Laplace transform

In mathematics and in particular, functional analysis, the Laplace transform of a function $f(t)$ defined for all real numbers t ≥ 0 is the function $F(s)$, defined by:

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

A sometimes convenient abuse of notation, prevailing especially among engineers and physicists, writes this in the following form:

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

The Laplace transform $F(s)$ typically exists for all real numbers $s>a$, where $a$ is a constant which depends on the growth behavior of $f(t)$.

The Laplace transform is named after its discoverer Pierre-Simon Laplace.

The transform has a number of properties that make it useful for analysing linear dynamic system.

Properties

Linearity

${\mathcal {L}}\left\{af(t)+bg(t)\right\}=a{\mathcal {L}}\left\{f(t)\right\}+b{\mathcal {L}}\left\{g(t)\right\}$

Differentiation

${\mathcal {L}}\{f'\}=s{\mathcal {L}}(f)-f(0)$

${\mathcal {L}}\{f''\}=s^{2}{\mathcal {L}}(f)-sf(0)-f'(0)$

${\mathcal {L}}\left\{f^{(n)}\right\}=s^{n}{\mathcal {L}}\{f\}-s^{n-1}f(0)-\cdots -f^{(n-1)}(0)$

${\mathcal {L}}\{tf(t)\}=-F'(s)$

${\mathcal {L}}\left\{{\frac {f(t)}{t}}\right\}$

$=\int _{s}^{\infty }F(\sigma )d\sigma$

Integration

${\mathcal {L}}\left\{\int _{0}^{t}f(\tau )d\tau \right\}={1 \over s}{\mathcal {L}}\{f\}$

$s$ 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 step function.

Convolution

${\mathcal {L}}\{f*g\}={\mathcal {L}}\{f\}{\mathcal {L}}\{g\}$

Laplace transform of a function with period $p$

${\mathcal {L}}\{f\}={1 \over 1-e^{-ps}}\int _{0}^{p}e^{-st}f(t)dt$

See also

• Fourier transform, transfer function.