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:

$F(s)={\mathcal {L}}\left\{f(t)\right\}=\int _{{0^{-}}}^{\infty }e^{{-st}}f(t)\,dt.$

The lower limit of $0^{-}$ is short notation to mean $\lim _{{\epsilon \rightarrow +0}}-\epsilon \$ 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:

$s=\sigma +i\omega .\,$

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 $s$. (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 Pierre-Simon Laplace, who used the transform in his work on probability theory. The Laplace transform was discovered originally by Leonhard Euler.

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.

Inverse Laplace transform

The inverse Laplace transform is the Bromwich integral, which is a complex integral given by:

$f(t)={\mathcal {L}}^{{-1}}\left\{F(s)\right\}={\frac {1}{2\pi \imath }}\int _{{\gamma -\imath \infty }}^{{\gamma +\imath \infty }}e^{{st}}F(s)\,ds,$

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 >\operatorname {Re}(s_{p})\$ for every singularity $s_{p}\$ of $F(s)\$ and $\imath ={\sqrt {-1}}$. If all singularities are in the left half-plane, that is $\operatorname {Re}(s_{p})<0\$ for every $s_{p}\$, then $\gamma \$ 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 two-sided 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 one-sided 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:

$F(s)={\mathcal {L}}\left\{f(t)\right\}=\int _{{-\infty }}^{{+\infty }}e^{{-st}}f(t)\,dt.$

Alternative Definition

Moreover, both transforms are sometimes defined slightly differently, by

$F(s)={\mathcal {L}}\left\{f(t)\right\}=s\int _{{0^{-}}}^{{+\infty }}e^{{-st}}f(t)\,dt.$

$F(s)={\mathcal {L}}\left\{f(t)\right\}=s\int _{{-\infty }}^{{+\infty }}e^{{-st}}f(t)\,dt.$

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 $s=i\omega$:

$F(\omega )={\mathcal {F}}\left\{f(t)\right\}$
$={\mathcal {L}}\left\{f(t)\right\}|_{{s=i\omega }}=F(s)|_{{s=i\omega }}$
$=\int _{{-\infty }}^{{+\infty }}e^{{-\imath \omega t}}f(t)\,{\mathrm {d}}t.$

Note that this expression excludes the scaling factor ${\frac {1}{{\sqrt {2\pi }}}}$, 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 two-sided Laplace transform by a simple change of variables. If in the Mellin transform

$G(s)={\mathcal {M}}\left\{g(\theta )\right\}=\int _{0}^{\infty }\theta ^{s}g(\theta ){\frac {d\theta }{\theta }}$

we set $\theta =\exp(-t)$ 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

$z\equiv e^{{sT}}\$
where $T=1/f_{s}\$ is the sampling period (in units of time e.g. seconds) and $f_{s}\$ is the sampling rate (in samples per second or hertz)

Let

$q(t)\equiv \sum _{{n=0}}^{{\infty }}\delta (t-nT)$

be a sampling impulse train (also called a Dirac comb) and

$x_{q}(t)\equiv x(t)q(t)=x(t)\sum _{{n=0}}^{{\infty }}\delta (t-nT)$
$=\sum _{{n=0}}^{{\infty }}x(nT)\delta (t-nT)=\sum _{{n=0}}^{{\infty }}x[n]\delta (t-nT)$

be the continuous-time representation of the sampled $x(t)\$.

$x[n]\equiv x(nT)\$ are the discrete samples of $x(t)\$.

The Laplace transform of the sampled signal $x_{q}(t)\$ is

$X_{q}(s)=\int _{{0^{-}}}^{{\infty }}x_{q}(t)e^{{-st}}\,dt$
$\ =\int _{{0^{-}}}^{{\infty }}\sum _{{n=0}}^{{\infty }}x[n]\delta (t-nT)e^{{-st}}\,dt$
$\ =\sum _{{n=0}}^{{\infty }}x[n]\int _{{0^{-}}}^{{\infty }}\delta (t-nT)e^{{-st}}\,dt$
$\ =\sum _{{n=0}}^{{\infty }}x[n]e^{{-nsT}}$.

This is precisely the definition of the Z-transform of the discrete function $x[n]\$

$X(z)=\sum _{{n=0}}^{{\infty }}x[n]z^{{-n}}$

with the substitution of $z\leftarrow e^{{sT}}\$.

Comparing the last two equations, we find the relationship between the Z-transform and the Laplace transform of the sampled signal:

$X_{q}(s)=X(z){\Big |}_{{z=e^{{sT}}}}$

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.

Properties and theorems

${\mathcal {L}}\left\{af(t)+bg(t)\right\}=a{\mathcal {L}}\left\{f(t)\right\}+b{\mathcal {L}}\left\{g(t)\right\}$
${\mathcal {L}}\{f'\}=s{\mathcal {L}}\{f\}-f(0)$
${\mathcal {L}}\{f''\}=s^{2}{\mathcal {L}}\{f\}-sf(0)-f'(0)$
${\mathcal {L}}\left\{f^{{(n)}}\right\}=s^{n}{\mathcal {L}}\{f\}-s^{{n-1}}f(0)-\cdots -f^{{(n-1)}}(0)$
${\mathcal {L}}\{tf(t)\}=-F'(s)$
${\mathcal {L}}\{t^{{n}}f(t)\}=(-1)^{{n}}F^{{(n)}}(s)$
${\mathcal {L}}\left\{{\frac {f(t)}{t}}\right\}=\int _{s}^{\infty }F(\sigma )\,d\sigma$
${\mathcal {L}}\left\{\int _{0}^{t}f(\tau )\,d\tau \right\}={\mathcal {L}}\left\{1*f(t)\right\}={1 \over s}{\mathcal {L}}\{f\}$
• Initial value theorem
$f(0^{+})=\lim _{{s\to \infty }}{sF(s)}$
• Final value theorem
$f(\infty )=\lim _{{s\to 0}}{sF(s)}$, 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 functions poles are in the right hand plane (e.g. $e^{t}$ or $\sin(t)$) the behaviour of this formula is undefined.
• $s$ shifting
${\mathcal {L}}\left\{e^{{at}}f(t)\right\}=F(s-a)$
${\mathcal {L}}^{{-1}}\left\{F(s-a)\right\}=e^{{at}}f(t)$
• $t$ shifting
${\mathcal {L}}\left\{f(t-a)u(t-a)\right\}=e^{{-as}}F(s)$
${\mathcal {L}}^{{-1}}\left\{e^{{-as}}F(s)\right\}=f(t-a)u(t-a)$
Note: $u(t)$ is the Heaviside step function.
• $n$th-power shifting
${\mathcal {L}}\{\,t^{n}f(t)\}=(-1)^{n}D_{s}^{n}[F(s)]$
${\mathcal {L}}\{f*g\}={\mathcal {L}}\{f\}{\mathcal {L}}\{g\}$

Common transforms

• $n$th power
${\mathcal {L}}\{\,t^{n}\}={\frac {n!}{s^{{n+1}}}}$
• Exponential
${\mathcal {L}}\{\,e^{{-at}}\}={\frac {1}{s+a}}$
• Sine
${\mathcal {L}}\{\,\sin(\omega t)\}={\frac {\omega }{s^{2}+\omega ^{2}}}$
• Cosine
${\mathcal {L}}\{\,\cos(\omega t)\}={\frac {s}{s^{2}+\omega ^{2}}}$
• Hyperbolic sine
${\mathcal {L}}\{\,\sinh(bt)\}={\frac {b}{s^{2}-b^{2}}}$
• Hyperbolic cosine
${\mathcal {L}}\{\,\cosh(bt)\}={\frac {s}{s^{2}-b^{2}}}$
• Natural logarithm
${\mathcal {L}}\{\,\ln(t)\}=-{\frac {\ln(s)+\gamma }{s}}$
• nth root
${\mathcal {L}}\{\,{\sqrt[ {n}]{t}}\}=s^{{-{\frac {n+1}{n}}}}\cdot \Gamma \left(1+{\frac {1}{n}}\right)$
• Bessel function of the first kind
${\mathcal {L}}\{\,J_{n}(t)\}={\frac {\left(s+{\sqrt {1+s^{2}}}\right)^{{-n}}}{{\sqrt {1+s^{2}}}}}$
• Modified Bessel function of the first kind
${\mathcal {L}}\{\,I_{n}(t)\}={\frac {\left(s+{\sqrt {s^{2}-1}}\right)^{{-n}}}{{\sqrt {s^{2}-1}}}}$
${\mathcal {L}}\{\,\operatorname {erf}(t)\}={e^{{s^{2}/4}}\operatorname {erfc}\left(s/2\right) \over s}$
• Periodic Function period $T$
${\mathcal {L}}\{f\}={1 \over 1-e^{{-Ts}}}\int _{0}^{T}e^{{-st}}f(t)\,dt$

 Laplace transform Time function $1$ $\delta (t)$, unit impulse
${\frac {1}{s}}$ $u(t)$, unit step
${\frac {1}{(s+a)^{n}}}$ ${\frac {t^{{n-1}}}{(n-1)!}}e^{{-at}}$
${\frac {a}{s(s+a)}}$ $1-e^{{-at}}$
${\frac {1}{(s+a)(s+b)}}$ ${\frac {1}{b-a}}\left(e^{{-at}}-e^{{-bt}}\right)$
${\frac {s+c}{(s+a)^{2}+b^{2}}}$ $e^{{-at}}\left(\cos {(bt)}+\left({\frac {c-a}{b}}\right)\sin {(bt)}\right)$
${\frac {b}{(s+a)^{2}+b^{2}}}$ $e^{{-at}}\sin {(bt)}$
${\frac {s+a}{(s+a)^{2}+b^{2}}}$ $e^{{-at}}\cos {(bt)}$
${\frac {s\sin \varphi +a\cos \varphi }{s^{2}+a^{2}}}$ $\sin {(at+\varphi )}$