From: David Kastrup
Newsgroups: sci.math
Subject: Re: About Laplace and Fourier transforms
Date: 22 Nov 1997 13:23:30 +0100
"Rodrigo Batres" writes:
> This is the third time I write this because this machine keeps crashing
> before I send the message, so this time I'll be real brief.
> I first thought the Laplace transform (LT) was an instance of the Fourier
> Transform (FT) because of its deduction from Fourier series and not from
> Laplace, but after answering a question and being corrected by a
> mathematician (I guess), something I am not, I find myself in doubt. So what
> I would like to know is this: Which was first? What is their relationship?
> Is the LT a fix on the FT?
> I've read the deduction of the FT but I've never had a grasp on the meaning
> of the real part of s in the LT. What is it? A term to make f(t) more
> convergent?
>
> If the tone of this message is rude, I am sorry but this machine is driving
> me nuts.
If you think the tone of your request rude, then it seems you have not
read anything at all on sci.math, and it is recommended practice to
participate passively a while on newsgroups before posting.
Just kidding.
The Laplace transform might be considered a fix on Fourier
transforms. It allows getting better converging and basically
holomorphic transforms for functinos defined predominantly on the
positive real axis, at the cost of sacrificing the negative real axis
almost entirely.
The real part of s is indeed a "term" to make f(t) more convergent for
t->+oo. But it also defines the Laplace transform as a function of a
complex variable, which makes complex analysis help quite a bit.
Also, it tends to make typical Laplace transforms have nicer forms:
while Fourier transforms often openly involve imaginary terms and
factors, Laplace transforms look more "straight", usually.
Consequently, it is often easier to manipulate and remember Laplace
transforms algebraically. It is no accident that before rigorous
mathematical definitions, the Laplace transform already existed as the
operator calculus of Heaviside.
--
David Kastrup Phone: +49-234-700-5570
Email: dak@neuroinformatik.ruhr-uni-bochum.de Fax: +49-234-709-4209
Institut für Neuroinformatik, Universitätsstr. 150, 44780 Bochum, Germany
==============================================================================
From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: About Laplace and Fourier transforms
Date: 29 Nov 1997 06:28:38 GMT
In article <655t3f$cll@news.mty.itesm.mx>,
Rodrigo Batres wrote:
>I first thought the Laplace transform (LT) was an instance of the Fourier
>Transform (FT) because of its deduction from Fourier series and not from
>Laplace, but after answering a question and being corrected by a
>mathematician (I guess), something I am not, I find myself in doubt. So what
>I would like to know is this: Which was first? What is their relationship?
Neither is more general than the other; rather, each is a particular
instance of a Fredholm integral operator, a transform of the form
T(f)=g, where
g(t) = \int_I k(t,s) f(s) ds,
the interval I and the kernel function k being fixed. Using k(t,s)=exp(-st)
on I=(0, oo) gives the Laplace transform; using k(s,t)=exp(-ist)/sqrt(2 pi)
on I=R gives the Fourier transform.
There is an important sense in which these are different, natural operators,
and not just "tricks": the Fourier transform (suitably normalized) is
unitary, which means it preserves norms (Parseval's identity), and is
inverted by its adjoint (that is, T^(-1)(f) = T^*(f), which is also a
Fredholm operator with k^*(s,t) = k(t,s)^\bar). The Laplace transform,
on the other hand, is self-adjoint: T^* = T; its inverse has to be rather
different. If your linear algebra experience introduced you to unitary
and aymmetric matrices, you know these kinds of operators are very
different, e.g. one's eigenvalues are all on the complex unit circle,
the other's are all on the real line.
I cannot comment authoritatively on the history of the transforms, but
I always imagined the Fourier transform was a modification of the Fourier
coefficients, which in turn came from Fourier's investigations of heat
propogation across the unit disk, ca. 1808. Laplace was among the first
to comment (with reservations) on Fourier's expansions of functions as
Fourier series. I suppose the Laplace transform came on the heels of
the Fourier transform rather than the other way around, but they were
probably close. [Unlike some other respondents to this thread I _am_ a
mathematician, but not a historian!]
dave