Editing Laplace transform (section)

== Examples and applications ==
<!--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 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>{{harvnb|Korn|Korn|1967|loc=§8.1}}</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.

=== Evaluating improper integrals ===
Let <math>\mathcal{L}\left\{f(t)\right\} = F(s)</math>. Then (see the table above)

<math display="block">\partial_s\mathcal{L} \left\{\frac{f(t)} t \right\} = \partial_s\int_0^\infty \frac{f(t)}{t}e^{-st}\, dt = -\int_0^\infty f(t)e^{-st}dt = - F(s) </math>

From which one gets:

<math display=block>
 \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#Final Value Theorem for improperly integrable functions (Abel's theorem for integrals)|final value theorem]]. Even when the interchange cannot be justified the calculation can be suggestive. For example, with {{math|''a'' ≠ 0 ≠ ''b''}}, proceeding formally one has
<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]].

Another example is [[Dirichlet integral]].

=== Complex impedance of 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 (with equations as for the [[International System of Units|SI]] unit system). Symbolically, this is expressed by the differential equation
<math display=block>i = C { dv \over dt} ,</math>
where {{math|''C''}} is the capacitance of the capacitor, {{math|1=''i'' = ''i''(''t'')}} is the [[electric current]] through the capacitor as a function of time, and {{math|1=''v'' = ''v''(''t'')}} is the [[electrostatic potential|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}
  I(s) &= \mathcal{L} \{ i(t) \},\\
  V(s) &= \mathcal{L} \{ v(t) \},
\end{align}</math>
and
<math display=block>V_0 = v(0). </math>

Solving for {{math|''V''(''s'')}} we have
<math display=block>V(s) = { I(s) \over sC } + { V_0 \over s }.</math>

The definition of the complex impedance {{math|''Z''}} (in [[ohm]]s) is the ratio of the complex voltage {{math|''V''}} divided by the complex current {{math|''I''}} while holding the initial state {{math|''V''<sub>0</sub>}} 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.

=== Impulse response ===
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
<!-- [[Partial fractions in Laplace transforms]] redirect here -->
To evaluate this inverse transform, we begin by expanding {{math|''H''(''s'')}} 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 {{math|''P''}} and {{math|''R''}} are the [[residue (complex analysis)|residues]] located at the corresponding poles 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 {{math|''P''}}, we multiply both sides of the equation by {{math|''s'' + ''α''}} to get
<math display=block>\frac{1}{s + \beta} = P  + { R (s + \alpha) \over s + \beta }.</math>

Then by letting {{math|1=''s'' = −''α''}}, the contribution from {{math|''R''}} 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 {{math|''R''}} 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 {{math|''R''}} and {{math|''P''}} into the expanded expression for {{math|''H''(''s'')}} 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 {{math|''H''(''s'')}} 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 theorem|convolution property]] as if the system is a series of filters with transfer functions {{math|1/(''s'' + ''α'')}} and {{math|1/(''s'' + ''β'')}}. That is, the inverse of
<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>

=== Phase delay ===
{| class="wikitable"
|-
 ! scope="col" | Time function
 ! scope="col" | Laplace transform
|-
 | <math>\sin{(\omega t + \varphi)}</math>
 | <math>\frac{s\sin(\varphi) + \omega \cos(\varphi)}{s^2 + \omega^2}</math>
|-
 | <math>\cos{(\omega t + \varphi)}</math>
 | <math>\frac{s\cos(\varphi) - \omega \sin(\varphi)}{s^2 + \omega^2}.</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}
  X(s) &= \frac{s \sin(\varphi)}{s^2 + \omega^2} + \frac{\omega \cos(\varphi)}{s^2 + \omega^2} \\
       &= \sin(\varphi) \left(\frac{s}{s^2 + \omega^2} \right) + \cos(\varphi) \left(\frac{\omega}{s^2 + \omega^2} \right).
\end{align}</math>

We are now able to take the inverse Laplace transform of our terms:
<math display=block>\begin{align}
  x(t) &= \sin(\varphi) \mathcal{L}^{-1}\left\{\frac{s}{s^2 + \omega^2} \right\} + \cos(\varphi) \mathcal{L}^{-1}\left\{\frac{\omega}{s^2 + \omega^2} \right\} \\
       &= \sin(\varphi)\cos(\omega t) + \cos(\varphi)\sin(\omega t).
\end{align}</math>

This is just the [[Trigonometric identity#Angle sum and difference identities|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>

=== Statistical mechanics ===
In [[statistical mechanics]], the Laplace transform of the density of states <math>g(E)</math> defines the [[partition function (statistical mechanics)|partition function]].<ref>{{cite book|author1=RK Pathria|author2=Paul Beal|title=Statistical mechanics|url=https://archive.org/details/statisticalmecha00path_911|url-access=limited|edition=2nd|publisher=Butterworth-Heinemann|year=1996|page=[https://archive.org/details/statisticalmecha00path_911/page/n66 56]|isbn=9780750624695 }}</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 [[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>{{citation |first1=M. |last1=Salem |first2=M. J. |last2=Seaton |year=1974 |title=I. Continuum spectra and brightness contours |journal=[[Monthly Notices of the Royal Astronomical Society]] |volume=167 |pages=493–510 |doi=10.1093/mnras/167.3.493|bibcode=1974MNRAS.167..493S |doi-access=free}}, and<br/>{{citation |first1=M. |last1=Salem |year=1974 |title=II. Three-dimensional models |journal=Monthly Notices of the Royal Astronomical Society |volume=167 |pages=511–516 |doi=10.1093/mnras/167.3.511|bibcode=1974MNRAS.167..511S |doi-access=free}}</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.

===Birth and death processes===
Consider a [[random walk]], with steps <math>\{+1,-1\}</math> occurring with probabilities <math>p,q=1-p</math>.<ref>{{cite book|author=Feller|title=Introduction to Probability Theory, volume II,pp=479-483}}</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
:<math>P_n(t) = \int_0^t\lambda e^{-\lambda(t-s)}(pP_{n-1}(s) + qP_{n+1}(s))\,ds\quad (+e^{-\lambda t}\quad\text{when}\ n=0).</math>
This leads to a system of [[integral equation]]s (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
:<math>\pi_n(s) = \mathcal L(P_n)(s),</math>
namely:
:<math>\pi_n(s) = \frac{\lambda}{\lambda+s}(p\pi_{n-1}(s) + q\pi_{n+1}(s))\quad (+\frac1{\lambda + s}\quad \text{when}\ n=0)</math>
which may now be solved by standard methods.

===Tauberian theory===
The Laplace transform of the measure <math>\mu</math> on <math>[0,\infty)</math> is given by
:<math>\mathcal L\mu(s) = \int_0^\infty e^{-st}d\mu(t).</math>
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:
:<math>\lim_{s\to 0^+}\mathcal L\mu(s) = \lim_{t\to\infty} M(t).</math>
[[Tauberian theorem]]s 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>{{cite book|author=Feller|title=Introduction to Probability Theory, volume II,pp=479-483}}</ref>

Two Tauberian theorems of note are the [[Hardy–Littlewood Tauberian theorem]] and  [[Wiener's Tauberian theorem]].  The Wiener theorem generalizes the [[Ikehara Tauberian theorem]], which is the following statement:

Let ''A''(''x'') be a non-negative, [[monotonic function|monotonic]] nondecreasing function of ''x'', defined for 0&nbsp;≤&nbsp;''x''&nbsp;<&nbsp;∞. Suppose that

:<math>f(s)=\int_0^\infty A(x) e^{-xs}\,dx</math>

converges for ℜ(''s'')&nbsp;>&nbsp;1 to the function ''&fnof;''(''s'') and that, for some non-negative number ''c'',

:<math>f(s) - \frac{c}{s-1}</math>

has an extension as a [[continuous function]] for ℜ(''s'')&nbsp;≥&nbsp;1.
Then the [[Limit of a function|limit]] as ''x'' goes to infinity of ''e''<sup>&minus;''x''</sup>&thinsp;''A''(''x'') is equal to&nbsp;c.

This statement can be applied in particular to the [[logarithmic derivative]] of [[Riemann zeta function]], and thus provides an extremely short way to prove the [[prime number theorem]].<ref>{{citation| author=S. Ikehara | authorlink=Shikao Ikehara | title=An extension of Landau's theorem in the analytic theory of numbers | journal=Journal of Mathematics and Physics of the Massachusetts Institute of Technology | year=1931 | volume=10 | issue=1–4 | pages=1–12 | doi=10.1002/sapm19311011 |zbl=0001.12902}}</ref>