Laplace transform
In mathematics and in particular, in functional analysis, the Laplace transform of a function f(t) defined for all real numbers t ≥ 0 is the function F(s), defined by:
The lower limit of is short notation to mean and assures the inclusion of the entire dirac delta function at 0 if there is such an impulse in f(t) at 0.
The parameter s is in general complex:
This integral transform has a number of properties that make it useful for analysing linear dynamical systems. The most significant advantage is that differentiation and integration become multiplication and division, respectively, with . (This is similar to the way that logarithms change an operation of multiplication of numbers to addition of their logarithms.) This changes integral equations and differential equations to polynomial equations, which are much easier to solve.
The Laplace transform is named in honor of PierreSimon Laplace, who used the transform in his work on probability theory. The Laplace transform was discovered originally by Leonhard Euler.
Contents
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 twosided 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 twosided 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.
Inverse Laplace transform
The inverse Laplace transform is the Bromwich integral, which is a complex integral given by:
where is a real number so that the contour path of integration is in the region of convergence of normally requiring for every singularity of and . If all singularities are in the left halfplane, that is for every , then can be set to zero and the above inverse integral formula above becomes identical to the inverse Fourier transform.
Bilateral Laplace transform
When one says "the Laplace transform" without qualification, the unilateral transform is normally intended. The Laplace transform can be extended to the twosided Laplace transform or bilateral Laplace transform by setting the range of integration to be the entire real axis; if that is done the ordinary or onesided transform becomes simply a special case consisting of those transforms making use of a Heaviside step function in the definition of the function being transformed.
The bilateral Laplace transform is defined as follows:
Alternative Definition
Moreover, both transforms are sometimes defined slightly differently, by
Applications
The Laplace transform is used frequently in engineering and physics; the output of a 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, which often makes matters easier. For more information, see control theory.
The Laplace transform can also be used to solve differential equations and is used extensively in electrical engineering.
Relation to other transforms
Fourier transform
The continuous Fourier transform is equivalent to evaluating the bilateral Laplace transform with complex argument :
Note that this expression excludes the scaling factor , 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 or dynamical system.
Mellin transform
The Mellin transform and its inverse are related to the twosided Laplace transform by a simple change of variables. If in the Mellin transform
we set we get a twosided Laplace transform.
Ztransform
The Ztransform is simply the Laplace transform of an ideally sampled signal with the substitution of
 where is the sampling period (in units of time e.g. seconds) and is the sampling rate (in samples per second or hertz)
Let
be a sampling impulse train (also called a Dirac comb) and
be the continuoustime representation of the sampled .
 are the discrete samples of .
The Laplace transform of the sampled signal is

 .
This is precisely the definition of the Ztransform of the discrete function
with the substitution of .
Comparing the last two equations, we find the relationship between the Ztransform and the Laplace transform of the sampled signal:
Fundamental relationships
Since an ordinary Laplace transform can be written as a special case of a twosided transform, and since the twosided transform can be written as the sum of two onesided transforms, the theory of the Laplace, Fourier, Mellin, and Ztransforms 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.
Properties and theorems
 Initial value theorem
 Final value theorem
 , all poles in lefthand plane.
 The final value theorem is useful because it gives the longterm behaviour without having to perform partial fraction decompositions or other difficult algebra. If a functions poles are in the right hand plane (e.g. or ) the behaviour of this formula is undefined.
 shifting
 shifting
 Note: is the Heaviside step function.
 thpower shifting
Common transforms
 th power
 Exponential
 Sine
 Cosine
 Hyperbolic sine
 Hyperbolic cosine
 Natural logarithm
 nth root
 Bessel function of the first kind
 Modified Bessel function of the first kind
 Periodic Function period
Laplace transform  Time function 
, unit impulse</tr>
<tr> <td> <td>, unit step</tr> <tr> <td> <td></tr> <tr> <td> <td></tr> <tr> <td> <td></tr> <tr> <td> <td></tr> <tr> <td> <td></tr> <tr> <td> <td></tr> <tr> <td> <td></tr> </table> External links
Bibliography
de:LaplaceTransformation es:Transformada de Laplace fr:Transformée de Laplace ko:라플라스 변환 it:Trasformata di Laplace nl:Laplacetransformatie ja:ラプラス変換 pl:Transformata Laplace'a pt:Transformada de Laplace sl:Laplaceova transformacija sv:Laplacetransform zh:拉普拉斯变换 