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Advanced Mathematics for Applications by Andrea Prosperetti

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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

image

(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 + , image. The Laplace transform of f is defined as considered as a function ...

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