</math>
where s is a complex number. It is related to many other transforms, most notably the Fourier transform and the Mellin transform. Formally, the Laplace transform is converted into a Fourier transform by the substitution <math>s = i\omega</math> where <math>\omega</math> is real. However, unlike the Fourier transform, which gives the decomposition of a function into its components in each frequency, the Laplace transform of a function with suitable decay is an analytic function, and so has a convergent power series, the coefficients of which give the decomposition of a function into its moments. Also unlike the Fourier transform, when regarded in this way as an analytic function, the techniques of complex analysis, and especially contour integrals, can be used for calculations.
These types of integrals seem first to have attracted Laplace's attention in 1782, where he was following in the spirit of Euler in using the integrals themselves as solutions of equations.<ref>Template:Harvnb</ref> However, in 1785, Laplace took the critical step forward when, rather than simply looking for a solution in the form of an integral, he started to apply the transforms in the sense that was later to become popular. He used an integral of the form
<math display=block> \int x^s \varphi (x)\, dx,</math>
akin to a Mellin transform, to transform the whole of a difference equation, in order to look for solutions of the transformed equation. He then went on to apply the Laplace transform in the same way and started to derive some of its properties, beginning to appreciate its potential power.<ref>Template:Harvnb</ref>
Laplace also recognised that Joseph Fourier's method of Fourier series for solving the diffusion equation could only apply to a limited region of space, because those solutions were periodic. In 1809, Laplace applied his transform to find solutions that diffused indefinitely in space.<ref>Template:Harvnb</ref> In 1821, Cauchy developed an operational calculus for the Laplace transform that could be used to study linear differential equations in much the same way the transform is now used in basic engineering. This method was popularized, and perhaps rediscovered, by Oliver Heaviside around the turn of the century.<ref>Template:Citation</ref>
In 1929, Vannevar Bush and Norbert Wiener published Operational Circuit Analysis as a text for engineering analysis of electrical circuits, applying both Fourier transforms and operational calculus, and in which they included one of the first predecessors of the modern table of Laplace transforms.
In 1934, Raymond Paley and Norbert Wiener published the important work Fourier transforms in the complex domain, about what is now called the Laplace transform (see below). Also during the 30s, the Laplace transform was instrumental in G H Hardy and John Edensor Littlewood's study of tauberian 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,<ref>An influential book was: Template:Citation</ref> replacing the earlier Heaviside operational calculus. The advantages of the Laplace transform had been emphasized by Gustav Doetsch,<ref>Template:Citation translation 1943</ref> to whom the name Laplace transform is apparently due.
File:Complex frequency s-domain negative.jpg<math>\Re(e^{-st})</math> for various complex frequencies in the s-domain <math>(s = \sigma + i \omega),</math> which can be expressed as <math>e^{-\sigma t} \cos(\omega t).</math> The <math>\sigma = 0</math> axis contains pure cosines. Positive <math>\sigma</math> contains damped cosines. Negative <math>\sigma</math> contains exponentially growing cosines.
An alternate notation for the Laplace transform is Template:Anchor<math>\mathcal{L}\{f\}</math> instead of Template:Math,<ref name=":1" /> often written as <math>
F(s) = \mathcal{L}\{f(t)\}</math> 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 Template:Mvar must be locally integrable on Template:Closed-open. For locally integrable functions that decay at infinity or are of exponential type (<math>|f(t)| \le Ae^{B|t|}</math>), the integral can be understood to be a (proper) Lebesgue integral. However, for many applications it is necessary to regard it as a conditionally convergentimproper integral at Template:Math. Still more generally, the integral can be understood in a weak sense, and this is dealt with below.
An important special case is where Template:Mvar 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 Template:Mvar. In that case, to avoid potential confusion, one often writes
<math display=block>
This limit emphasizes that any point mass located at Template:Math 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.
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 Template:Math is defined as follows:
Template:Equation box 1
An alternate notation for the bilateral Laplace transform is <math>\mathcal{B}\{f\}</math>, instead of Template:Mvar.
Template:Main article
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 Template:Math, or more generally tempered distributions on Template:Open-open. 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):
Template:Equation box 1
where Template:Mvar is a real number so that the contour path of integration is in the region of convergence of Template:Math. 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.
Of particular use is the ability to recover the cumulative distribution function of a continuous random variable Template:Mvar by means of the Laplace transform as follows:<ref>The cumulative distribution function is the integral of the probability density function.</ref>
<math display=block>
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).<ref>Template:Cite book</ref>
Template:See also
If Template:Math is a locally integrable function (or more generally a Borel measure locally of bounded variation), then the Laplace transform Template:Math of Template:Math converges provided that the limit
<math display=block>\lim_{R\to\infty}\int_0^R f(t)e^{-st}\,dt</math>
exists.
The Laplace transform converges absolutely if the integral
<math display=block>\int_0^\infty \left|f(t)e^{-st}\right|\,dt</math>
exists as a proper Lebesgue integral. The Laplace transform is usually understood as conditionally convergent, meaning that it converges in the former but not in the latter sense.
Similarly, the set of values for which Template:Math 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 Template:Math, then it automatically converges for all Template:Math with Template:Math. Therefore, the region of convergence is a half-plane of the form Template:Math, possibly including some points of the boundary line Template:Math.
In the region of convergence Template:Math, the Laplace transform of Template:Math can be expressed by integrating by parts as the integral
<math display=block>F(s) = (s-s_0)\int_0^\infty e^{-(s-s_0)t}\beta(t)\,dt, \quad \beta(u) = \int_0^u e^{-s_0t}f(t)\,dt.</math>
That is, Template:Math 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 Template:Math, 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 Template:Math. 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 Template:Math in the Laplace domain. Thus, the Laplace variable Template:Math is also known as an operator variable in the Laplace domain: either the derivative operator or (for Template:Math the integration operator.
Given the functions Template:Math and Template:Math, and their respective Laplace transforms Template:Math and Template:Math,
<math display=block>\begin{align}
f(t) &= \mathcal{L}^{-1}\{F(s)\},\\
g(t) &= \mathcal{L}^{-1}\{G(s)\},
\end{align}</math>
the following table is a list of properties of unilateral Laplace transform:<ref>Template:Harvnb</ref>
Template:Math is assumed to be a differentiable function, and its derivative is assumed to be of exponential type. This can then be obtained by integration by parts
Template:Math is assumed twice differentiable and the second derivative to be of exponential type. Follows by applying the Differentiation property to Template:Math.
<math>f(\infty)=\lim_{s\to 0}{sF(s)}</math>, if all poles of <math>sF(s)</math> are in the left half-plane.
The final value theorem is useful because it gives the long-term behaviour without having to perform partial fraction decompositions (or other difficult algebra). If Template:Math has a pole in the right-hand plane or poles on the imaginary axis (e.g., if <math>f(t) = e^t</math> or <math>f(t) = \sin(t)</math>), then the behaviour of this formula is undefined.
Changing the base of the power from Template:Math to Template:Math gives
<math display=block>\int_{0}^{\infty} f(t) \left(e^{\ln{x}}\right)^t\, dt</math>
For this to converge for, say, all bounded functions Template:Math, it is necessary to require that Template:Math. Making the substitution Template:Math gives just the Laplace transform:
<math display=block>\int_{0}^{\infty} f(t) e^{-st}\, dt</math>
In other words, the Laplace transform is a continuous analog of a power series, in which the discrete parameter Template:Math is replaced by the continuous parameter Template:Math, and Template:Math is replaced by Template:Math.
Template:Main article
The quantities
<math display=block>\mu_n = \int_0^\infty t^nf(t)\, dt</math>
are the moments of the function Template:Math. If the first Template:Math moments of Template:Math converge absolutely, then by repeated differentiation under the integral,
<math display=block>(-1)^n(\mathcal L f)^{(n)}(0) = \mu_n .</math>
This is of special significance in probability theory, where the moments of a random variable Template:Math are given by the expectation values <math>\mu_n=\operatorname{E}[X^n]</math>. Then, the relation holds
<math display=block>\mu_n = (-1)^n\frac{d^n}{ds^n}\operatorname{E}\left[e^{-sX}\right](0).</math>
It is often convenient to use the differentiation property of the Laplace transform to find the transform of a function's derivative. This can be derived from the basic expression for a Laplace transform as follows:
<math display=block>\begin{align}
\end{align}</math>
yielding
<math display=block>\mathcal{L} \{ f'(t) \} = s\cdot\mathcal{L} \{ f(t) \}-f(0^-), </math>
and in the bilateral case,
<math display=block> \mathcal{L} \{ f'(t) \} = s \int_{-\infty}^\infty e^{-st} f(t)\,dt = s \cdot \mathcal{L} \{ f(t) \}. </math>
The general result
<math display=block>\mathcal{L} \left\{ f^{(n)}(t) \right\} = s^n \cdot \mathcal{L} \{ f(t) \} - s^{n - 1} f(0^-) - \cdots - f^{(n - 1)}(0^-),</math>
where <math>f^{(n)}</math> denotes the Template:Mathth derivative of Template:Math, can then be established with an inductive argument.
A useful property of the Laplace transform is the following:
<math display=block>\int_0^\infty f(x)g(x)\,dx = \int_0^\infty(\mathcal{L} f)(s)\cdot(\mathcal{L}^{-1}g)(s)\,ds </math>
under suitable assumptions on the behaviour of <math>f,g</math> in a right neighbourhood of <math>0</math> and on the decay rate of <math>f,g</math> in a left neighbourhood of <math>\infty</math>. The above formula is a variation of integration by parts, with the operators
<math>\frac{d}{dx}</math> and <math>\int \,dx</math> being replaced by <math>\mathcal{L}</math> and <math>\mathcal{L}^{-1}</math>. Let us prove the equivalent formulation:
<math display=block>\int_0^\infty(\mathcal{L} f)(x)g(x)\,dx = \int_0^\infty f(s)(\mathcal{L}g)(s)\,ds. </math>
By plugging in <math>(\mathcal{L}f)(x)=\int_0^\infty f(s)e^{-sx}\,ds</math> the left-hand side turns into:
<math display=block>\int_0^\infty\int_0^\infty f(s)g(x) e^{-sx}\,ds\,dx, </math>
but assuming Fubini's theorem holds, by reversing the order of integration we get the wanted right-hand side.
This method can be used to compute integrals that would otherwise be difficult to compute using elementary methods of real calculus. For example,
<math display=block>\int_0^\infty\frac{\sin x}{x}dx = \int_0^\infty \mathcal{L}(1)(x)\sin x dx = \int_0^\infty 1 \cdot \mathcal{L}(\sin)(x)dx = \int_0^\infty \frac{dx}{x^2 + 1} = \frac{\pi}{2}. </math>
then the Laplace–Stieltjes transform of Template:Mvar and the Laplace transform of Template:Mvar coincide. In general, the Laplace–Stieltjes transform is the Laplace transform of the Stieltjes measure associated to Template:Mvar. So in practice, the only distinction between the two transforms is that the Laplace transform is thought of as operating on the density function of the measure, whereas the Laplace–Stieltjes transform is thought of as operating on its cumulative distribution function.<ref>Template:Harvnb</ref>
Let <math>f</math> be a complex-valued Lebesgue integrable function supported on <math>[0,\infty)</math>, and let <math>F(s) = \mathcal Lf(s)</math> be its Laplace transform. Then, within the region of convergence, we have
which is the Fourier transform of the function <math>f(t)e^{-\sigma t}</math>.<ref>Template:Cite book, p 224.</ref>
Indeed, the Fourier transform is a special case (under certain conditions) of the bilateral Laplace transform. The main difference is that the Fourier transform of a function is a complex function of a real variable (frequency), the Laplace transform of a function is a complex function of a complex variable. The Laplace transform is usually restricted to transformation of functions of Template:Math with Template:Math. A consequence of this restriction is that the Laplace transform of a function is a holomorphic function of the variable Template:Math. 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 Template:Math<ref>Template:Citation</ref><ref>Template:Harvnb</ref> when the condition explained below is fulfilled,
This convention of the Fourier transform (<math>\hat f_3(\omega)</math> in Template:Section link) requires a factor of Template:Math 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.
However, a relation of the form
<math display="block">\lim_{\sigma\to 0^+} F(\sigma+i\omega) = \hat{f}(\omega)</math>
holds under much weaker conditions. For instance, this holds for the above example provided that the limit is understood as a weak limit of measures (see vague topology). General conditions relating the limit of the Laplace transform of a function on the boundary to the Fourier transform take the form of Paley–Wiener theorems.
Template:Main
The Mellin transform and its inverse are related to the two-sided Laplace transform by a simple change of variables.
If in the Mellin transform
<math display=block>G(s) = \mathcal{M}\{g(\theta)\} = \int_0^\infty \theta^s g(\theta) \, \frac{d\theta} \theta </math>
we set Template:Math we get a two-sided Laplace transform.
The unilateral or one-sided Z-transform is simply the Laplace transform of an ideally sampled signal with the substitution of
<math display=block> z \stackrel{\mathrm{def} }{ {}={} } e^{sT} ,</math>
where Template:Math is the sampling interval (in units of time e.g., seconds) and Template:Math is the sampling rate (in samples per second or hertz).
Let
<math display=block> \Delta_T(t) \ \stackrel{\mathrm{def}}{=}\ \sum_{n=0}^{\infty} \delta(t - n T) </math>
be a sampling impulse train (also called a Dirac comb) and
<math display=block>\begin{align}
x_q(t) &\stackrel{\mathrm{def} }{ {}={} } x(t) \Delta_T(t) = x(t) \sum_{n=0}^{\infty} \delta(t - n T) \\
&= \sum_{n=0}^{\infty} x(n T) \delta(t - n T) = \sum_{n=0}^{\infty} x[n] \delta(t - n T)
\end{align}</math>
be the sampled representation of the continuous-time Template:Math
<math display=block> x[n] \stackrel{\mathrm{def} }{ {}={} } x(nT) ~.</math>
The Laplace transform of the sampled signal Template:Math is
<math display=block>\begin{align}
This is the precise definition of the unilateral Z-transform of the discrete function Template:Math
<math display=block> X(z) = \sum_{n=0}^{\infty} x[n] z^{-n} </math>
with the substitution of Template:Math.
Comparing the last two equations, we find the relationship between the unilateral Z-transform and the Laplace transform of the sampled signal,
<math display=block>X_q(s) = X(z) \Big|_{z=e^{sT}}.</math>
The similarity between the Z- and Laplace transforms is expanded upon in the theory of time scale calculus.
The integral form of the Borel transform
<math display=block>F(s) = \int_0^\infty f(z)e^{-sz}\, dz</math>
is a special case of the Laplace transform for Template:Math an entire function of exponential type, meaning that
<math display=block>|f(z)|\le Ae^{B|z|}</math>
for some constants Template:Math and Template:Math. 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.
The following table provides Laplace transforms for many common functions of a single variable.<ref>Template:Citation</ref><ref>Template:Citation</ref> For definitions and explanations, see the Explanatory Notes at the end of the table.
Because the Laplace transform is a linear operator,
The Laplace transform of a sum is the sum of Laplace transforms of each term.<math display=block>\mathcal{L}\{f(t) + g(t)\} = \mathcal{L}\{f(t)\} + \mathcal{L}\{ g(t)\} </math>
The Laplace transform of a multiple of a function is that multiple times the Laplace transformation of that function.<math display=block>\mathcal{L}\{a f(t)\} = a \mathcal{L}\{ f(t)\}</math>
Using this linearity, and various 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, Template:Math.
The Laplace transform is often used in circuit analysis, and simple conversions to the Template:Math-domain of circuit elements can be made. Circuit elements can be transformed into impedances, very similar to phasor impedances.
Note that the resistor is exactly the same in the time domain and the Template:Math-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 Template:Math-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.<ref>Template:Harvnb</ref>
The Laplace transform can also be used to solve differential equations and is used extensively in mechanical engineering and electrical engineering. The Laplace transform reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of algebra. The original differential equation can then be solved by applying the inverse Laplace transform. English electrical engineer Oliver Heaviside first proposed a similar scheme, although without using the Laplace transform; and the resulting operational calculus is credited as the Heaviside calculus.
\mathcal{L} \left\{\frac{f(t)} t \right\} = \int_s^\infty F(p)\, dp.</math>
In the limit <math>s \rightarrow 0</math>, one gets
<math display=block>\int_0^\infty \frac{f(t)} t \, dt = \int_0^\infty F(p)\, dp,</math>
provided that the interchange of limits can be justified. This is often possible as a consequence of the final value theorem. Even when the interchange cannot be justified the calculation can be suggestive. For example, with Template:Math, proceeding formally one has
<math display=block>
\begin{align}
\int_0^\infty \frac{ \cos(at) - \cos(bt) }{t} \, dt
&=\int_0^\infty \left(\frac p {p^2 + a^2} - \frac{p}{p^2 + b^2}\right)\, dp \\[6pt]
&=\left[ \frac{1}{2} \ln\frac{p^2 + a^2}{p^2 + b^2} \right]_0^\infty = \frac{1}{2} \ln \frac{b^2}{a^2} = \ln \left| \frac {b}{a} \right|.
\end{align}
</math>
The validity of this identity can be proved by other means. It is an example of a Frullani integral.
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
<math display=block>i = C { dv \over dt} ,</math>
where Template:Math is the capacitance of the capacitor, Template:Math is the electric current through the capacitor as a function of time, and Template:Math is the voltage across the terminals of the capacitor, also as a function of time.
Taking the Laplace transform of this equation, we obtain
<math display=block>I(s) = C(s V(s) - V_0),</math>
where
<math display=block>\begin{align}
\end{align}</math>
and
<math display=block>V_0 = v(0). </math>
Solving for Template:Math we have
<math display=block>V(s) = { I(s) \over sC } + { V_0 \over s }.</math>
The definition of the complex impedance Template:Math (in ohms) is the ratio of the complex voltage Template:Math divided by the complex current Template:Math while holding the initial state Template:Math at zero:
<math display=block>Z(s) = \left. { V(s) \over I(s) } \right|_{V_0 = 0}.</math>
Using this definition and the previous equation, we find:
<math display=block>Z(s) = \frac{1}{sC}, </math>
which is the correct expression for the complex impedance of a capacitor. In addition, the Laplace transform has large applications in control theory.
Consider a linear time-invariant system with transfer function
<math display=block>H(s) = \frac{1}{(s + \alpha)(s + \beta)}.</math>
The impulse response is simply the inverse Laplace transform of this transfer function:
<math display=block>h(t) = \mathcal{L}^{-1}\{H(s)\}.</math>
Partial fraction expansion
To evaluate this inverse transform, we begin by expanding Template:Math using the method of partial fraction expansion,
<math display=block>\frac{1}{(s + \alpha)(s + \beta)} = { P \over s + \alpha } + { R \over s+\beta }.</math>
The unknown constants Template:Math and Template:Math 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 Template:Math, we multiply both sides of the equation by Template:Math to get
<math display=block>\frac{1}{s + \beta} = P + { R (s + \alpha) \over s + \beta }.</math>
Then by letting Template:Math, the contribution from Template:Math vanishes and all that is left is
<math display=block>P = \left.{1 \over s+\beta}\right|_{s=-\alpha} = {1 \over \beta - \alpha}.</math>
Similarly, the residue Template:Math is given by
<math display=block>R = \left.{1 \over s + \alpha}\right|_{s=-\beta} = {1 \over \alpha - \beta}.</math>
Note that
<math display=block>R = {-1 \over \beta - \alpha} = - P</math>
and so the substitution of Template:Math and Template:Math into the expanded expression for Template:Math gives
<math display=block>H(s) = \left(\frac{1}{\beta - \alpha} \right) \cdot \left( { 1 \over s + \alpha } - { 1 \over s + \beta } \right).</math>
Finally, using the linearity property and the known transform for exponential decay (see Item #3 in the Table of Laplace Transforms, above), we can take the inverse Laplace transform of Template:Math to obtain
<math display=block>h(t) = \mathcal{L}^{-1}\{H(s)\} = \frac{1}{\beta - \alpha}\left(e^{-\alpha t} - e^{-\beta t}\right),</math>
which is the impulse response of the system.
Convolution
The same result can be achieved using the convolution property as if the system is a series of filters with transfer functions Template:Math and Template:Math. That is, the inverse of
<math display=block>H(s) = \frac{1}{(s + \alpha)(s + \beta)} = \frac{1}{s+\alpha} \cdot \frac{1}{s + \beta}</math>
is
<math display=block> \mathcal{L}^{-1}\! \left\{ \frac{1}{s + \alpha} \right\} * \mathcal{L}^{-1}\! \left\{\frac{1}{s + \beta} \right\} = e^{-\alpha t} * e^{-\beta t} = \int_0^t e^{-\alpha x}e^{-\beta (t - x)}\, dx = \frac{e^{-\alpha t}-e^{-\beta t}}{\beta - \alpha}.</math>
Starting with the Laplace transform,
<math display=block>X(s) = \frac{s\sin(\varphi) + \omega \cos(\varphi)}{s^2 + \omega^2}</math>
we find the inverse by first rearranging terms in the fraction:
<math display=block>\begin{align}
This is just the sine of the sum of the arguments, yielding:
<math display=block>x(t) = \sin (\omega t + \varphi).</math>
We can apply similar logic to find that
<math display=block>\mathcal{L}^{-1} \left\{ \frac{s\cos\varphi - \omega \sin\varphi}{s^2 + \omega^2} \right\} = \cos{(\omega t + \varphi)}.</math>
In statistical mechanics, the Laplace transform of the density of states <math>g(E)</math> defines the partition function.<ref>Template:Cite book</ref> That is, the canonical partition function <math>Z(\beta)</math> is given by
<math display=block> Z(\beta) = \int_0^\infty e^{-\beta E}g(E)\,dE</math>
and the inverse is given by
<math display=block> g(E) = \frac{1}{2\pi i} \int_{\beta_0-i\infty}^{\beta_0+i\infty} e^{\beta E}Z(\beta) \, d\beta</math>
Spatial (not time) structure from astronomical spectrum
The wide and general applicability of the Laplace transform and its inverse is illustrated by an application in astronomy which provides some information on the spatial distribution of matter of an astronomical source of radiofrequencythermal radiation too distant to resolve as more than a point, given its flux densityspectrum, 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.<ref>Template:Citation, and Template:Citation</ref> When independent information on the structure of an object is available, the inverse Laplace transform method has been found to be in good agreement.
Consider a random walk, with steps <math>\{+1,-1\}</math> occurring with probabilities <math>p,q=1-p</math>.<ref>Template:Cite book</ref> Suppose also that the time step is an Poisson process, with parameter <math>\lambda</math>. Then the probability of the walk being at the lattice point <math>n</math> at time <math>t</math> is
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
It is intuitively clear that, for small <math>s>0</math>, the exponentially decaying integrand will become more sensitive to the concentration of the measure <math>\mu</math> on larger subsets of the domain. To make this more precise, introduce the distribution function:
<math>M(t) = \mu([0,t)).</math>
Formally, we expect a limit of the following kind:
Tauberian theorems are theorems relating the asymptotics of the Laplace transform, as <math>s\to 0^+</math>, to those of the distribution of <math>\mu</math> as <math>t\to\infty</math>. They are thus of importance in asymptotic formulae of probability and statistics, where often the spectral side has asymptotics that are simpler to infer.<ref>Template:Cite book</ref>
