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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.

Linearity

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


Differentiation

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

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

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


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

$\mathcal{L}\left\{ \frac{f(t)}{t} \right\} = \int_s^\infty F(\sigma) d\sigma$

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 $p$

$\mathcal{L}\{ f \}  = {1 \over 1 - e^{-ps}} \int_0^p e^{-st} f(t) dt$


All Wikipedia text is available under the terms of the GNU Free Documentation License

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