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In the branch of [[mathematics]] called [[functional analysis]], the '''Laplace transform''', <math>\scriptstyle\mathcal{L} \left\{f(t)\right\}</math>, is a [[linear operator]] on a function ''f''(''t'') (''original'' ) with a real argument ''t'' (''t'' ≥ 0) that transforms it to a function ''F''(''s'') (''image'') with a complex argument ''s''. This transformation is essentially [[bijection|bijective]] for the majority of practical uses; the respective pairs of ''f(t)'' and ''F(s)'' are matched in tables. The Laplace transform has the useful property that many relationships and operations over the originals ''f''(''t'') correspond to simpler relationships and operations over the images ''F''(''s'')<ref>Korn and Korn, Section 8.1</ref>.
 
The Laplace transform has many important applications in mathematics, [[physics]], [[optics]], [[electrical engineering]], [[control engineering]], [[signal processing]], and [[probability theory]]. In mathematics, it is used for solving differential and integral equations. In physics, it is used for analysis of [[LTI system|linear time-invariant]] [[dynamical system|systems]] such as [[electrical circuit]]s, [[harmonic oscillator]]s, [[optical device]]s, and mechanical systems.  In this analysis, 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]], or [[radians]] 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.

The Laplace transform is named in honor of [[mathematician]] and [[astronomer]] [[Pierre-Simon Laplace]], who used the transform in his work on [[probability theory]].   

==Formal definition==

The Laplace transform of a [[function (mathematics)|function]] ''f''(''t''), defined for all [[real number]]s ''t'' &ge; 0, is the function ''F''(''s''), defined by:

:<math>F(s) = \mathcal{L} \left\{f(t)\right\}=\int_{0^-}^\infty e^{-st} f(t) \,dt. </math>

The lower limit of 0<sup>&minus;</sup> is short notation to mean

:<math> \lim_{\varepsilon\to 0+}\int_{-\varepsilon;}^\infty </math>

and assures the inclusion of the entire [[Dirac delta]] function &delta;(''t'') at 0 if there is such an impulse in ''f''(''t'') at 0.

The parameter ''s'' is in general [[complex number|complex]]:

:<math>s = \sigma + i \omega \, </math>

This [[integral 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''. (This is similar to the way that [[logarithm]]s change an operation of multiplication of numbers to addition of their logarithms.)  This changes [[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 back to the time domain.

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

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]].

The bilateral Laplace transform is defined as follows:

: <math>F(s)  = \mathcal{L}\left\{f(t)\right\}  =\int_{-\infty}^{+\infty} e^{-st} f(t)\,dt.</math>

===Inverse Laplace transform===
The inverse Laplace transform is the [[Bromwich integral]], which is a [[complex number|complex]] integral given by

: <math>f(t) = \mathcal{L}^{-1} \{F(s)\} = \frac{1}{2 \pi i} \int_{ \gamma - i \cdot \infty}^{ \gamma + i \cdot \infty} e^{st} F(s)\,ds,</math>

where &gamma; is a real number so that the contour path of integration is in the ''[[region of convergence]]'' of  ''F''(''s'') normally requiring &gamma; > Re(''s''<sub>p</sub>) for every [[mathematical singularity|singularity]] ''s''<sub>p</sub> of ''F''(''s'') and ''i''<sup>2</sup> = &minus;1. If all singularities are in the left half-plane, that is Re(''s''<sub>p</sub>) < 0 for every ''s''<sub>p</sub>, then &gamma; can be set to zero and the above inverse integral formula above becomes identical to the inverse [[Fourier transform]].

An alternative formula for the inverse Laplace transform is given by [[Post's inversion formula]].

==Region of convergence==

The Laplace transform ''F''(''s'') typically  exists for all complex numbers such that Re{''s''} > ''a'', where ''a'' is a real constant which depends on the growth behavior of ''f''(''t''), whereas the two-sided transform is defined in a range
''a'' < Re{''s''} < ''b''.  The subset of values of ''s'' for which the Laplace transform exists is called the ''region of convergence'' (ROC) or the ''domain of convergence''.  In the two-sided case, it is sometimes called the ''strip of convergence.''

There are no specific conditions that one can check a function against to know in all cases if its Laplace transform can be taken,  other than to say the defining integral converges. It is however easy to give theorems on cases where it may or may not be taken.

== Properties and theorems==


Given the functions ''f''(''t'') and ''g''(''t''), and their respective Laplace transforms ''F''(''s'') and ''G''(''s''):
:<math> f(t) = \mathcal{L}^{-1} \{  F(s) \} </math>
:<math> g(t) = \mathcal{L}^{-1} \{  G(s) \} </math>

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

{| class="wikitable"
 |+ '''Properties of the unilateral Laplace transform'''
 !
 	! Time domain
	! Frequency domain
 	! Comment
 |-
 ! [[Linearity]]
	| <math>a f(t) + b g(t) \ </math>
	| <math>a F(s)  + b G(s) \ </math>
	| Can be proved using basic rules of integration.
 |-
 ! [[Frequency|Frequency differentiation]]
	| <math> t f(t) \ </math>
	| <math> -F'(s) \ </math>
	|
 |-
 ! Frequency differentiation 
	| <math> t^{n} f(t) \ </math>
	| <math>  (-1)^{n} F^{(n)}(s) \ </math>
	| more general
 |-
 ! [[Derivative|Differentiation]]
	| <math> f'(t) \ </math>
	| <math>  s F(s) - f(0^-) \ </math>
	| Write the exact integral form of the given function, and add another integral to complement the former to deduce the sum to indefinite integration of a differential. Next few steps are simple.
 |-
 ! Second [[Derivative|Differentiation]]
	| <math> f''(t) \ </math>
	| <math>  s^2 F(s) - s f(0^-) - f'(0^-) \ </math>
	| Apply the Differentiation property to <math> f'(t) </math>. 
 |-
 ! General [[Derivative|Differentiation]]
	| <math> f^{(n)}(t)  \ </math>
	| <math>  s^n F(s) - s^{n - 1} f(0^-) - \cdots - f^{(n - 1)}(0^-) \ </math>
	| Follow the process briefed for the Second Differentiation.
 |-
 ! [[Frequency|Frequency integration]]
	| <math> \frac{f(t)}{t}  \ </math>
	| <math>  \int_s^\infty F(\sigma)\, d\sigma \ </math>
	|
 |-
 ! [[Integral|Integration]]
	| <math> \int_0^t f(\tau)\, d\tau  =  u(t) * f(t)</math>
	| <math>  {1 \over s} F(s) </math>
	| <math>u(t)</math> is the [[Heaviside step function]].
 |-
 ! Scaling
	| <math> f(at) \ </math>
	| <math>  {1 \over |a|} F \left ( {s \over a} \right )</math>
	|
|-
 ! Frequency shifting
	| <math> e^{at} f(t)  \ </math>
	| <math> F(s - a) \ </math>
	|
|-
 ! Time shifting
	| <math> f(t - a) u(t - a) \ </math>
	| <math>  e^{-as} F(s) \ </math>
	| <math>u(t)</math> is the [[Heaviside step function]]
|-
 ! [[Convolution]]
	| <math> (f * g)(t) \ </math>
	| <math> F(s) \cdot G(s) \ </math>
	| 
|-
 ! Periodic Function
	| <math> f(t) \ </math>
	| <math>{1 \over 1 - e^{-Ts}} \int_0^T e^{-st} f(t)\,dt </math>
	| <math>f(t)</math> is a periodic function of  [[periodic function|period]] <math>T</math> so that <math>f(t) = f(t + T), \; \forall t</math>
|}


* '''Initial value theorem''':
: <math>f(0^+)=\lim_{s\to \infty}{sF(s)}</math>

* '''Final value theorem''':
: <math>f(\infty)=\lim_{s\to 0}{sF(s)}</math>, all poles in left-hand 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 a function's poles are in the right hand plane (e.g. <math>e^t</math> or <math>\sin(t)</math>) the behaviour of this formula is undefined.

=== Proof of the Laplace transform of a function's derivative===

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>\mathcal{L} \left\{f(t)\right\}  =\int_{0^-}^{+\infty} e^{-st} f(t)\,dt</math>

:<math> ~~ = \left[ \frac{f(t)e^{-st}}{-s} \right]_{0^-}^{+\infty} - 
\int_{0^-}^{+\infty} \frac{e^{-st}}{-s} f'(t)dt</math> (by parts)

:<math> ~~ = \left[-\frac{f(0)}{-s}\right] + 
\frac{1}{s}\mathcal{L}\left\{f'(t)\right\},</math>

yielding

:<math>\mathcal{L}\left\{\frac{df}{dt}\right\} = s\cdot\mathcal{L} \left\{ f(t) \right\}-f(0), </math>

and in the bilateral case, we have

: <math> \mathcal{L}\left\{ { df \over dt }  \right\}
  = s \int_{-\infty}^{+\infty} e^{-st} f(t)\,dt  = s \cdot \mathcal{L} \{ f(t) \}. </math>

=== Relationship to other transforms ===
====Fourier transform====
The [[continuous Fourier transform]] is equivalent to evaluating the bilateral Laplace transform with complex argument ''s'' = ''i''&omega;:

:<math>
\begin{array}{rcl}
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^{-\imath \omega t} f(t)\,\mathrm{d}t.\\
\end{array}
</math>

Note that this expression excludes the scaling factor <math>\frac{1}{\sqrt{2 \pi}}</math>, which is often included in definitions of the 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]].

====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 &theta; = e<sup>-t</sup> we get a two-sided Laplace transform.

====Z-transform====
The [[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 <math>T = 1/f_s \ </math> is the [[Sampling theorem|sampling]] period (in units of time e.g. seconds) and <math> f_s \ </math> 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> x_q(t) \ \stackrel{\mathrm{def}}{=}\  x(t) \Delta_T(t) = x(t) \sum_{n=0}^{\infty}  \delta(t - n T)  </math>
:: <math> = \sum_{n=0}^{\infty} x(n T) \delta(t - n T) = \sum_{n=0}^{\infty} x[n] \delta(t - n T) </math>

be the continuous-time representation of the sampled <math> x(t) \ </math>.
: <math> x[n] \ \stackrel{\mathrm{def}}{=}\  x(nT) \ </math> are the discrete samples of <math> x(t) \ </math>.

The Laplace transform of the sampled signal <math> x_q(t) \ </math> is
:<math>X_q(s) = \int_{0^-}^{\infty} x_q(t) e^{-s t} \,dt </math>
::<math> \ = \int_{0^-}^{\infty} \sum_{n=0}^{\infty} x[n] \delta(t - n T) e^{-s t} \, dt </math>
::<math> \ = \sum_{n=0}^{\infty} x[n] \int_{0^-}^{\infty} \delta(t - n T) e^{-s t} \, dt </math>
::<math> \ = \sum_{n=0}^{\infty} x[n] e^{-n s T}.</math>

This is precisely the definition of the [[Z-transform]] of the discrete function <math> x[n] \ </math>
:<math> X(z) = \sum_{n=0}^{\infty} x[n] z^{-n} </math>

with the substitution of <math> z \leftarrow e^{s T} \ </math>.

Comparing the last two equations, we find the relationship between the Z-transform and the Laplace transform of the sampled signal:
:<math>X_q(s) =  X(z) \Big|_{z=e^{sT}}.</math>

====Borel transform====
The integral form of the [[Borel transform]] is identical to the Laplace transform; indeed, these are sometimes mistakenly assumed to be synonyms. The [[generalized Borel transform]] generalizes the Laplace transform for functions not of [[exponential type]].

==== 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. 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>

The unilateral Laplace transform is only valid when t is non-negative, which is why all of the time domain functions in the table below are multiples of the Heaviside step function, u(''t'').  

{| class="wikitable"
|-
! ID || Function || Time domain <br> <math>x(t) = \mathcal{L}^{-1} \left\{ X(s) \right\}</math> || Frequency domain <br> <math>X(s) = \mathcal{L}\left\{ x(t) \right\}</math> || Region of convergence <br> ''for [[causal system|causal]] systems''
|-  align="center"
| 1 || ideal delay || <math> \delta(t-\tau) \ </math> || <math> e^{-\tau s} \ </math> ||
|-  align="center"
| 1a || [[Dirac delta function|unit impulse]] || <math> \delta(t) \ </math> || <math> 1 \ </math> || <math> \mathrm{all} \  s \,</math>
|-  align="center"
| 2 || delayed ''n''th power <br /> with frequency shift ||  <math>\frac{(t-\tau)^n}{n!} e^{-\alpha (t-\tau)} \cdot u(t-\tau) </math>  ||  <math> \frac{e^{-\tau s}}{(s+\alpha)^{n+1}} </math> ||  <math> s > 0 \, </math>
|-  align="center"
| 2a ||  ''n''th power <br /> ( for integer ''n'' )  ||  <math>{  t^n \over n! } \cdot u(t) </math>  ||  <math> { 1 \over s^{n+1} } </math> ||  <math> s > 0 \, </math>
|- align="center"
|-  align="center"
| 2a.1 ||  ''q''th power <br /> ( for real ''q'' )  ||  <math>{  t^q \over \Gamma(q+1) } \cdot u(t) </math>  ||  <math> { 1 \over s^{q+1} } </math> ||  <math> s > 0 \, </math>
|- align="center"

| 2a.2 || [[Heaviside step function|unit step]] || <math> u(t) \ </math> || <math> { 1 \over s } </math> ||  <math> s > 0 \, </math>
|-  align="center"
| 2b || delayed unit step || <math> u(t-\tau) \ </math> || <math> { e^{-\tau s} \over s } </math> || <math> s > 0 \, </math>
|-  align="center"
| 2c || [[ramp function|ramp]] || <math> t \cdot u(t)\ </math> || <math>\frac{1}{s^2}</math> || <math> s > 0 \, </math>
|- align="center"
| 2d ||  ''n''th power with frequency shift  || <math>\frac{t^{n}}{n!}e^{-\alpha t} \cdot u(t) </math>  || <math>\frac{1}{(s+\alpha)^{n+1}}</math>  ||  <math> s > - \alpha \, </math>
|-  align="center"
| 2d.1  || [[exponential decay]]  || <math> e^{-\alpha t} \cdot u(t)  \ </math>  || <math> { 1 \over s+\alpha } </math>  || <math>  s > - \alpha \ </math>
|-  align="center"
| 3 || exponential approach  || <math>( 1-e^{-\alpha t})  \cdot u(t)  \ </math> || <math>\frac{\alpha}{s(s+\alpha)} </math> || <math>  s > 0\ </math>
|-  align="center"
| 4  || [[sine]] || <math> \sin(\omega t) \cdot u(t) \ </math> || <math> { \omega \over s^2 + \omega^2  } </math> || <math> s > 0  \ </math>
|-  align="center"
| 5 || [[cosine]] || <math> \cos(\omega t) \cdot u(t) \ </math>  || <math> { s \over s^2 + \omega^2  } </math>    || <math> s > 0 \ </math>
|-  align="center"
| 6 || [[hyperbolic sine]]  ||  <math> \sinh(\alpha t) \cdot u(t) \ </math>  || <math> { \alpha \over s^2 - \alpha^2 } </math>   ||  <math> s > | \alpha | \ </math>
|-  align="center"
| 7 ||  [[hyperbolic cosine]] ||   <math> \cosh(\alpha t) \cdot u(t) \ </math>  ||  <math> { s \over s^2 - \alpha^2  } </math>    ||  <math> s > | \alpha | \ </math>
|-  align="center"
| 8 ||  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>  ||  <math> s > \alpha \ </math>
|-  align="center"
| 9 ||  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>  ||  <math> s > \alpha \ </math>
|-  align="center"
| 10 ||   ''n''th root  || <math> \sqrt[n]{t} \cdot u(t) </math>  ||  <math> s^{-(n+1)/n} \cdot \Gamma\left(1+\frac{1}{n}\right)</math>  || <math> s > 0 \, </math>
|-  align="center"
| 11 ||  [[natural logarithm]] ||  <math> \ln \left (  { t \over t_0 } \right ) \cdot u(t) </math>  ||  <math> - { t_0 \over s} \  [ \  \ln(t_0 s)+\gamma \ ] </math>   ||  <math> s > 0 \, </math>
|-  align="center"
| 12 ||  [[Bessel function]] <br> of the first kind, <br /> of order ''n'' ||  <math> J_n( \omega t) \cdot u(t)</math>  || <math>\frac{ \omega^n \left(s+\sqrt{s^2+ \omega^2}\right)^{-n}}{\sqrt{s^2 + \omega^2}}</math>  ||  <math> s > 0 \, </math>   <br /> <math> (n > -1) \, </math>
|-  align="center"
| 13 ||  [[Bessel function|Modified Bessel function]] <br /> of the first kind, <br /> of order ''n'' ||  <math>I_n(\omega t) \cdot u(t)</math>  ||  <math> \frac{ \omega^n \left(s+\sqrt{s^2-\omega^2}\right)^{-n}}{\sqrt{s^2-\omega^2}} </math>   ||   <math> s > | \omega | \, </math>
|-  align="center"
| 14 ||  [[Bessel function]] <br /> of the second kind, <br /> of order 0 ||  <math> Y_0(\alpha t) \cdot u(t)</math>  || <math>-{2 \sinh^{-1}(s/\alpha) \over \pi \sqrt{s^2+\alpha^2}}</math> ||  <math>s > 0 \, </math>
|-  align="center"
| 15 || Modified [[Bessel function]] <br /> of the second kind, <br /> of order 0 ||  <math> K_0(\alpha t) \cdot u(t)</math>  ||  &nbsp;  || &nbsp;
|-  align="center"
| 16 ||  [[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>   ||  <math> s > 0 \, </math>
|- 
|colspan=5|'''Explanatory notes:'''
{{col-begin}}
{{col-break}}
* <math> u(t) \, </math> represents the [[Heaviside step function]].
* <math> \delta(t) \, </math> represents the [[Dirac delta function]].
* <math> \Gamma (z) \, </math> represents the [[Gamma function]].
* <math> \gamma \, </math> is the [[Euler-Mascheroni constant]].
{{col-break}}
* <math>t \, </math>, a real number, typically represents ''time'', <br />although it can represent ''any'' independent dimension.
* <math>s \, </math> is the [[complex number|complex]] [[angular frequency]].
* <math> \alpha \,</math>, <math> \beta \,</math>, <math> \tau \, </math>, and <math>\omega \,</math> are [[real numbers]]. 
* <math>n \, </math>, is an [[integer]].
{{col-end}}
* 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 ROC for causal systems is not the same as the ROC for [[anticausal system]]s.  See also [[causality (physics)|causality]].
|}

==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]] impedances.

Here is a summary of equivalents:
:[[Image:S-domain_circuit_equivalents.png]]

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==
{{Copy to Wikibooks}}

The Laplace transform is used frequently in [[engineering]] and [[physics]]; the output of a [[LTI system theory|linear dynamic 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 method of using the Laplace Transform to solve [[differential equations]] was developed by the English electrical engineer [[Oliver Heaviside]].

:''The following examples, derived from applications in [[physics]] and [[engineering]], will use [[SI]] units of measure.  SI is based on [[meters]] for distance, [[kilograms]] for mass, [[second]]s for time, and [[ampere]]s for electric current.''

===Example #1: Solving a differential equation===

:''The following example is based on concepts from [[nuclear physics]]''.

Consider the following first-order, linear differential equation:

:<math>\frac{dN}{dt} = -\lambda N.</math>

This equation is the fundamental relationship describing [[radioactive decay]], where

:<math> N \ = \ N(t) </math>

represents the number of undecayed atoms remaining in a sample of a radioactive [[isotope]] at time ''t'' (in seconds), and <math>\ \lambda </math> is the [[decay constant]].

We can use the Laplace transform 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>N(t) \ = \mathcal{L}^{-1} \{\tilde{N}(s)\} = \mathcal{L}^{-1}  \left\{ \frac{N_o}{s + \lambda} \right\}</math>

::: <math> = \ N_o e^{-\lambda t},</math>

which is indeed the correct form for radioactive decay.

===Example #2: Deriving the complex impedance for a capacitor===

:''This example is based on the principles of [[electrical circuit]] theory.''

The constitutive relation governing the dynamic behavior of a [[capacitor]] is the following differential equation:

:<math> i = C { dv \over dt} </math>

where ''C'' is the capacitance (in [[farads]]) of the capacitor, ''i'' = ''i''(''t'') is the [[electrical current]] (in [[amperes]]) flowing 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

:<math>  V(s) = { I(s) \over sC }  +  { V_o  \over s }.  </math>

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>

Using this definition and the previous equation, we find:

:<math>Z(s) = \frac{1}{sC}, </math>

which is the correct expression for the complex impedance of a capacitor.

===Example #3: Finding the transfer function from the impulse response===

[[Image:LTI.png|thumb|Relationship between the '''time domain''' and the '''frequency domain'''.  Note the <math>*</math> in the time domain, denoting convolution.|right|300px]]

:''This example is based on concepts from [[signal processing]], and describes the dynamic behavior of a [[damped harmonic oscillator]].  See also [[RLC circuit]].''

Consider a linear time-invariant system with [[impulse response]]

:<math> h(t) = A e^{- \alpha t} \cos(\omega_d t - \phi_d) \, </math>

such that

:<math>\omega_d t - \phi_d \ge 0 \, </math>

where ''t'' is the time (in seconds), and

:<math>  0 \le \phi_d \le 2 \pi </math>

is the [[phase (waves)|phase delay]] (in [[radians]]).

Suppose that we want to find the [[transfer function]] of the system.  We begin by noting that

:<math> h(t) = A e^{- \alpha t} \cos \left[ \omega_d (t - t_d) \right] \cdot u(t - t_d) \, </math>

where

:<math>t_d = { \phi_d \over \omega_d }</math>

is the time delay of the system (in seconds), and <math> \ u(t) \ </math> is the [[Heaviside step function]].

The transfer function is simply the Laplace transform of the impulse response:

:<math>H(s) \ = \ \mathcal{L} \{ h(t) \}  = A e^{-s t_d} {(s + \alpha) \over (s + \alpha)^2 + \omega_d^2 }</math>

::::<math>  = \ A e^{-s t_d} {(s + \alpha) \over (s^2 + 2 \alpha s + \alpha^2) + \omega_d^2 }  </math>

::::<math>  = \ A e^{-s t_d} {(s + \alpha) \over (s^2 + 2 \alpha s + \omega_0^2 ) }  </math>

where

:<math> \omega_0 = \sqrt{\alpha^2 + \omega_d^2}</math>

is the (undamped) [[resonance|natural frequency]] or [[resonance]] of the system (in [[radians per second]]).

=== Example #4: Method of partial fraction expansion ===

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|partial fraction expansion]]:

:<math>H(s) = \frac{1}{(s+\alpha)(s+\beta)} =   { P \over (s+\alpha) } + { R  \over (s+\beta) }  </math>

for unknown constants ''P'' and ''R''.  To find these constants, we evaluate

:<math>P = \left.{1 \over (s+\beta)}\right|_{s=-\alpha} = {1 \over (\beta - \alpha)} </math>

and

:<math>R = \left.{1 \over (s+\alpha)}\right|_{s=-\beta} = {1 \over (\alpha - \beta)} = {-1 \over (\beta - \alpha)} = - P . </math>

Substituting these values into the expression for ''H''(''s''), we find

:<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 #5: Mixing sines, cosines, and exponentials ===

{| class="wikitable"
|-
! Time function
! Laplace transform
|-
| <math>e^{-\alpha t}\left[ \cos{(\omega t)}+\left(\frac{\beta-\alpha}{\omega}\right)\sin{(\omega t)}\right]u(t) </math>
| <math>\frac{s+\beta}{(s+\alpha)^2+\omega^2}</math>
|}

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 &alpha; 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> x(t) = e^{-\alpha t} \mathcal{L}^{-1} \left\{   {s \over s^2 + \omega^2}  +  {  \beta - \alpha \over s^2 + \omega^2  } \right\}</math>

:::<math> = 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\}</math>

:::<math> = 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].  </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 #6: 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>X(s) \,\!</math> || <math> {} = \frac{s \sin \phi}{s^2 + \omega^2} + \frac{\omega \cos \phi}{s^2 + \omega^2}</math>
|-
| || <math> {} = (\sin \phi) \left(\frac{s}{s^2 + \omega^2} \right) + (\cos \phi) \left(\frac{\omega}{s^2 + \omega^2} \right).</math>
|}

We are now able to take the inverse Laplace transform of our terms:

:{|
| <math>x(t) \,\!</math> || <math> {} = (\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\}</math>
|-
| || <math> {}=(\sin \phi)(\cos \omega t) + (\sin \omega t)(cos \phi). \,\!</math>
|}

To simplify this answer, we must recall the [[trigonometric identity]] that

:<math>a \sin \omega t + b \cos \omega t = \sqrt{a^2+b^2} \cdot \sin \left(\omega t + \arctan (b/a) \right)</math>

and apply it to our value for x(''t''):

:{|
| <math>x(t) \,\!</math> || <math> {} = \sqrt{\cos^2 \phi + \sin^2 \phi} \cdot \sin \left( \omega t + \arctan \left(\frac{\sin \phi}{\cos \phi} \right) \right)</math>
|-
| || <math>{}= \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>

==History==
From [[1744]], [[Leonhard Euler]] investigated integrals of the form:

:<math> z = \int X(x) e^{ax} dx</math>  and  <math> z = \int X(x) x^A dx</math>,

&mdash; as solutions of differential equations but did not pursue the matter very far.<ref>Euler (1744), (1753) and (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:

:<math> \int X(x) e^{- a x } a^x dx</math>

&mdash; which some modern historians have interpreted within modern Laplace transform theory.<ref>Lagrange (1773)</ref><ref>Grattan-Guiness (1997) ''p.''260</ref>

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>Grattan-Guiness (1997) ''p.''261</ref> However, in [[1785]], Laplace took the critical step forward when, rather than just look 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> \int x^s \phi (s) dx</math>,

&mdash; 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>Grattan-Guiness (1997) ''p.''261-262</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 as the solutions were periodic. In [[1809]], Laplace applied the eponymous transform to find solutions that diffused indefinitely in space.<ref>Grattan-Guiness (1997) ''p.''262-266</ref>

==References==
{{Reflist}}

==Bibliography==
===Modern===
* G.A. Korn and T.M. Korn, ''Mathematical Handbook for Scientists and Engineers'', McGraw-Hill Companies; 2nd edition (June 1967). {{ISBN|0-0703-5370-0}} 
* A. D. Polyanin and A. V. Manzhirov, ''Handbook of Integral Equations'', CRC Press, Boca Raton, 1998. {{ISBN|0-8493-2876-4}}
* William McC. Siebert, ''Circuits, Signals, and Systems'', MIT Press, Cambridge, Massachusetts, 1986. {{ISBN|0-262-19229-2}}
* Davies, Brian, ''Integral transforms and their applications'' (Springer, New York, 1978). {{ISBN|0-387-90313-5}}

===Historical===
*{{ cite journal | author=Deakin, 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 }}
*{{ cite journal | author=— | year=1982 | title=The development of the Laplace transform | journal=Archive for the History of the Exact Sciences | volume=26 | pages=351-381 }}
*Euler, L. (1744) "De constructione aequationum", ''Opera omnia'' 1st series, '''22''':150-161
*&mdash; (1753) "Methodus aequationes differentiales", ''Opera omnia'' 1st series, '''22''':181-213
*&mdash; (1769) ''Institutiones calculi integralis'' '''2''', Chs.3-5, in ''Opera omnia'' 1st series, '''12'''
*Grattan-Guiness, I (1997) "Laplace's integral solutions to partial differential equations", in Gillispie, C. C. ''Pierre Simon Laplace 1749-1827: A Life in Exact Science'', Princeton: Princeton University Press, {{ISBN|0-691-01185-0}}
*Lagrange, J. L. (1773) "Mémoire sur l'utilité de la méthode", ''Œuvres de Lagrange'', '''2''':171-234