11 |
The Laplace Transform |
11.1 Direct and inverse Laplace transform
Consider a function f (t) such that f = 0 for −∞ < t < 0, and let e^{−σt} f (t), with σ a real number, be absolutely integrable. This condition ensures the existence of the Fourier transform of e^{−σt} f (t) given by
(11.1.1) |
As we have seen in discussing the one-sided Fourier transform in §10.6, the integral can be considered as the Fourier transform of f (t) evaluated for the complex argument ω + iσ, . The Laplace transform of f is defined as considered as a function ...
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