Fourier, Laplace…

Again this is quasi-mathematical imprecise thoughts about blah blah transforms.

I saw a book that has both Fourier and Laplace transforms in its title.  The book says that Laplace transform is a generalisation of Fourier transform, in something like this way…

Let’s say x(t) is a function that has such and such nice properties.  Then there is another function X(\omega) such that

\displaystyle X(\omega) = \int_{-\infty}^\infty x(t) e^{-i \omega t}dt ,

which is called Fourier transform of x(t).

Then letting i \omega = s, we have

\displaystyle X(s) = \int_{-\infty}^\infty x(t) e^{-st}dt ,

which is called Lapalce trasnform.  —I think X(\omega) should have been X(-i \omega) to make the things more straight.

Okay.  The point is taken.  s is simpler than i\omega.  I must admit Laplace transform is more general.  This seemed to be all spoken about relation between Fourier and Laplace transforms in the book.

But wait a second.  Usually Laplace transform is introduced to us as a means for solving initial value problems of differential equations, isn’t it?  I thought Fourier transform also should be talked about as a similar method for solving IVPs, to make relationship between these trasnforms clearer…

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2 Responses to Fourier, Laplace…

  1. Pingback: Fourier… 2 | 0909

  2. Pingback: Answering my own question on the Fourier Series | cartesian product

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