Editing Laplace transform (section)

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

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

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

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

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

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

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

=== Bilateral Laplace transform ===
{{Main article|Two-sided Laplace transform}}

When one says "the Laplace transform" without qualification, the unilateral or one-sided transform is usually intended. The Laplace transform can be alternatively defined as the ''bilateral Laplace transform'', or [[two-sided Laplace transform]], by extending the limits of integration to be the entire real axis. If that is done, the common unilateral transform simply becomes a special case of the bilateral transform, where the definition of the function being transformed is multiplied by the [[Heaviside step function]].

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

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

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

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

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

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

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

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

=== Algebraic construction ===

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