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== inverse laplace transform == | == inverse laplace transform == | ||
| + | The [[inverse Laplace transform_ECE301Fall2008mboutin]] is given by the following [[complex number_ECE301Fall2008mboutin|complex]] integral, which is known by various names (the '''Bromwich integral''', the '''Fourier-Mellin integral''', and '''Mellin's inverse formula'''): | ||
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| + | : <math>f(t) = \mathcal{L}^{-1} \{F(s)\} = \frac{1}{2 \pi i} \int_{ \gamma - i \cdot \infty}^{ \gamma + i \cdot \infty} e^{st} F(s)\,ds,</math> | ||
Revision as of 16:22, 24 November 2008
definition
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_{\varepsilon\to 0+}\int_{-\varepsilon}^\infty $
and assures the inclusion of the entire Dirac delta function δ(t) at 0 if there is such an impulse in f(t) at 0.
The parameter s is in general complex number:
- $ s = \sigma + i \omega \, $
inverse laplace transform
The inverse Laplace transform_ECE301Fall2008mboutin is given by the following complex integral, which is known by various names (the Bromwich integral, the Fourier-Mellin integral, and Mellin's inverse formula):
- $ f(t) = \mathcal{L}^{-1} \{F(s)\} = \frac{1}{2 \pi i} \int_{ \gamma - i \cdot \infty}^{ \gamma + i \cdot \infty} e^{st} F(s)\,ds, $
