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Laplace transform

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

<math>F(s)
  = \left\{\mathcal{L} f\right\}(s)
  =\int_0^\infty e^{-st} f(t)\,dt.</math>

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

<math>F(s)
  = \mathcal{L} \left\{f(t)\right\}
  =\int_0^\infty e^{-st} f(t)\,dt.</math>

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

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.

Table of contents

Properties

Linearity

<math>\mathcal{L}\left\{a f(t) + b g(t) \right\}
  = a \mathcal{L}\left\{ f(t) \right\} +
    b \mathcal{L}\left\{ g(t) \right\}</math>

Differentiation

<math>\mathcal{L}\{f'\}
  = s \mathcal{L}(f) - f(0)</math>
<math>\mathcal{L}\{f\}
  = s^2 \mathcal{L}(f) - s f(0) - f'(0)</math>
<math>\mathcal{L}\left\{ f^{(n)} \right\}
  = s^n \mathcal{L}\{f\} - s^{n - 1} f(0) - \cdots - f^{(n - 1)}(0)</math>

<math>\mathcal{L}\{ t f(t)\}
  = -F'(s)</math>
<math>\mathcal{L}\left\{ \frac{f(t)}{t} \right\} = \int_s^\infty F(\sigma) d\sigma</math>

Integration

<math>\mathcal{L}\left\{ \int_0^t f(\tau) d\tau \right\}
  = {1 \over s} \mathcal{L}\{f\}</math>

<math>s</math> shifting

<math>\mathcal{L}\left\{ e^{at} f(t) \right\}
  = F(s - a)</math>
<math>\mathcal{L}^{-1} \left\{ F(s - a) \right\}
  = e^{at} f(t)</math>

<math>t</math> shifting

<math>\mathcal{L}\left\{ f(t - a) u(t - a) \right\}
  = e^{-as} F(s)</math>
<math>\mathcal{L}^{-1} \left\{ e^{-as} F(s) \right\}
  = f(t - a) u(t - a)</math>
Note: <math>u(t)</math> is the step function.

Convolution

<math>\mathcal{L}\{f * g\}
  = \mathcal{L}\{ f \} \mathcal{L}\{ g \}</math>

Laplace transform of a function with period <math>p</math>

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

See also



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