## 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…