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| The '''Laplace transform''' is a widely used [[integral transform]] in [[mathematics]] with many applications in [[physics]] and [[engineering]]. It is a [[linear operator]] of a function ''f''(''t'') with a real argument ''t'' (''t'' ≥ 0) that transforms ''f''(''t'') to a function ''F''(''s'') with complex argument ''s'', given by the [[integral]]
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| :<math>F(s) = \int_0^\infty f(t) e^{-st}\,dt.</math>
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| This transformation is [[bijection|bijective]] for the majority of practical uses; the most-common pairs of ''f''(''t'') and ''F''(''s'') are often given in tables for easy reference. The Laplace transform has the useful property that many relationships and operations over the original ''f''(''t'') correspond to simpler relationships and operations over its image ''F''(''s'').<ref>{{harvnb|Korn|Korn|1967|loc=§8.1}}</ref>
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| It is named after [[Pierre-Simon Laplace]] ({{IPAc-en|l|ə|ˈ|p|l|ɑː|s}}), who introduced the transform in his work on [[probability theory]].
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| The Laplace transform is related to the [[Fourier transform]], but whereas the Fourier transform expresses a function or signal as a series of modes of [[vibration]] (frequencies), the Laplace transform resolves a function into its [[moment (mathematics)|moments]]. Like the Fourier transform, the Laplace transform is used for solving differential and integral equations. In physics and engineering it is used for analysis of [[LTI system|linear time-invariant]] [[dynamic system|systems]] such as [[electrical circuit]]s, [[harmonic oscillator]]s, [[optical device]]s, and mechanical systems. In such analyses, the Laplace transform is often interpreted as a transformation from the ''[[time-domain]]'', in which inputs and outputs are functions of time, to the ''[[frequency-domain]]'', where the same inputs and outputs are functions of [[complex number|complex]] [[angular frequency]], in [[radian]]s per unit time. 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.
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| == History ==
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| The Laplace transform is named after mathematician and astronomer [[Pierre-Simon Laplace]], who used a similar transform (now called [[z transform]]) in his work on [[probability theory]]. The current widespread use of the transform came about soon after World War II although it had been used in the 19th century by [[Niels Henrik Abel|Abel]], [[Mathias Lerch|Lerch]], [[Oliver Heaviside|Heaviside]], and [[Thomas John I'Anson Bromwich|Bromwich]]. The older history of similar transforms is as follows. From 1744, [[Leonhard Euler]] investigated integrals of the form
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| : <math> z = \int X(x) e^{ax}\, dx \quad\text{ and }\quad z = \int X(x) x^A \, dx</math>
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| as solutions of differential equations but did not pursue the matter very far.<ref>{{harvnb|Euler|1744}}, {{harvnb|Euler|1753}}, {{harvnb|Euler|1769}}</ref> [[Joseph Louis Lagrange]] was an admirer of Euler and, in his work on integrating [[probability density function]]s, investigated expressions of the form
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| : <math> \int X(x) e^{- a x } a^x\, dx,</math>
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| which some modern historians have interpreted within modern Laplace transform theory.<ref>{{harvnb|Lagrange|1773}}</ref><ref>{{harvnb|Grattan-Guinness| 1997|p=260}}</ref>{{Clarify|date=May 2010}}
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| These types of integrals seem first to have attracted Laplace's attention in 1782 where he was following in the spirit of Euler in using the integrals themselves as solutions of equations.<ref>{{harvnb|Grattan-Guinness|1997|p=261}}</ref> However, in 1785, Laplace took the critical step forward when, rather than just 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:
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| : <math> \int x^s \phi (x)\, dx,</math>
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| akin to a [[Mellin transform]], to transform the whole of a [[difference equation]], in order to look for solutions of the transformed equation. He then went on to apply the Laplace transform in the same way and started to derive some of its properties, beginning to appreciate its potential power.<ref>{{harvnb|Grattan-Guinness|1997|pp=261–262}}</ref>
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| 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 as the solutions were periodic. In 1809, Laplace applied his transform to find solutions that diffused indefinitely in space.<ref>{{harvnb|Grattan-Guinness|1997|pp=262–266}}</ref>
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| == Formal definition ==
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| The Laplace transform of a [[function (mathematics)|function]] ''f''(''t''), defined for all [[real number]]s ''t'' ≥ 0, is the function ''F''(''s''), defined by:
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| : <math>F(s) =\int_0^{\infty} e^{-st} f(t) \,dt</math>
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| The parameter ''s'' is a [[complex number]]:
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| : <math>s = \sigma + i \omega, \, </math> with real numbers σ and ω.
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| Other notations for the Laplace transform include <math>\displaystyle\mathcal{L}f</math> or alternatively <math>\displaystyle\mathcal{L}_t\left\{f(t)\right\}</math> instead of ''F''.
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| The meaning of the integral depends on types of functions of interest. A necessary condition for existence of the integral is that ''f'' must be [[locally integrable]] on {{closed-open|0,∞}}. For locally integrable functions that decay at infinity or are of [[exponential type]], the integral can be understood as 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 [[distribution (mathematics)|weak sense]], and this is dealt with below.
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| One can define the Laplace transform of a finite [[Borel measure]] μ by the [[Lebesgue integral]]<ref>{{harvnb|Feller|1971|loc=§XIII.1}}</ref>
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| : <math>(\mathcal{L}\mu)(s) = \int_{[0,\infty)} e^{-st}d\mu(t).</math>
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| An important special case is where μ is a [[probability measure]] or, even more specifically, the Dirac delta function. In [[operational calculus]], the Laplace transform of a measure is often treated as though the measure came from a [[distribution function]] ''f''. In that case, to avoid potential confusion, one often writes
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| : <math>(\mathcal{L}f)(s) = \int_{0^-}^\infty e^{-st}f(t)\,dt</math>
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| where the lower limit of 0<sup>−</sup> is shorthand notation for
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| : <math>\lim_{\varepsilon\downarrow 0}\int_{-\varepsilon}^\infty.</math>
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| This limit emphasizes that any point mass located at 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]].
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| === Probability theory ===
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| In [[probability theory|pure]] and [[applied probability]], the Laplace transform is defined as an [[expected value]]. If ''X'' is a [[random variable]] with [[probability density function]] ''f'', then the Laplace transform of ''f'' is given by the expectation
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| : <math>(\mathcal{L}f)(s) = E\left[e^{-sX} \right] \, </math>
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| By [[abuse of notation|abuse of language]], this is referred to as the Laplace transform of the random variable ''X'' itself. Replacing ''s'' by −''t'' gives the [[moment generating function]] of ''X''. The Laplace transform has applications throughout probability theory, including [[first passage time]]s of [[stochastic processes]] such as [[Markov chain]]s, and [[renewal theory]].
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| Of particular use is the ability to recover the [[cumulative distribution function]] of a random variable ''X'' by means of the Laplace transform as follows<ref>The cumulative distribution function is the integral of the probability density function.</ref>
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| : <math> F_X(x) = \mathcal{L}^{-1}_s \left\lbrace \frac{E\left[e^{-sX}\right]}{s}\right\rbrace (x) = \mathcal{L}^{-1}_s \left\lbrace \frac{\left(\mathcal{L} f\right)(s)}{s} \right\rbrace (x) \, </math>
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| === Bilateral Laplace transform ===
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| {{Main|Two-sided Laplace transform}}
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| When one says "the Laplace transform" without qualification, the unilateral or one-sided transform is normally 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]].
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| The bilateral Laplace transform is defined as follows:
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| : <math>F(s) = \mathcal{L}\left\{f(t)\right\}(s) =\int_{-\infty}^{\infty} e^{-st} f(t)\,dt.</math>
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| === Inverse Laplace transform ===
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| {{details|Inverse Laplace transform}}
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| The [[inverse Laplace transform]] is given by the following [[complex number|complex]] integral, which is known by various names (the '''Bromwich integral''', the '''Fourier-Mellin integral''', and '''Mellin's inverse formula'''):
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| : <math>f(t) = \mathcal{L}^{-1} \{F\} = \mathcal{L}^{-1}_s \{F(s)\} := \frac{1}{2 \pi i} \lim_{T\to\infty}\int_{ \gamma - i T}^{ \gamma + i T} e^{st} F(s)\,ds,</math>
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| where γ is a real number so that the contour path of integration is in the ''[[region of convergence]]'' of ''F''(''s''). An alternative formula for the inverse Laplace transform is given by [[Post's inversion formula]].
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| == Region of convergence ==
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| If ''f'' is a [[locally integrable]] function (or more generally a Borel measure locally of bounded variation), then the Laplace transform ''F''(''s'') of ''f'' converges provided that the limit
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| : <math>\lim_{R\to\infty}\int_0^R f(t)e^{-st}\,dt</math>
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| exists. The Laplace transform converges absolutely if the integral
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| : <math>\int_0^\infty |f(t)e^{-st}|\,dt</math>
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| exists (as a proper Lebesgue integral). The Laplace transform is usually understood as conditionally convergent, meaning that it converges in the former instead of the latter sense.
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| The set of values for which ''F''(''s'') converges absolutely is either of the form Re(''s'') > ''a'' or else Re(''s'') ≥ ''a'', where ''a'' is an [[extended real number|extended real constant]], −∞ ≤ ''a'' ≤ ∞. (This follows from the [[dominated convergence theorem]].) The constant ''a'' is known as the abscissa of absolute convergence, and depends on the growth behavior of ''f''(''t'').<ref>{{harvnb|Widder|1941|loc=Chapter II, §1}}</ref> Analogously, the two-sided transform converges absolutely in a strip of the form ''a'' < Re(''s'') < ''b'', and possibly including the lines Re(''s'') = ''a'' or Re(''s'') = ''b''.<ref>{{harvnb|Widder|1941|loc=Chapter VI, §2}}</ref> The subset of values of ''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 function|analytic]] in the region of absolute convergence.
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| Similarly, the set of values for which ''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 ''s'' = ''s''<sub>0</sub>, then it automatically converges for all ''s'' with Re(''s'') > Re(''s''<sub>0</sub>). Therefore the region of convergence is a half-plane of the form Re(''s'') > ''a'', possibly including some points of the boundary line Re(''s'') = ''a''. In the region of convergence Re(''s'') > Re(''s''<sub>0</sub>), the Laplace transform of ''f'' can be expressed by [[integration by parts|integrating by parts]] as the integral
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| : <math>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>
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| That is, in the region of convergence ''F''(''s'') can effectively be expressed as the absolutely convergent Laplace transform of some other function. In particular, it is analytic.
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| A variety of theorems, in the form of [[Paley–Wiener theorem]]s, exist concerning the relationship between the decay properties of ''f'' and the properties of the Laplace transform within the region of convergence.
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| In engineering applications, a function corresponding to a [[LTI system|linear time-invariant (LTI) system]] is ''stable'' if every bounded input produces a bounded output. This is equivalent to the absolute convergence of the Laplace transform of the impulse response function in the region Re(''s'') ≥ 0. As a result, LTI systems are stable provided the poles of the Laplace transform of the impulse response function have negative real part.
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| == Properties and theorems ==
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| The Laplace transform has a number of properties that make it useful for analyzing linear [[dynamical system]]s. The most significant advantage is that [[derivative|differentiation]] and [[integral|integration]] become multiplication and division, respectively, by ''s'' (similarly to [[logarithm]]s changing multiplication of numbers to addition of their logarithms). Because of this property, the Laplace variable ''s'' is also known as ''operator variable'' in the L domain: either ''derivative operator'' or (for ''s''<sup>−1</sup>) ''integration operator''. The transform turns [[integral equation]]s and [[differential equation]]s to [[polynomial equation]]s, which are much easier to solve. Once solved, use of the inverse Laplace transform reverts to the time domain.
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| Given the functions ''f''(''t'') and ''g''(''t''), and their respective Laplace transforms ''F''(''s'') and ''G''(''s''):
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| : <math> f(t) = \mathcal{L}^{-1} \{ F(s) \} </math>
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| : <math> g(t) = \mathcal{L}^{-1} \{ G(s) \} </math>
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| the following table is a list of properties of unilateral Laplace transform:<ref>{{harvnb|Korn|Korn|1967|pp=226–227}}</ref>
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| {| class="wikitable"
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| |+ Properties of the unilateral Laplace transform
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| !
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| ! Time domain
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| ! 's' domain
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| ! Comment
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| |-
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| ! [[Linearity]]
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| | <math> a f(t) + b g(t) \ </math>
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| | <math> a F(s) + b G(s) \ </math>
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| | Can be proved using basic rules of integration.
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| |-
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| ! [[Frequency|Frequency differentiation]]
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| | <math> t f(t) \ </math>
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| | <math> -F'(s) \ </math>
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| | ''F''′ is the first [[derivative]] of ''F''.
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| |-
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| ! [[Frequency|Frequency differentiation]]
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| | <math> t^{n} f(t) \ </math>
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| | <math> (-1)^{n} F^{(n)}(s) \ </math>
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| | More general form, ''n''th derivative of ''F''(''s'').
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| |-
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| ! [[Derivative|Differentiation]]
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| | <math> f'(t) \ </math>
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| | <math> s F(s) - f(0) \ </math>
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| | ''f'' is assumed to be a [[differentiable function]], and its derivative is assumed to be of [[exponential type]]. This can then be obtained by [[integration by parts]]
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| |-
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| ! Second [[Derivative|Differentiation]]
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| | <math> f''(t) \ </math>
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| | <math> s^2 F(s) - s f(0) - f'(0) \ </math>
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| | ''f'' is assumed twice differentiable and the second derivative to be of exponential type. Follows by applying the Differentiation property to ''f''′(''t'').
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| |-
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| ! General [[Derivative|Differentiation]]
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| | <math> f^{(n)}(t) \ </math>
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| | <math> s^n F(s) - \sum_{k=1}^{n} s^{k-1} f^{(n-k)}(0) \ </math>
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| | ''f'' is assumed to be ''n''-times differentiable, with ''n''th derivative of exponential type. Follow by [[mathematical induction]].
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| |-
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| ! [[Frequency|Frequency integration]]
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| | <math> \frac{f(t)}{t} \ </math>
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| | <math> \int_s^\infty F(\sigma)\, d\sigma \ </math>
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| | This is deduced using the nature of frequency differentiation and conditional convergence.
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| |-
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| ! [[Integral|Integration]]
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| | <math> \int_0^t f(\tau)\, d\tau = (u * f)(t)</math>
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| | <math> {1 \over s} F(s) </math>
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| | ''u''(''t'') is the [[Heaviside step function]]. Note (''u'' ∗ ''f'')(''t'') is the [[convolution]] of ''u''(''t'') and ''f''(''t'').
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| |-
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| ! Time scaling
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| | <math>f(at)</math>
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| | <math> \frac{1}{|a|} F \left ( {s \over a} \right )</math>
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| |-
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| ! Frequency shifting
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| | <math> e^{at} f(t) \ </math>
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| | <math> F(s - a) \ </math>
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| |-
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| ! Time shifting
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| | <math> f(t - a) u(t - a) \ </math>
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| | <math> e^{-as} F(s) \ </math>
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| | ''u''(''t'') is the [[Heaviside step function]]
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| |-
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| ! [[Multiplication]]
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| | <math>f(t)g(t)</math>
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| | <math> \frac{1}{2\pi i}\lim_{T\to\infty}\int_{c-iT}^{c+iT}F(\sigma)G(s-\sigma)\,d\sigma \ </math>
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| | the integration is done along the vertical line Re(σ) = ''c'' that lies entirely within the region of convergence of ''F''.<ref>{{harvnb|Bracewell|2000|loc=Table 14.1, p. 385}}</ref>
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| |-
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| ! [[Convolution]]
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| | <math> (f * g)(t) = \int_0^t f(\tau)g(t-\tau)\,d\tau</math>
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| | <math> F(s) \cdot G(s) \ </math>
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| | ''f''(''t'') and ''g''(''t'') are extended by zero for ''t'' < 0 in the definition of the convolution.
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| |-
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| ! [[Complex conjugation]]
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| | <math> f^*(t) </math>
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| | <math> F^*(s^*) </math>
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| |-
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| ! [[Cross-correlation]]
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| | <math> f(t)\star g(t) </math>
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| | <math> F^*(-s^*)\cdot G(s) </math>
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| |-
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| ! [[Periodic Function]]
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| | <math>f(t)</math>
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| | <math>{1 \over 1 - e^{-Ts}} \int_0^T e^{-st} f(t)\,dt </math>
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| | ''f''(''t'') is a periodic function of [[periodic function|period]] ''T'' so that ''f''(''t'') = ''f''(''t'' + ''T''), for all ''t'' ≥ 0. This is the result of the time shifting property and the [[geometric series]].
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| |}
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| * '''[[Initial value theorem]]''':
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| : <math>f(0^+)=\lim_{s\to \infty}{sF(s)}.</math>
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| * '''[[Final value theorem]]''':
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| : <math>f(\infty)=\lim_{s\to 0}{sF(s)}</math>, if all [[Pole (complex analysis)|poles]] of ''sF''(''s'') are in the left half-plane.
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| : 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 a function has poles in the right-hand plane or on the imaginary axis, (e.g. <math>e^t</math> or sin(''t''), respectively) the behaviour of this formula is undefined.
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| === Relation to power series ===
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| The Laplace transform can be viewed as a continuous analogue of a power series.
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| Replacing summation with integration, a continuous version of a power series becomes
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| :<math>\int f(t) x^t \; dt</math>
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| For this to have nicer convergence properties, we would like for 0 < x < 1.
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| Changing the base of the power, this becomes
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| :<math>\int f(t) \exp{t(\log x)} \; dt</math>
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| and making the substitution -s=log x, we obtain the definition of the Laplace transform.
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| === Relation to moments ===
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| {{main|Moment generating function}}
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| The quantities
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| :<math>\mu_n = \int_0^\infty t^nf(t)\,dt</math>
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| are the ''moments'' of the function ''f''. Note by repeated [[differentiation under the integral]], <math>(-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 ''X'' are given by the expectation values <math>\mu_n=E[X^n]</math>. Then the relation holds:
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| :<math>\mu_n = (-1)^n\frac{d^n}{ds^n}E[e^{-sX}].</math>
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| === Proof of the Laplace transform of a function's derivative ===
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| It is often convenient to use the differentiation property of the Laplace transform to find the transform of a function's derivative. This can be derived from the basic expression for a Laplace transform as follows:
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| : <math>\begin{align}
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| \mathcal{L} \left\{f(t)\right\} & = \int_{0^-}^{\infty} e^{-st} f(t)\,dt \\[8pt]
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| & = \left[\frac{f(t)e^{-st}}{-s} \right]_{0^-}^{\infty} -
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| \int_{0^-}^\infty \frac{e^{-st}}{-s} f'(t) \, dt\quad \text{(by parts)} \\[8pt]
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| & = \left[-\frac{f(0^-)}{-s}\right] +
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| \frac{1}{s}\mathcal{L}\left\{f'(t)\right\},
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| \end{align}</math>
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| yielding
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| : <math>\mathcal{L}\left\{ f'(t) \right\} = s\cdot\mathcal{L} \left\{ f(t) \right\}-f(0^-), </math>
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| and in the bilateral case,
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| : <math> \mathcal{L}\left\{ { f'(t) } \right\} = s \int_{-\infty}^\infty e^{-st} f(t)\,dt = s \cdot \mathcal{L} \{ f(t) \}. </math>
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| The general result
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| : <math>\mathcal{L} \left\{ f^{(n)}(t) \right\} = s^n \cdot \mathcal{L} \left\{ f(t) \right\} - s^{n - 1} f(0^-) - \cdots - f^{(n - 1)}(0^-),</math>
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| where ''f''<sup>''n''</sup> is the ''n''th derivative of ''f'', can then be established with an [[mathematical induction|inductive]] argument.
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| === Evaluating improper integrals ===
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| Let <math>\mathcal{L}\left\{f(t)\right\}=F(s)</math>, then (see the table above)
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| : <math>\mathcal{L}\left\{\frac{f(t)}{t}\right\}=\int_{s}^{\infty}F(p)\, dp,</math>
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| or
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| : <math>\int_{0}^{\infty}\frac{f(t)}{t}e^{-st}\, dt=\int_{s}^{\infty}F(p)\, dp.</math>
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| Letting ''s'' → 0, we get the identity
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| : <math>\int_{0}^{\infty}\frac{f(t)}{t}\, dt=\int_{0}^{\infty}F(p)\, dp.</math>
| |
| | |
| For example,
| |
| | |
| :<math>\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=\frac{1}{2}\left.\ln\frac{p^{2}+a^{2}}{p^{2}+b^{2}} \right|_{0}^{\infty} =\ln b-\ln a.</math>
| |
| | |
| Another example is [[Dirichlet integral]].
| |
| | |
| === Relationship to other transforms ===
| |
| | |
| ==== Laplace–Stieltjes transform ====
| |
| The (unilateral) [[Laplace–Stieltjes transform]] of a function ''g'' : '''R''' → '''R''' is defined by the [[Lebesgue–Stieltjes integral]]
| |
| | |
| : <math>\{\mathcal{L}^*g\}(s) = \int_0^\infty e^{-st}dg(t).</math>
| |
| | |
| The function ''g'' is assumed to be of [[bounded variation]]. If ''g'' is the [[antiderivative]] of ''f'':
| |
| | |
| : <math>g(x) = \int_0^x f(t)\,dt</math>
| |
| | |
| then the Laplace–Stieltjes transform of ''g'' and the Laplace transform of ''f'' coincide. In general, the Laplace–Stieltjes transform is the Laplace transform of the [[Stieltjes measure]] associated to ''g''. So in practice, the only distinction between the two transforms is that the Laplace transform is thought of as operating on the density function of the measure, whereas the Laplace–Stieltjes transform is thought of as operating on its [[cumulative distribution function]].<ref>{{harvnb|Feller|1971|p=432}}</ref>
| |
| | |
| ==== Fourier transform ====
| |
| The [[continuous Fourier transform]] is equivalent to evaluating the bilateral Laplace transform with imaginary argument ''s'' = ''i''ω or ''s = 2πfi'':<ref>{{harvnb|Takacs|1953|p=93}}</ref>
| |
| | |
| :<math>\begin{align}
| |
| \hat{f}(\omega) & = \mathcal{F}\left\{f(t)\right\} \\[1em]
| |
| & = \mathcal{L}\left\{f(t)\right\}|_{s = i\omega} = F(s)|_{s = i \omega}\\[1em]
| |
| & = \int_{-\infty}^{\infty} e^{-i \omega t} f(t)\,\mathrm{d}t.\\
| |
| \end{align}</math>
| |
| | |
| This definition of the Fourier transform requires a prefactor of 1/2π on the reverse Fourier transform. This relationship between the Laplace and Fourier transforms is often used to determine the [[frequency spectrum]] of a [[signal (information theory)|signal]] or [[dynamical system]].
| |
| | |
| The above relation is valid as stated if and only if the region of convergence (ROC) of ''F''(''s'') contains the imaginary axis, σ = 0. For example, the function ''f''(''t'') = cos(ω<sub>0</sub>''t'') has a Laplace transform ''F''(''s'') = ''s''/(''s''<sup>2</sup> + ω<sub>0</sub><sup>2</sup>) whose ROC is Re(''s'') > 0. As ''s'' = ''i''ω is a pole of ''F''(''s''), substituting ''s'' = ''i''ω in ''F''(''s'') does not yield the Fourier transform of ''f''(''t'')''u''(''t''), which is proportional to the [[Dirac delta-function]] δ(ω-ω<sub>0</sub>).
| |
| | |
| However, a relation of the form
| |
| | |
| : <math>\lim_{\sigma\to 0^+} F(\sigma+i\omega) = \hat{f}(\omega)</math>
| |
| | |
| holds under much weaker conditions. For instance, this holds for the above example provided that the limit is understood as a [[weak limit]] of measures (see [[vague topology]]). General conditions relating the limit of the Laplace transform of a function on the boundary to the Fourier transform take the form of [[Paley-Wiener theorem]]s.
| |
| | |
| ==== Mellin transform ====
| |
| 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>G(s) = \mathcal{M}\left\{g(\theta)\right\} = \int_0^\infty \theta^s g(\theta) \frac{d\theta}{\theta}</math>
| |
| | |
| we set θ = e<sup>−t</sup> we get a two-sided Laplace transform.
| |
| | |
| ==== Z-transform ====
| |
| The unilateral or one-sided [[Z-transform]] is simply the Laplace transform of an ideally sampled signal with the substitution of
| |
| | |
| : <math> z \ \stackrel{\mathrm{def}}{=}\ e^{s T} \ </math>
| |
| | |
| : where ''T'' = 1/''f<sub>s</sub>'' is the [[Sampling theorem|sampling]] period (in units of time e.g., seconds) and ''f<sub>s</sub>'' is the [[sampling rate]] (in [[sample (signal)|samples per second]] or [[hertz]])
| |
| | |
| Let
| |
| | |
| : <math> \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>\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 ''x''(''t'')
| |
| | |
| : <math> x[n] \ \stackrel{\mathrm{def}}{=}\ x(nT) \ </math>
| |
| | |
| The Laplace transform of the sampled signal <math> x_q(t) \ </math> is
| |
| | |
| : <math>\begin{align}
| |
| X_q(s) & = \int_{0^-}^\infty x_q(t) e^{-s t} \,dt \\
| |
| & = \int_{0^-}^\infty \sum_{n=0}^\infty x[n] \delta(t - n T) e^{-s t} \, dt \\
| |
| & = \sum_{n=0}^\infty x[n] \int_{0^-}^\infty \delta(t - n T) e^{-s t} \, dt \\
| |
| & = \sum_{n=0}^\infty x[n] e^{-n s T}.
| |
| \end{align}</math>
| |
| | |
| This is precisely the definition of the unilateral [[Z-transform]] of the discrete function ''x''[''n'']
| |
| | |
| : <math> X(z) = \sum_{n=0}^{\infty} x[n] z^{-n} </math>
| |
| | |
| with the substitution of ''z'' → e<sup>''sT''</sup>.
| |
| | |
| Comparing the last two equations, we find the relationship between the unilateral [[Z-transform]] and the Laplace transform of the sampled signal:
| |
| | |
| : <math>X_q(s) = X(z) \Big|_{z=e^{sT}}.</math>
| |
| | |
| The similarity between the Z and Laplace transforms is expanded upon in the theory of [[time scale calculus]].
| |
| | |
| ==== Borel transform ====
| |
| The integral form of the [[Borel summation|Borel transform]]
| |
| | |
| : <math>F(s) = \int_0^\infty f(z)e^{-sz}\,dz</math>
| |
| | |
| is a special case of the Laplace transform for ''f'' an [[entire function]] of [[exponential type]], meaning that
| |
| | |
| : <math>|f(z)|\le Ae^{B|z|}</math>
| |
| | |
| for some constants ''A'' and ''B''. The generalized Borel transform allows a different weighting function to be used, rather than the exponential function, to transform functions not of exponential type. [[Nachbin's theorem]] gives necessary and sufficient conditions for the Borel transform to be well defined.
| |
| | |
| ==== Fundamental relationships ====
| |
| Since an ordinary Laplace transform can be written as a special case of a two-sided transform, and since the two-sided transform can be written as the sum of two one-sided transforms, the theory of the Laplace-, Fourier-, Mellin-, and Z-transforms are at bottom the same subject. However, a different point of view and different characteristic problems are associated with each of these four major integral transforms.
| |
| | |
| == Table of selected Laplace transforms ==
| |
| The following table provides Laplace transforms for many common functions of a single variable.<ref>{{Citation|edition=3rd| page=455|author=K.F. Riley, M.P. Hobson, S.J. Bence| title=Mathematical methods for physics and engineering| publisher=Cambridge University Press| year=2010 | isbn=978-0-521-86153-3}}</ref><ref>{{Citation| author=J.J.Distefano, A.R. Stubberud, I.J. Williams|page=78| title=Feedback systems and control|edition=2nd| publisher=Schaum's outlines| year=1995 |isbn=0-07-017052-5}}</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>\mathcal{L}\left\{f(t) + g(t) \right\} = \mathcal{L}\left\{f(t)\right\} + \mathcal{L}\left\{ g(t) \right\} </math>
| |
| | |
| * The Laplace transform of a multiple of a function is that multiple times the Laplace transformation of that function.
| |
| | |
| :: <math>\mathcal{L}\left\{a f(t)\right\} = a \mathcal{L}\left\{ f(t)\right\}</math>
| |
| | |
| Using this [[linearity]], and various [[List of trigonometric identities|trigonometric]], [[Hyperbolic function|hyperbolic]], and [[complex number]] (etc.) properties and/or identities, some Laplace transforms can be obtained from others quicker 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]], u(''t''). The entries of the table that involve a time delay τ are required to be [[causal system|causal]] (meaning that τ > 0). A causal system is a system where the [[impulse response]] ''h''(''t'') is zero for all time ''t'' prior to ''t'' = 0. In general, the region of convergence for causal systems is not the same as that of [[anticausal system]]s.
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| ! Function
| |
| ! Time domain <br> <math>f(t) = \mathcal{L}^{-1} \left\{ F(s) \right\}</math>
| |
| ! Laplace s-domain <br> <math>F(s) = \mathcal{L}\left\{ f(t) \right\}</math>
| |
| ! Region of convergence
| |
| ! Reference
| |
| | |
| |- style="text-align:center;"
| |
| | [[Dirac delta function|unit impulse]]
| |
| || <math> \delta(t) \ </math>
| |
| || <math> 1 </math>
| |
| || <math> \mathrm{all} \ s \,</math>
| |
| || inspection
| |
| | |
| |- style="text-align:center;"
| |
| | delayed impulse
| |
| || <math> \delta(t-\tau) \ </math>
| |
| || <math> e^{-\tau s} \ </math>
| |
| ||
| |
| || time shift of<br>unit impulse
| |
| | |
| |- style="text-align:center;"
| |
| | [[Heaviside step function|unit step]]
| |
| || <math> u(t) \ </math>
| |
| || <math> { 1 \over s } </math>
| |
| || Re(''s'') > 0
| |
| || integrate unit impulse
| |
| | |
| |- style="text-align:center;"
| |
| | delayed unit step
| |
| || <math> u(t-\tau) \ </math>
| |
| || <math> { e^{-\tau s} \over s } </math>
| |
| || Re(''s'') > 0
| |
| || time shift of<br>unit step
| |
| | |
| |- style="text-align:center;"
| |
| | [[ramp function|ramp]]
| |
| || <math> t \cdot u(t)\ </math>
| |
| || <math>\frac{1}{s^2}</math>
| |
| || Re(''s'') > 0
| |
| || integrate unit<br>impulse twice
| |
| | |
| |- style="text-align:center;"
| |
| | ''n''th power <br /> ( for integer ''n'' )
| |
| || <math> t^n \cdot u(t) </math>
| |
| || <math> { n! \over s^{n+1} } </math>
| |
| || Re(''s'') > 0 <br /> (''n'' > −1)
| |
| || Integrate unit<br>step n times
| |
| | |
| |- style="text-align:center;"
| |
| | ''q''th power <br /> (for complex ''q'')
| |
| || <math> t^q \cdot u(t) </math>
| |
| || <math> { \Gamma(q+1) \over s^{q+1} } </math>
| |
| || Re(''s'') > 0 <br /> Re(''q'') > −1
| |
| || <ref>Mathematical Handbook of Formulas and Tables (3rd edition), S. Lipschutz, M.R. Spiegel, J. Liu, Schuam's Outline Series, p.183, 2009, ISBN 978-0-07-154855-7 - provides the case for real ''q''.</ref><ref>http://mathworld.wolfram.com/LaplaceTransform.html - Wolfram Mathword provides case for complex q</ref>
| |
| | |
| |- style="text-align:center;"
| |
| | ''n''th root
| |
| || <math> \sqrt[n]{t} \cdot u(t) </math>
| |
| || <math> { \Gamma(\frac{1}{n}+1) \over s^{\frac{1}{n}+1} } </math>
| |
| || Re(''s'') > 0
| |
| || Set ''q'' = 1/''n'' above.
| |
| | |
| |- style="text-align:center;"
| |
| | ''n''th power with frequency shift
| |
| || <math>t^{n} e^{-\alpha t} \cdot u(t) </math>
| |
| || <math>\frac{n!}{(s+\alpha)^{n+1}}</math>
| |
| || Re(''s'') > −α
| |
| || Integrate unit step,<br>apply frequency shift
| |
| | |
| |- style="text-align:center;"
| |
| | delayed ''n''th power <br /> with frequency shift
| |
| || <math>(t-\tau)^n e^{-\alpha (t-\tau)} \cdot u(t-\tau) </math>
| |
| || <math> \frac{n! \cdot e^{-\tau s}}{(s+\alpha)^{n+1}} </math> || Re(''s'') > −α
| |
| || Integrate unit step,<br>apply frequency shift,<br>apply time shift
| |
| | |
| |- style="text-align:center;"
| |
| | [[exponential decay]]
| |
| || <math> e^{-\alpha t} \cdot u(t) </math>
| |
| || <math> { 1 \over s+\alpha } </math>
| |
| || Re(''s'') > −α
| |
| || Frequency shift of<br>unit step
| |
| | |
| |- style="text-align:center;"
| |
| | [[Two-sided Laplace transform|two-sided]] exponential decay
| |
| || <math> e^{-\alpha|t|} \ </math>
| |
| || <math> { 2\alpha \over \alpha^2 - s^2 } </math>
| |
| || −α < Re(''s'') < α
| |
| || Frequency shift of<br>unit step
| |
| | |
| |- style="text-align:center;"
| |
| | exponential approach
| |
| || <math>( 1-e^{-\alpha t}) \cdot u(t) \ </math>
| |
| || <math>\frac{\alpha}{s(s+\alpha)} </math>
| |
| || Re(''s'') > 0
| |
| || Unit step minus<br>exponential decay
| |
| | |
| |- style="text-align:center;"
| |
| | [[sine]]
| |
| || <math> \sin(\omega t) \cdot u(t) \ </math>
| |
| || <math> { \omega \over s^2 + \omega^2 } </math>
| |
| || Re(''s'') > 0
| |
| || {{Harvnb|Bracewell|1978|p=227}}
| |
| | |
| |- style="text-align:center;"
| |
| | [[cosine]]
| |
| || <math> \cos(\omega t) \cdot u(t) \ </math>
| |
| || <math> { s \over s^2 + \omega^2 } </math>
| |
| || Re(''s'') > 0
| |
| || {{Harvnb|Bracewell|1978|p=227}}
| |
| | |
| |- style="text-align:center;"
| |
| | [[hyperbolic sine]]
| |
| || <math> \sinh(\alpha t) \cdot u(t) \ </math>
| |
| || <math> { \alpha \over s^2 - \alpha^2 } </math>
| |
| || Re(''s'') > |α|
| |
| || {{Harvnb|Williams|1973|p=88}}
| |
| | |
| |- style="text-align:center;"
| |
| | [[hyperbolic cosine]]
| |
| || <math> \cosh(\alpha t) \cdot u(t) \ </math>
| |
| || <math> { s \over s^2 - \alpha^2 } </math>
| |
| || Re(''s'') > |α|
| |
| || {{Harvnb|Williams|1973|p=88}}
| |
| | |
| |- style="text-align:center;"
| |
| | exponentially decaying <br /> sine wave
| |
| || <math>e^{-\alpha t} \sin(\omega t) \cdot u(t) \ </math>
| |
| || <math> { \omega \over (s+\alpha )^2 + \omega^2 } </math>
| |
| || Re(''s'') > −α
| |
| || {{Harvnb|Bracewell|1978|p=227}}
| |
| | |
| |- style="text-align:center;"
| |
| | exponentially decaying <br /> cosine wave
| |
| || <math>e^{-\alpha t} \cos(\omega t) \cdot u(t) \ </math>
| |
| || <math> { s+\alpha \over (s+\alpha )^2 + \omega^2 } </math>
| |
| || Re(''s'') > −α
| |
| || {{Harvnb|Bracewell|1978|p=227}}
| |
| | |
| |- style="text-align:center;"
| |
| | [[natural logarithm]]
| |
| || <math> \ln (t) \cdot u(t) </math>
| |
| || <math> - { 1 \over s}\, \left[ \ln(s)+\gamma \right] </math>
| |
| || Re(''s'') > 0
| |
| || {{Harvnb|Williams|1973|p=88}}
| |
| | |
| |- style="text-align:center;"
| |
| | [[Bessel function]] <br> of the first kind, <br /> of order ''n''
| |
| || <math> J_n( \omega t) \cdot u(t)</math>
| |
| || <math>\frac{ \left(\sqrt{s^2+ \omega^2}-s\right)^{n}}{\omega^n \sqrt{s^2 + \omega^2}}</math>
| |
| || Re(''s'') > 0 <br /> (''n'' > −1)
| |
| || {{Harvnb|Williams|1973|p=89}}
| |
| | |
| |- style="text-align:center;"
| |
| | [[Error function]]
| |
| || <math> \mathrm{erf}(t) \cdot u(t) </math>
| |
| || <math> {e^{s^2/4} \left(1 - \operatorname{erf} \left(s/2\right)\right) \over s}</math>
| |
| || Re(''s'') > 0
| |
| || {{Harvnb|Williams|1973|p=89}}
| |
| | |
| |-
| |
| | colspan=5|'''Explanatory notes:'''
| |
| {{col-begin}}
| |
| {{col-break}}
| |
| | |
| * ''u''(''t'') represents the [[Heaviside step function]].
| |
| * <math> \delta(t) \, </math> represents the [[Dirac delta function]].
| |
| * Γ(''z'') represents the [[Gamma function]].
| |
| * γ is the [[Euler–Mascheroni constant]].
| |
| | |
| {{col-break}}
| |
| | |
| * ''t'', a real number, typically represents ''time'', <br />although it can represent ''any'' independent dimension.
| |
| * ''s'' is the [[complex number|complex]] [[angular frequency]], and Re(''s'') is its [[real part]].
| |
| * α, β, τ, and ω are [[real numbers]].
| |
| * ''n'' is an [[integer]].
| |
| | |
| {{col-end}}
| |
| |}
| |
| | |
| == s-Domain equivalent circuits and impedances ==
| |
| The Laplace transform is often used in circuit analysis, and simple conversions to the s-Domain of circuit elements can be made. Circuit elements can be transformed into [[Electrical impedance|impedance]]s, very similar to [[Phasor (sine waves)|phasor]] impedances.
| |
| | |
| Here is a summary of equivalents:
| |
| | |
| : [[File:S-Domain circuit equivalency.svg]]
| |
| | |
| Note that the resistor is exactly the same in the time domain and the 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 s-Domain account for that.
| |
| | |
| The equivalents for current and voltage sources are simply derived from the transformations in the table above.
| |
| | |
| == Examples: How to apply the properties and theorems ==
| |
| <!--A few worked examples are provided here to enable the reader to assess comprehension of the factual presentation. Elaboration beyond the role of supporting factual comprehension belongs at [[v:|Wikiversity]] or [[b:|Wikibooks]].-->
| |
| | |
| The Laplace transform is used frequently in [[engineering]] and [[physics]]; the output of a [[linear time invariant]] system can be calculated by convolving its unit [[impulse response]] with the input signal. Performing this calculation in Laplace space turns the [[convolution]] into a [[multiplication]]; the latter being easier to solve because of its algebraic form. For more information, see [[control theory]].
| |
| | |
| The Laplace transform can also be used to [[Laplace transform applied to differential equations|solve differential equations]] and is used extensively in [[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. The 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.
| |
| | |
| === Example 1: Solving a differential equation ===
| |
| In [[nuclear physics]], the following fundamental relationship governs [[radioactive decay]]: the number of radioactive atoms ''N'' in a sample of a radioactive [[isotope]] decays at a rate proportional to ''N''. This leads to the first order linear differential equation
| |
| | |
| : <math>\frac{dN}{dt} = -\lambda N</math>
| |
| | |
| where λ is the [[decay constant]]. The Laplace transform can be used to solve this equation.
| |
| | |
| Rearranging the equation to one side, we have
| |
| | |
| : <math>\frac{dN}{dt} + \lambda N = 0. </math>
| |
| | |
| Next, we take the Laplace transform of both sides of the equation:
| |
| | |
| : <math>\left( s \tilde{N}(s) - N_o \right) + \lambda \tilde{N}(s) \ = \ 0 </math>
| |
| | |
| where
| |
| | |
| : <math>\tilde{N}(s) = \mathcal{L}\{N(t)\}</math>
| |
| | |
| and
| |
| | |
| : <math>N_o \ = \ N(0).</math>
| |
| | |
| Solving, we find
| |
| | |
| : <math>\tilde{N}(s) = { N_o \over s + \lambda }.</math>
| |
| | |
| Finally, we take the inverse Laplace transform to find the general solution
| |
| | |
| : <math>\begin{align}
| |
| N(t) & = \mathcal{L}^{-1} \{\tilde{N}(s)\} = \mathcal{L}^{-1} \left\{ \frac{N_o}{s + \lambda} \right\} \\
| |
| & = \ N_o e^{-\lambda t},
| |
| \end{align}</math>
| |
| | |
| which is indeed the correct form for radioactive decay.
| |
| | |
| === Example 2: Deriving the complex impedance for a capacitor ===
| |
| In the theory of [[electrical circuit]]s, the current flow in a [[capacitor]] is proportional to the capacitance and rate of change in the electrical potential (in [[SI]] units). Symbolically, this is expressed by the differential equation
| |
| | |
| : <math> i = C { dv \over dt} </math>
| |
| | |
| where ''C'' is the capacitance (in [[farads]]) of the capacitor, ''i'' = ''i''(''t'') is the [[electric current]] (in [[amperes]]) through the capacitor as a function of time, and ''v'' = ''v''(''t'') is the [[electrostatic potential|voltage]] (in [[volts]]) across the terminals of the capacitor, also as a function of time.
| |
| | |
| Taking the Laplace transform of this equation, we obtain
| |
| | |
| : <math> I(s) = C \left( s V(s) - V_o \right) </math>
| |
| | |
| where
| |
| | |
| : <math> I(s) = \mathcal{L} \{ i(t) \}, \, </math>
| |
| : <math> V(s) = \mathcal{L} \{ v(t) \}, \, </math>
| |
| | |
| and
| |
| | |
| : <math> V_o \ = \ v(t)|_{t=0}. \, </math>
| |
| | |
| Solving for ''V''(''s'') we have
| |
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| : <math> V(s) = { I(s) \over sC } + { V_o \over s }. </math>
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| The definition of the [[complex number|complex]] [[Electrical impedance|impedance]] ''Z'' (in [[ohm (unit)|ohms]]) is the ratio of the complex voltage ''V'' divided by the complex current ''I'' while holding the initial state ''V''<sub>o</sub> at zero:
| |
| | |
| : <math>Z(s) = { V(s) \over I(s) } \bigg|_{V_o = 0}.</math>
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| | |
| Using this definition and the previous equation, we find:
| |
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| : <math>Z(s) = \frac{1}{sC}, </math>
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| | |
| which is the correct expression for the complex impedance of a capacitor.
| |
| | |
| === Example 3: Method of partial fraction expansion ===<!-- [[Partial fractions in Laplace transforms]] redirect here -->
| |
| Consider a linear time-invariant system with [[transfer function]]
| |
| | |
| : <math>H(s) = \frac{1}{(s+\alpha)(s+\beta)}.</math>
| |
| | |
| The [[impulse response]] is simply the inverse Laplace transform of this transfer function:
| |
| | |
| : <math>h(t) = \mathcal{L}^{-1}\{H(s)\}.</math>
| |
| | |
| To evaluate this inverse transform, we begin by expanding ''H''(''s'') using the method of [[partial fraction]] expansion:
| |
| | |
| : <math>\frac{1}{(s+\alpha)(s+\beta)} = { P \over s+\alpha } + { R \over s+\beta }.</math>
| |
| | |
| The unknown constants ''P'' and ''R'' are the [[residue (complex analysis)|residue]]s located at the corresponding [[pole (complex analysis)|pole]]s of the transfer function. Each residue represents the relative contribution of that [[mathematical singularity|singularity]] to the transfer function's overall shape. By the [[residue theorem]], the inverse Laplace transform depends only upon the poles and their residues. To find the residue ''P'', we multiply both sides of the equation by ''s'' + ''α'' to get
| |
| | |
| : <math>\frac{1}{s+\beta} = P + { R (s+\alpha) \over s+\beta }.</math>
| |
| | |
| Then by letting ''s'' = −''α'', the contribution from ''R'' vanishes and all that is left is
| |
| | |
| : <math>P = \left.{1 \over s+\beta}\right|_{s=-\alpha} = {1 \over \beta - \alpha}. </math>
| |
| | |
| Similarly, the residue ''R'' is given by
| |
| | |
| : <math>R = \left.{1 \over s+\alpha}\right|_{s=-\beta} = {1 \over \alpha - \beta}.</math>
| |
| | |
| Note that
| |
| | |
| : <math>R = {-1 \over \beta - \alpha} = - P</math>
| |
| | |
| and so the substitution of ''R'' and ''P'' into the expanded expression for ''H''(''s'') gives
| |
| | |
| : <math>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 ''H''(''s'') to obtain:
| |
| | |
| : <math>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.
| |
| | |
| === Example 3.2: Convolution ===
| |
| The same result can be achieved using the convolution property as if the system is a series of filters with transfer functions of 1/(s+a) and 1/(s+b). That is, the inverse of
| |
| | |
| : <math>H(s) = \frac{1}{(s+a)(s+b)} = \frac{1}{s+a} \cdot \frac{1}{s+b} </math>
| |
| | |
| is
| |
| | |
| : <math> \mathcal{L}^{-1} \left \{ \frac{1}{s+a} \right \} \, * \, \mathcal{L}^{-1} \left \{ \frac{1}{s+b} \right \} = e^{-at} \, * \, e^{-bt} = \int_0^t e^{-ax}e^{-b(t-x)} \, dx = \frac{e^{-a t}-e^{-b t}}{b-a}.</math>
| |
| | |
| === Example 4: Mixing sines, cosines, and exponentials ===
| |
| {| class="wikitable"
| |
| |-
| |
| ! Time function !! Laplace transform
| |
| |-
| |
| |{{pad|1em}} <math>e^{-\alpha t}\left[\cos{(\omega t)}+\left(\frac{\beta-\alpha}{\omega}\right)\sin{(\omega t)}\right]u(t) </math>{{pad|2em}}
| |
| | |
| | {{pad|1em}}<math> \frac{s+\beta}{(s+\alpha)^2+\omega^2} </math>{{pad|1em}}
| |
| |}
| |
| | |
| Starting with the Laplace transform
| |
| | |
| : <math>X(s) = \frac{s+\beta}{(s+\alpha)^2+\omega^2}, </math>
| |
| | |
| we find the inverse transform by first adding and subtracting the same constant α to the numerator:
| |
| | |
| : <math>X(s) = \frac{s+\alpha } { (s+\alpha)^2+\omega^2} + \frac{\beta - \alpha }{(s+\alpha)^2+\omega^2}. </math>
| |
| | |
| By the shift-in-frequency property, we have
| |
| | |
| : <math>\begin{align}
| |
| x(t) & = e^{-\alpha t} \mathcal{L}^{-1} \left\{ {s \over s^2 + \omega^2} + { \beta - \alpha \over s^2 + \omega^2 } \right\} \\[8pt]
| |
| & = e^{-\alpha t} \mathcal{L}^{-1} \left\{ {s \over s^2 + \omega^2} + \left( { \beta - \alpha \over \omega } \right) \left( { \omega \over s^2 + \omega^2 } \right) \right\} \\[8pt]
| |
| & = e^{-\alpha t} \left[\mathcal{L}^{-1} \left\{ {s \over s^2 + \omega^2} \right\} + \left( { \beta - \alpha \over \omega } \right) \mathcal{L}^{-1} \left\{ { \omega \over s^2 + \omega^2 } \right\} \right].
| |
| \end{align}</math>
| |
| | |
| Finally, using the Laplace transforms for sine and cosine (see the table, above), we have
| |
| | |
| : <math>x(t) = e^{-\alpha t} \left[\cos{(\omega t)}u(t)+\left(\frac{\beta-\alpha}{\omega}\right)\sin{(\omega t)}u(t)\right].</math>
| |
| : <math>x(t) = e^{-\alpha t} \left[\cos{(\omega t)}+\left(\frac{\beta-\alpha}{\omega}\right)\sin{(\omega t)}\right]u(t).</math>
| |
| | |
| === Example 5: Phase delay ===
| |
| {| class="wikitable"
| |
| |-
| |
| ! Time function
| |
| ! Laplace transform
| |
| |-
| |
| | <math>\sin{(\omega t+\phi)} \ </math>
| |
| | <math>\frac{s\sin\phi+\omega \cos\phi}{s^2+\omega^2} \ </math>
| |
| |-
| |
| | <math>\cos{(\omega t+\phi)} \ </math>
| |
| | <math>\frac{s\cos\phi - \omega \sin\phi}{s^2+\omega^2} \ </math>
| |
| |}
| |
| | |
| Starting with the Laplace transform,
| |
| | |
| : <math>X(s) = \frac{s\sin\phi+\omega \cos\phi}{s^2+\omega^2}</math>
| |
| | |
| we find the inverse by first rearranging terms in the fraction:
| |
| | |
| : <math>\begin{align}
| |
| X(s) & = \frac{s \sin \phi}{s^2 + \omega^2} + \frac{\omega \cos \phi}{s^2 + \omega^2} \\
| |
| & = (\sin \phi) \left(\frac{s}{s^2 + \omega^2} \right) + (\cos \phi) \left(\frac{\omega}{s^2 + \omega^2} \right).
| |
| \end{align}</math>
| |
| | |
| We are now able to take the inverse Laplace transform of our terms:
| |
| | |
| : <math>\begin{align}
| |
| x(t) & = (\sin \phi) \mathcal{L}^{-1}\left\{\frac{s}{s^2 + \omega^2} \right\} + (\cos \phi) \mathcal{L}^{-1}\left\{\frac{\omega}{s^2 + \omega^2} \right\} \\
| |
| & =(\sin \phi)(\cos \omega t) + (\sin \omega t)(\cos \phi).
| |
| \end{align}</math>
| |
| | |
| This is just the [[Trigonometric identity#Angle sum and difference identities|sine of the sum]] of the arguments, yielding:
| |
| | |
| :<math>x(t) = \sin (\omega t + \phi). \ </math>
| |
| | |
| We can apply similar logic to find that
| |
| | |
| : <math>\mathcal{L}^{-1} \left\{ \frac{s\cos\phi - \omega \sin\phi}{s^2+\omega^2} \right\} = \cos{(\omega t+\phi)}. \ </math>
| |
| | |
| === {{Anchor|Inferring spatial structure from spectrum}}Example 6: Determining structure of astronomical object from spectrum ===
| |
| The wide and general applicability of the Laplace transform and its inverse is illustrated by an application in astronomy which provides some information on the ''spatial distribution'' of matter of an [[Astronomy|astronomical]] source of [[radiofrequency]] [[thermal radiation]] too distant to [[Angular resolution|resolve]] as more than a point, given its [[flux density]] [[spectrum]], rather than relating the ''time'' domain with the spectrum (frequency domain).
| |
| | |
| Assuming certain properties of the object, e.g. spherical shape and constant temperature, calculations based on carrying out an inverse Laplace transformation on the spectrum of the object can produce the only possible [[Mathematical model|model]] of the distribution of matter in it (density as a function of distance from the center) consistent with the spectrum.<ref>On the interpretation of continuum flux observations from thermal radio sources: [http://adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1974MNRAS.167..493S&link_type=ARTICLE&db_key=AST&high= I. Continuum spectra and brightness contours, M Salem and MJ Seaton, Monthly Notices of the Royal Astronomical Society (MNRAS), Vol. 167, p. 493-510 (1974)] [http://adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1974MNRAS.167..511S&link_type=ARTICLE&db_key=AST&high= II. Three-dimensional models, M Salem, MNRAS Vol. 167, p. 511-516 (1974)]</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.
| |
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| == See also ==
| |
| | |
| * [[Analog signal processing]]
| |
| * [[Bernstein's theorem on monotone functions]]
| |
| * [[Continuous-repayment mortgage#Ordinary time differential equation|Continuous-repayment mortgage]]
| |
| * [[Fourier transform]]
| |
| * [[Hamburger moment problem]]
| |
| * [[Hardy–Littlewood tauberian theorem]]
| |
| * [[Laplace transform applied to differential equations]]
| |
| * [[Moment-generating function]]
| |
| * [[Post's inversion formula]]
| |
| * [[N-transform]]
| |
| * [[Pierre-Simon Laplace]]
| |
| * [[Sumudu transform]] or [[Laplace–Carson transform]]
| |
| * [[Symbolic integration]]
| |
| * [[Z-transform]] (discrete equivalent of the Laplace transform)
| |
| | |
| == Notes ==
| |
| {{Reflist|30em}}
| |
| | |
| == References ==
| |
| | |
| === Modern ===
| |
| | |
| * {{citation|first1=Wolfgang|last1=Arendt|first2=Charles J.K.|last2=Batty|first3=Matthias|last3=Hieber|first4=Frank|last4=Neubrander|title=Vector-Valued Laplace Transforms and Cauchy Problems|publisher=Birkhäuser Basel|year=2002|isbn=3-7643-6549-8}}.
| |
| * {{Citation |last=Bracewell |first=Ronald N. |title=The Fourier Transform and its Applications |edition=2nd |year=1978 |publisher=McGraw-Hill Kogakusha |isbn=0-07-007013-X }}<!-- This edition is used for pinpoint citations in the transform table. -->
| |
| * {{citation|first=R. N.|last=Bracewell|title=The Fourier Transform and Its Applications|edition=3rd|publication-place=Boston|publisher=McGraw-Hill|year=2000|isbn=0-07-116043-4}}.
| |
| * {{citation|last=Davies|first=Brian|title=Integral transforms and their applications|edition=Third|publisher=Springer|publication-place=New York|year=2002|isbn= 0-387-95314-0}}.
| |
| * {{Citation | last1=Feller | first1=William | author1-link=William Feller | title=An introduction to probability theory and its applications. Vol. II. | publisher=[[John Wiley & Sons]] | location=New York | series=Second edition | mr=0270403 | year=1971}}.
| |
| * {{citation |first1=G. A. |last1=Korn |first2=T. M. |last2=Korn |title=Mathematical Handbook for Scientists and Engineers |publisher=McGraw-Hill Companies |edition=2nd |year=1967 |isbn=0-07-035370-0 }}.
| |
| * {{citation|first1=A. D.|last1=Polyanin|first2=A. V.|last2=Manzhirov|title=Handbook of Integral Equations|publisher=CRC Press|publication-place=Boca Raton|year=1998|isbn=0-8493-2876-4}}.
| |
| * {{Citation | last1=Schwartz | first1=Laurent | title=Transformation de Laplace des distributions | mr=0052555 | year=1952 | journal=Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] | volume=1952 | pages=196–206 |language=French }}.
| |
| * {{citation|first=William McC.|last=Siebert|title=Circuits, Signals, and Systems|publisher=MIT Press|publication-place=Cambridge, Massachusetts|year=1986|isbn=0-262-19229-2}}.
| |
| * {{Citation | last1=Widder | first1=David Vernon | title=The Laplace Transform | publisher=[[Princeton University Press]] | series=Princeton Mathematical Series, v. 6 | mr=0005923 | year=1941}}.
| |
| * {{Citation | last1=Widder | first1=David Vernon | title=What is the Laplace transform? | mr=0013447 | year=1945 | journal=[[American Mathematical Monthly|The American Mathematical Monthly]] | issn=0002-9890 |volume=52 |issue=8 | pages=419–425 | doi=10.2307/2305640 |publisher=The American Mathematical Monthly | jstor=2305640}}.
| |
| * {{Citation | last=Williams |first=J. |title=Laplace Transforms |series=Problem Solvers |volume= |publisher=George Allen & Unwin |year=1973 |isbn= 0-04-512021-8 }}
| |
| * {{Citation | last=Takacs | first= J.|title=Fourier amplitudok meghatarozasa operatorszamitassal | year=1953 | journal=Magyar Hiradastechnika | volume=IV | issue=7–8|pages=93–96 |language=Hungarian }}
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| | |
| === Historical ===
| |
| * {{citation | last=Deakin|first= M. A. B. | year=1981 | title=The development of the Laplace transform | journal=Archive for the History of the Exact Sciences | volume=25 | pages=343–390 | doi=10.1007/BF01395660 | issue=4 }}
| |
| * {{citation | last=Deakin|first= M. A. B. | year=1982 | title=The development of the Laplace transform | journal=Archive for the History of the Exact Sciences | volume=26 | pages=351–381 | doi=10.1007/BF00418754 | issue=4}}
| |
| * {{citation|last=Euler|first=L.|authorlink=Leonhard Euler|year=1744|title=De constructione aequationum|journal=Opera omnia|series=1st series|volume=22|pages=150–161}}.
| |
| * {{citation|last=Euler|first=L.|authorlink=Leonhard Euler|year=1753|title=Methodus aequationes differentiales|journal=Opera omnia|series=1st series|volume=22|pages=181–213}}.
| |
| * {{citation|last=Euler|first=L.|authorlink=Leonhard Euler|year= 1769|title=Institutiones calculi integralis, Volume 2|journal=Opera omnia|series=1st series|volume=12}}, Chapters 3–5.
| |
| * {{citation|last=Grattan-Guinness|first=I|authorlink=Ivor Grattan-Guinness|year=1997|contribution=Laplace's integral solutions to partial differential equations|editor=Gillispie, C. C.|title=Pierre Simon Laplace 1749–1827: A Life in Exact Science|publication-place=Princeton|publisher=Princeton University Press|isbn=0-691-01185-0}}.
| |
| * {{citation|last=Lagrange|first=J. L.|authorlink=Joseph Louis Lagrange|year=1773|title=Mémoire sur l'utilité de la méthode|series=Œuvres de Lagrange|volume=2|pages=171–234}}.
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| | |
| == External links ==
| |
| * {{springer|title=Laplace transform|id=p/l057540}}
| |
| * [http://wims.unice.fr/wims/wims.cgi?lang=en&+module=tool%2Fanalysis%2Ffourierlaplace.en 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://math.fullerton.edu/mathews/c2003/LaplaceTransformMod.html Laplace Transform Module by John H. Mathews]
| |
| * [http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/ Good explanations of the initial and final value theorems]
| |
| * [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.
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| {{DEFAULTSORT:Laplace Transform}}
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| [[Category:Integral transforms]]
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| [[Category:Differential equations]]
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| [[Category:Fourier analysis]]
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| [[Category:Transforms]]
| |