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Last 10 Posts [ In reverse order ]
Strider Posted on Aug 10 2005, 07:02 PM
  The Fourier transform is a special case of the Laplace transform. The Laplace is the more general form. The Fourier transform is when the real part of the transform variable is equal to zero (where the s in the Laplace transform is equal to jwt: s=0+jwt or s=jwt). The Laplace transform can be divided into unilateral (from 0 to inf) and the bilateral transforms (-inf to +inf). The unilateral Laplace transform is used for initial value problems.

Heres some useful information for uses of both from http://www.everything2.com/index.pl?node_id=996318:

QUOTE
Laplace versus Fourier
Both Laplace Transforms and Fourier Transforms can be used to solve differential equations, so a natural question to ask is "which one is better?".

Because of the negative exponential term in the Laplace Transform integration, convergence is a lot stronger, and polynomial expressions which could not be handled with the Fourier Transform can be happily considered. However Fourier Transforms make up for this with a greater mathematical elegance, and tie in beautifully with many areas of applied mathematics, particularly quantum mechanics. The Fourier Transform also has many applications beyond the solution of differential equations.

The Laplace Transform picks out precisely the solution to a differential equation that obeys certain initial conditions at t=0. Hence it is ideally suited to initial value problems, like the one given above.

The Fourier Transform picks out precisely the solution to a differential equation which decays to zero at large distances. In many applied problems this is exactly the solution we want, since a function which grows at large distances would be considered unphysical.

Often a mixture of the two methods proves the most effective. We are often interested in functions that depend on time and space. The most physical solution to an equation is one which obeys initial conditions at t=0, and which tends to zero at large distances. This directly corresponds to Fourier transforming with respect space and Laplace transforming with respect to time.


Hopes this helps, and if I have anything wrong, somebody corrent me, as I haven't done any transformation with Laplace or Fourier in awhile.
philip347 Posted on Aug 9 2005, 05:14 PM
  Lapace is a transformation equation.Can be used to adjust values.

Fourier, is a boundary series of terms, that can be held internally within a database.

Fourier does not adequately describe the processes occurring within a new proposed android type of brain, as in order to function as a human brain would, there must be a ret rest probable failure mode, to how this brain would have operated.

The guess by the Star Trek series, that neural synaptic connections or communications of self within this brain would have to be cascade by nature, was a very good guess.
ap Posted on Aug 9 2005, 03:21 PM
  I am novice in this field- may be this is a stupid question

impedance is calculated taking laplace transforms of voltage and current and is shown as impedance as such using argand diagrams.

here the inverse laplace transform is not taken- to get the true relationship between current and voltage isn't it necessary to take the inverse


William R. Frensley Posted on Aug 29 2004, 07:39 PM
  In a nutshell: The Fourier transform is optimum when dealing
with boundary-value problems and the Laplace transform is optimum
when dealing with initial-value problems. Engineers have
favored the latter because the systems they deal with clearly do
not have a history that goes back to t = - infinity.

By the way, if you set up a Green's function formalism for the
Schroedinger equation in terms of Laplace, rather than Fourier
transforms, you will automatically get the retarded (or causal)
case, without having to worry about "adiabatic switching on"
and the resulting +/- i epsilon terms in the denominator.

- Bill Frensley
Stuart Wilson Posted on Aug 29 2004, 07:38 PM
  Being a physicist, I am comfortable with using Fourier transforms
to swap from time to frequency spaces. This involves (to within a
constant or so..)taking the integral from negative infinity to
positive infinity.

F(w) ~ int f(t)e^(iwt) dt

However, reading the engineering literature, it seems the Laplace
transform is used in a similar way. ( Note that the limits of the
integral are now 0 to infinity. )

L(s) ~ int f(t)e^(-st) dt

Although I understand that the resulting Laplace transform is a
complex function, how are the two related? Is one `more general'?
What are the restrictions on the function in the time domain?

Thanks in advance,
Stu
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