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== definition == | == definition == | ||
| − | The Laplace transform of a | + | The Laplace transform of a function ''f''(''t''), defined for all real numbers ''t'' ≥ 0, is the function ''F''(''s''), defined by: |
:<math>F(s) = \mathcal{L} \left\{f(t)\right\}=\int_{0^-}^{\infty} e^{-st} f(t) \,dt. </math> | :<math>F(s) = \mathcal{L} \left\{f(t)\right\}=\int_{0^-}^{\infty} e^{-st} f(t) \,dt. </math> | ||
| Line 8: | Line 8: | ||
:<math>\lim_{\varepsilon\to 0+}\int_{-\varepsilon}^\infty </math> | :<math>\lim_{\varepsilon\to 0+}\int_{-\varepsilon}^\infty </math> | ||
| − | and assures the inclusion of the entire | + | 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 | + | The parameter ''s'' is in general complex number: |
:<math>s = \sigma + i \omega \, </math> | :<math>s = \sigma + i \omega \, </math> | ||
Revision as of 16:19, 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 \, $
