Difference between revisions of "Laplace transform"
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:<math>\mathcal{L} \left\{f(t)\right\}=\int_0^{\infty} e^{-st} f(t) \,dt </math> | :<math>\mathcal{L} \left\{f(t)\right\}=\int_0^{\infty} e^{-st} f(t) \,dt </math> | ||
Given the integral converges. A necessary condition for this integral to converge is | Given the integral converges. A necessary condition for this integral to converge is | ||
| − | \lim_{b\to\infty} e^{-sb}f(b)=0 | + | :<math>\lim_{b\to\infty} e^{-sb}f(b)=0</math> |
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Revision as of 18:31, June 30, 2009
Laplace transforms are one of the ways of solving linear ordinary differential equations (Linear ODEs) with constant coefficients. This technique allows us to transform a Linear ODE into a linear algebraic equation.
Definition
The unilateral Laplace transform is defined by
- <math>\mathcal{L} \left\{f(t)\right\}=\int_0^{\infty} e^{-st} f(t) \,dt </math>
Given the integral converges. A necessary condition for this integral to converge is
- <math>\lim_{b\to\infty} e^{-sb}f(b)=0</math>
References
- D. Lomen and D. Lovelock, Differential Equations Graphics. Model. Data., John Wiley and Sons, Toronto, 1999.
- Laplace transform on Wolfram Mathworld