Laplace Transform (Numeric Computation?)

1. Sep 1, 2009

John Creighto

I was exploring the connection between the Laplace transform and the Fourier transform (see "=[PLAIN]http://earthcubed.wordpress.com/2009/08/30/using-the-fft-to-calculate-the-laplace-transform/"[/URL])and [Broken] it occurred to me that the from the definition of the Laplace transform:

$$F(s) = \mathcal{L} \left\{f(t)\right\}=\int_0^{\infty} e^{-st} f(t) \,dt.$$

You can not compute it numerically for all values of "s". If the envelop of your function grows faster then $$e^{st}$$ then the integral will not converge even though the value of the Laplace transform will actually decrease when you move away from the pole. Here is what Wikipedia has to say:

http://en.wikipedia.org/wiki/Laplace_transform#Formal_definition
http://demonstrations.wolfram.com/RiemannVersusLebesgue/
Right now I'm trying to learn enough about measure theory to hopefully understand this and would appreciate any good references. I'm still wondering if it is possible to compute it numerically from the deffinition. I know it can't be done using the http://en.wikipedia.org/wiki/Riemann_integral" [Broken]

Here are some sources:
Does[/PLAIN] [Broken] there exist the Lebesgue measure in the infinite-dimensional Space
Borel-Laplace Transform and Asymptotic Theory: Introduction to
http://www.worldscibooks.com/etextbook/p245/p245_chap1.pdf [Broken]
http://demonstrations.wolfram.com/RiemannVersusLebesgue/
http://demonstrations.wolfram.com/LebesgueIntegration/
http://mathworld.wolfram.com/LebesgueIntegral.html
http://mathworld.wolfram.com/Measure.html

This Journal Articles Also sound interesting:

http://imamat.oxfordjournals.org/cgi/content/abstract/26/2/151

Last edited by a moderator: May 4, 2017
2. Sep 3, 2009

NJunJie

Laplace Transform is used very much in Systems Theory.
Normally a system model in time domain - finding some characteristics is complicated eg) convolution is needed (complex steps). But if we deal in frequency domain laplace transform it into "s-domain" - we can do it in a "multipication" process.
Laplace is fun! :)