Pierre Simon de Laplace introduced his transform in

Analytic Theory of Probability (Theorie Analytique des Probabilites , 1812

Laplace put together the first major blending of calculus with probability theory.

Jean Baptiste Joseph Fourier (1768-1830) introduced his transform in

Analytic Theory of Heat (Theorie analytique de la chaleur), 1822.

See

http://mathwiki.ucdavis.edu/Analysis/Ordinary_Differential_Equations/Differential_Equations_for_Engineers/The_Laplace_transform/The_Laplace_transform

and

http://en.wikiversity.org/wiki/Laplace_transform

http://books.google.ca/books?id=TDQJAAAAIAAJ&redir_esc=y

Laplace introduced his transform 10 years before Fourier introduced his transform.

The two transforms are different in ways they are defined and intended purposes. The Laplace transform is defined for functions that are defined over the non-negative section of the set of all real numbers [0,∞) while the Fourier transform is defined years later in 1822  by Fourier for functions that are integrable over (-∞,∞)  for a different purpose. We see that the Fourier transform is an extension (in some sense) of the Laplace transform. There is also a third transform called  Z-transform which is  similar to the Fourier transform but in discrete version. They all have purposes of their own and their domain of definitions are different as well.

My doubt is that in cases where fourier transform exists we can also find the laplace transform.But the reverse is not true.Then what is the need of fourier transform? And how fourier transform became this much popular?

Fourier transform is more general, since the corresponding Laplace transform can be easy found by substituting at the Fourier transformed function X the next: X(jw)=X(s), see:

http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node1.html

A question for historians could be why the Laplace trans. was found before the more general Fourier trans...

Dear Demetris,

I am sorry to disagree with your statement that Fourier transform is more general. Indeed, as you state you can always find the corresponding Laplace transform for a given Fourier transform but not the other way around. In fact, Fourier transform is a subset of Laplace transform.

In Fourier transform the only phenomenon under study is the variation of frequency. Laplace transform studies variations in frequency and scale, so, to me, Laplace is more general in that sense.

Cheers,

Fernando

I agree with Mr.Fernando and that is what is the reason why I posted my question.

Dear Renoh,

I think that one of the reasons of the wide acceptance of Fourier is that for most real-world problems (time-domain signals) Fourier is easier to visualize and to reason about than Laplace. If you transform a real-valued, time-domain signal to a complex valued, frequency domain signal you have to think in 3D to understand what is going on. Using Laplace you have to think in 4D: a complex valued signal over a complex domain, too much complexity!

Other than that, I believe that their mathematical properties are equivalent, maybe except for the existence of the inverse transform, but someone more knowledgeable than me should confirm this.

Cheers,

Fernando

Dear Renoh, I find Fourier Transform very interesting because of the visualization that FT offers. Splitting the understanding into a magnitude ( gain? ) and phase angle ( delay) components gives a beautiful understanding of the system which acts on an input to produce an output in the signals and system approach. Another beauty I see is the Bode plot which is an outcome where number crunching ( decibels) and frequency crunching ( log omega) gives a compact set of information. From the Bode plot  BW information, half power frequencies, 3db points.....

Laplace Transform (LT) is more general than Fourier Transform (FT), despite the date they were introduced. However, FT deals only with the frequency content of signals and systems, without attention to the magnitude variations, which LT cares for. Hence, LT is more suitable to study the system stability, while FT is more suitable to analyze signals.