors of the modern table of Laplace transforms. In 1934, Raymond Paley and Norbert Wiener published the important work Fourier transforms in the complex domain, about what is now called the Laplace transform (see below). Also during the 30s, the Laplace transform was instrumental in G H Hardy and John Edensor Littlewood's study of tauberian theorems, and this application was later expounded on by , 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: replacing the earlier Heaviside operational calculus. The advantages of the Laplace transform had been emphasized by Gustav Doetsch, translation 1943 to whom the name Laplace transform is apparently due.

where s is a Complex number frequency-domain parameter $$$$

s = \sigma + i \omega
     

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

The meaning of the integral depends on types of functions of interest. A necessary condition for existence of the integral is that must be locally integrable on . 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 . 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 by the Lebesgue integral. $$$$

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

An important special case is where 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 . In that case, to avoid potential confusion, one often writes $$$$

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

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

This limit emphasizes that any point mass located at 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.

The bilateral Laplace transform is defined as follows:

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

Typical function spaces in which this is true include the spaces of bounded continuous functions, the space , or more generally tempered distributions on . 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):

where is a real number so that the contour path of integration is in the region of convergence of . 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.

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

By convention, this is referred to as the Laplace transform of the random variable itself. Here, replacing by gives the moment generating function of . The Laplace transform has applications throughout probability theory, including first passage times of stochastic processes such as , and renewal theory.

Of particular use is the ability to recover the cumulative distribution function of a continuous random variable 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 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 converges absolutely is either of the form or , where is an extended real constant with (a consequence of the dominated convergence theorem). The constant is known as the abscissa of absolute convergence, and depends on the growth behavior of . Analogously, the two-sided transform converges absolutely in a strip of the form , and possibly including the lines or . The subset of values of 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 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 , then it automatically converges for all with . Therefore, the region of convergence is a half-plane of the form , possibly including some points of the boundary line .

In the region of convergence , the Laplace transform of 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, 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 , 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 . 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.

Given the functions and , and their respective Laplace transforms and ,

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

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 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.

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Changing the base of the power from to 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 , it is necessary to require that . Making the substitution 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 is replaced by the continuous parameter , and is replaced by .

Analogously to a power series, if $a(n)=O(\backslash rho^\{-n\})$, then the power series converges to an analytic function in , if $f(t)=O(e^\{-\backslash sigma\; t\})$, the Laplace transform converges to an analytic function in $\backslash Re(s)\; >\; \backslash sigma.$

are the moments of the function . If the first moments of 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 are given by the expectation values $\backslash mu\_n=\backslash operatorname\{E\}X^n$. Then, the relation holds $$\backslash mu\_n\; =\; (-1)^n\backslash frac\{d^n\}\{ds^n\}\backslash operatorname\{E\}\backslash lefte^\{-sX\}\backslash right(0).$$

 \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\},
     

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 th derivative of , can then be established with an inductive argument.

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

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

The function is assumed to be of bounded variation. If is the antiderivative of :

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

then the Laplace–Stieltjes transform of and the Laplace transform of coincide. In general, the Laplace–Stieltjes transform is the Laplace transform of the Stieltjes measure associated to . 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.

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

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 $\backslash tau$), the Laplace transform of a function is a complex function of a complex variable (damping factor $\backslash sigma$ and frequency $\backslash tau$). The Laplace transform is usually restricted to transformation of functions of with . A consequence of this restriction is that the Laplace transform of a function is a holomorphic function of the variable . 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 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~.
     

This convention of the Fourier transform ($\backslash hat\; f\_3(\backslash omega)$ in ) requires a factor of 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 contains the imaginary axis, .

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

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.

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 we get a two-sided Laplace transform.

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)
     

The Laplace transform of the sampled signal 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}~.
     

This is the precise definition of the unilateral Z-transform of the discrete function

$$X(z)\; =\; \backslash sum\_\{n=0\}^\{\backslash infty\}\; xn\; z^\{-n\}$$ with the substitution of .

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.

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, .

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

!rectangular impulse

! 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]]
| $\backslash sin(\backslash omega\; t)\; \backslash cdot\; u(t)\; \backslash$
| $\{\; \backslash omega\; \backslash over\; s^2\; +\; \backslash omega^2\; \}$
| $\backslash operatorname\{Re\}(s)\; >\; 0$
|
     

! scope="row" | [[cosine]]
| $\backslash cos(\backslash omega\; t)\; \backslash cdot\; u(t)\; \backslash$
| $\{\; s\; \backslash over\; s^2\; +\; \backslash omega^2\; \}$
| $\backslash operatorname\{Re\}(s)\; >\; 0$
|
     

! scope="row" | [[hyperbolic sine]]
| $\backslash sinh(\backslash alpha\; t)\; \backslash cdot\; u(t)\; \backslash$
| $\{\; \backslash alpha\; \backslash over\; s^2\; -\; \backslash alpha^2\; \}$
| $\backslash operatorname\{Re\}(s)\; >\; \backslash left|\; \backslash alpha\; \backslash right|$
|
     

! scope="row" | hyperbolic cosine
| $\backslash cosh(\backslash alpha\; t)\; \backslash cdot\; u(t)\; \backslash$
| $\{\; s\; \backslash over\; s^2\; -\; \backslash alpha^2\; \}$
|  $\backslash operatorname\{Re\}(s)\; >\; \backslash left|\; \backslash alpha\; \backslash right|$
|
     

! scope="row" | exponentially decaying 
 sine wave
| $e^\{-\backslash alpha\; t\}\; \backslash sin(\backslash omega\; t)\; \backslash cdot\; u(t)\; \backslash$
| $\{\; \backslash omega\; \backslash over\; (s+\backslash alpha)^2\; +\; \backslash omega^2\; \}$
| $\backslash operatorname\{Re\}(s)\; >\; -\; \backslash alpha$
|
     

! scope="row" | exponentially decaying 
 cosine wave
| $e^\{-\backslash alpha\; t\}\; \backslash cos(\backslash omega\; t)\; \backslash cdot\; u(t)\; \backslash$
| $\{\; s+\backslash alpha\; \backslash over\; (s+\backslash alpha)^2\; +\; \backslash omega^2\; \}$
| $\backslash operatorname\{Re\}(s)\; >\; -\; \backslash alpha$
|
     

! scope="row" | natural logarithm
| $\backslash ln(t)\; \backslash cdot\; u(t)$
| $-\{1\; \backslash over\; s\}\; \backslash left[\; \backslash ln(s)+\backslash gamma\; \backslash right]$
|  $\backslash operatorname\{Re\}(s)\; >\; 0$
|
     

! scope="row" | [[Bessel function]] 
 of the first kind, 
 of order
| $J\_n(\backslash omega\; t)\; \backslash cdot\; u(t)$
| $\backslash frac\{\; \backslash left(\backslash sqrt\{s^2+\; \backslash omega^2\}-s\backslash right)^\{\backslash !n\}\}\{\backslash omega^n\; \backslash sqrt\{s^2\; +\; \backslash omega^2\}\}$
| $\backslash operatorname\{Re\}(s)\; >\; 0$ 
 ()
|
     

! scope="row" | [[Error function]]
| $\backslash operatorname\{erf\}(t)\; \backslash cdot\; u(t)$
| $\backslash frac\{1\}\{s\}\; e^\{s^2\; /\; 4\}\; \backslash !\backslash left(1\; -\; \backslash operatorname\{erf\}\; \backslash frac\{s\}\{2\}\; \backslash right)$
| $\backslash operatorname\{Re\}(s)\; >\; 0$
|
     

| colspan=5 style="text-align: left;" |'''Explanatory notes:'''
     

• represents the Heaviside step function.
• represents the Dirac delta function.
• represents the gamma function.
• is the Euler–Mascheroni constant.

• , a real number, typically represents time, although it can represent any independent dimension.
• is the complex number frequency domain parameter, and is its real part.
• are real numbers.
• is an integer.

|}

Here is a summary of equivalents:

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

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

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.

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

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 , proceeding formally one has

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) \},
     

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

The definition of the complex impedance (in ) is the ratio of the complex voltage divided by the complex current while holding the initial state 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.

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 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 and 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 , we multiply both sides of the equation by to get $$\backslash frac\{1\}\{s\; +\; \backslash beta\}\; =\; P\; +\; \{\; R\; (s\; +\; \backslash alpha)\; \backslash over\; s\; +\; \backslash beta\; \}.$$

Then by letting , the contribution from 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 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 and into the expanded expression for 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 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

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).
     

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).
     

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

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

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

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

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

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

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

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

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.

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

• , Chapters 3–5

• .
• Mathews, Jon; Walker, Robert L. (1970), Mathematical methods of physics (2nd ed.), New York: W. A. Benjamin,
• - See Chapter VI. The Laplace transform.
• J.A.C.Weidman and Bengt Fornberg: "Fully numerical Laplace transform methods" .

• Online Computation of the transform or inverse transform, wims.unice.fr
• Tables of Integral Transforms at EqWorld: The World of Mathematical Equations.
• Good explanations of the initial and final value theorems
• Laplace Transforms at MathPages
• Computational Knowledge Engine allows to easily calculate Laplace Transforms and its inverse Transform.
• Laplace Calculator to calculate Laplace Transforms online easily.
• Code to visualize Laplace Transforms and many example videos.