Re: Why Fourier and Laplace transforms?



Timothy Murphy wrote:
> > why does it seem to me you don't understand this stuff too well? you
> > still haven't responded to my question. furthermore, there are losts of
> > functions for which F(w) does not exist, yet G(s) does exists. do i
> > don't see what the "sigh" is supposed to indicate. now please answer my
> > question (David Kastrup raised the same point as well) and show us you
> > know what you are talking about...
>
> What exactly is "your question"?
> I've answered the only question I could see you asked,
> namely what is the relation between the Fourier transform F(w) of f(x)
> and the Laplace transform G(s) of the same function, viz
>
> G(s) = F(-is) or F(w) = G(iw).
>

and i told you this is not correct. and as i told you as well, F(s)
may exist when g(ix) does not. more on this below.

>
> Why don't you give an example of a function f(x) with f(x) = 0 for x <= 0,
> such that F(w) exists but G(iw) does not exist?
> Or even a single value of w where the first integral converges
> but not the second.
>

first of all, i said F(s) (*not* F(w)) may exist while G(iw) may
not. this is very easy and i am completely surprised you do not know
this. just take f(x)=0 for x<0 and f(x)=sin x for x>0. then you can
easily that F(s) exists for all s with Re{s}>0 while G(iw) obviously
does not. since the fourier transform does not exist, fourier analysis
doesn't apply to f(x).

> To add another idea to a not very interesting thread,

indeed it is not very interesting. nobody understands what issue you
are trying to make out of the Laplace and Fourier transform.

> the Mellin transform
>
> M(s) = int_0^infty x^{-s} f(x) dx/x
>
> seems to me a third form of exactly the same transform,
> since the change of variable x = e^t transforms M(s) into G(s).
>

now it doesn't! c'mon Timothy don't be so sloppy. if you insert
x=e^t
then you would get:

M(s)= int_0^infty e^{-st}f(e^t) dt

notice the f(e^t) inside the integral, now do you still think that
M(s)=G(s)???
i would like to hear a yes or a no.

.

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